# Historical Weather Records

Document Sample

```					This worksheet tries to reconstruct Baltimore weather, circa 2001, based on historical records of my daily ice cream consumptio
chooses to use a Hidden Markov Model (HMM) with just 2 states, C and H.
The red numbers below are initial model parameters and the historical records. Try changing them (making sure each column
probabilities sums to 1) and see what happens! Double click any other cell to see its formula, with references to other cells colo
coded (then press Esc to avoid editing the formula).
Scroll to the bottom to see a graph of what states and transitions the model thinks are likely on each day. Those likely states a
transitions can be used to reestimate the red probabilities (this is the "forward-backward" or Baum-Welch algorithm), increasing
likelihood of the training data. Scroll right to see the results of doing so, and scroll all the way right to see the results of doing so
11 iterations.
p(…|C)           p(…|H)           p(…|START)
p(1|…) 0.7                  0.1
p(2|…) 0.2                  0.2                             If today is cold (C) or hot (H),
p(3|…) 0.1                  0.7                             how many cones did I prob. eat?
p(C|…) 0.8                   0.1             0.5
p(H|…) 0.1                   0.8             0.5             If today is cold or hot, what will
p(STOP|…) 0.1                     0.1             0               tomorrow probably be?

Total prob of all paths from
Total prob of all paths from      Total prob of all paths from        START to STOP that pass
START that emit the ice           state C (or H) that emit the rest   through state C (or H) after
cream data up through today       of the ice cream data and then      emitting the ice cream data to
and end up in state C (or H).     STOP. Found just like a but         date and before emitting the
Ice        (See diagram.)                    working backwards.                  rest of it.
Day #         Creams     a(C)              a(H)            b(C)              b(H)              a(C)*b(C)
1       2      0.1               0.1             1.17798E-18       7.94958E-18       1.17798E-19
2       3      0.009             0.063           2.34015E-18       1.41539E-17       2.10613E-20
3       3      0.00135           0.03591         7.24963E-18       2.51453E-17       9.78701E-21
4       2      0.0009342         0.0057726       2.60728E-17       1.53899E-16       2.43572E-20
5       3      0.000132462 0.00329805            8.67985E-17       2.7327E-16        1.14975E-20
6       2      8.71549E-05 0.000530337           3.3422E-16        1.66616E-15       2.91289E-20
7       3      1.22758E-05 0.00030309            1.59937E-15       2.94673E-15       1.96334E-20
8       2      8.02592E-06 4.87399E-05           7.81603E-15       1.744E-14         6.27308E-20
9       2      2.25894E-06 7.9589E-06            3.57843E-14       1.04527E-13       8.08347E-20
10       3      2.60304E-07 4.61511E-06           2.88487E-13       1.81504E-13       7.50946E-20
11       1      4.68828E-07 3.71812E-07           4.82176E-13       1.8469E-12        2.26058E-19
12       3      4.12244E-08 2.41033E-07           3.19128E-12       3.24105E-12       1.31559E-19
13       3      5.70827E-09 1.37864E-07           3.53797E-11       5.15581E-12       2.01957E-19
14       1      1.28471E-08 1.10862E-08           6.30117E-11       9.31231E-12       8.09518E-19
15       1      7.97041E-09 1.01537E-09           1.12195E-10       1.8233E-11        8.94243E-19
16       1      4.53451E-09 1.60933E-10           1.99394E-10       5.34426E-11       9.04156E-19
17       2      7.2874E-10        1.1644E-10      1.22358E-09       1.81068E-10       8.91673E-19
18       1      4.16245E-10 1.66026E-11           2.17859E-09       3.57087E-10       9.06828E-19
19       1      2.34259E-10 5.49066E-12           3.87112E-09       1.07636E-09       9.06847E-19
20       1      1.3157E-10        2.78185E-12     6.77837E-09       7.52345E-09       8.91827E-19
21       3      1.05534E-11 1.07677E-11           7.41325E-08       1.21109E-08       7.82349E-19
22       1      6.66364E-12 9.66951E-13           1.31735E-07       3.6119E-08        8.77831E-19
23       2      1.08552E-12 2.87985E-13           8.07744E-07       1.24776E-07       8.76823E-19
24       1      6.28051E-13 3.3894E-14            1.437E-06         3.0232E-07        9.0251E-19
25   1   3.54081E-13        8.99203E-15          2.53825E-06   1.55803E-06   8.98747E-19
26   1   1.98915E-13        4.26017E-15          4.25124E-06   1.57555E-05   8.45635E-19
27   2   3.19116E-14        4.65992E-15          1.44877E-05   9.66609E-05   4.62324E-19
28   3   2.59952E-15        4.84337E-15          3.05402E-05   0.000172063   7.93901E-20
29   3   2.56396E-16        2.89425E-15          0.000114696   0.000305208   2.94076E-20
30   2   9.89083E-17        4.68208E-16          0.000486      0.0018468     4.80695E-20
31   3   1.25948E-17        2.6912E-16           0.00324       0.00324       4.0807E-20
32   2   7.39756E-18        4.33111E-17          0.018         0.018         1.33156E-19
33   2   2.04983E-18        7.07773E-18          0.1           0.1           2.04983E-19

Scroll down to see comments below the graphs.

Weather States that Best Explain Ice Cream Consumption
3.5

3

2.5

2

1.5

1

0.5

0
1   2   3   4   5   6    7   8    9 10 11 12 13 14 15 16 17 18 19 20 21 22
Diary Day

This graph shows the model's reconstruction of the hidden state. We show the probability that the state
H (meaning that it was a hot day). Note that p(C) = 1-p(H) so we don't bother to show it. How did the m
1 was probably hot even though it wasn't preceded by a hot day and I ate only a medium amount of ice

Notice that the weather graph looks like a smoother version of the ice cream graph. The smoothing eff
has inertia: the model thinks explanations without a lot of weather changes are more likely. What happ
inertia by setting p(C | C) = p(H | C) = 0.45 and p(C | H) = p(H | H) = 0.45? How about putting in "anti-in
s of my daily ice cream consumption. It       a

g them (making sure each column of
a, with references to other cells color-

on each day. Those likely states and
Baum-Welch algorithm), increasing the
y right to see the results of doing so for

Prob given all ice cream
Total prob    Prob we reached                                                           today we transitioned fro
of START-     state C (or H) at                                                         state H to state C, i.e., th
al prob of all paths from       STOP          the end of this                                                           yesterday was hot and to
ART to STOP that pass           paths =       day - meaning it    Probability given all ice cream data that we reached This is tricky. We must f
ugh state C (or H) after       prob of ice   was a cold (or      state C (or H) at the end of this day and ate 1 ice   total p(paths passing thro
tting the ice cream data to    cream         hot) day - given    cream (or 2 or 3) today. We need this number so we the end of yesterday and
e and before emitting the       data.         all of the ice      can reestimate the emission probabilities such as p(1 end of today), then divide
of it.                         a(C)*b(C)    cream data.         | C).                                                 paths)=p(ice cream data
a(H)*b(H)       +a(H)*b(H)    p(C) p(H)         p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)
7.94958E-19     9.128E-19     0.129     0.871     0         0.129     0        0         0.871   0      #N/A
8.91695E-19     9.128E-19     0.023     0.977     0         0         0.023    0         0       0.977  0.021
9.02969E-19     9.128E-19     0.011     0.989     0         0         0.011    0         0       0.989  0.006
8.88399E-19     9.128E-19     0.027     0.973     0         0.027     0        0         0.973   0      0.006
9.01259E-19     9.128E-19     0.013     0.987     0         0         0.013    0         0       0.987  0.007
8.83628E-19     9.128E-19     0.032     0.968     0         0.032     0        0         0.968   0      0.008
8.93123E-19     9.128E-19     0.022     0.978     0         0         0.022    0         0       0.978  0.012
8.50026E-19     9.128E-19     0.069     0.931     0         0.069     0        0         0.931   0      0.017
8.31922E-19     9.128E-19     0.089     0.911     0         0.089     0        0         0.911   0      0.05
8.37662E-19     9.128E-19     0.082     0.918     0         0         0.082    0         0       0.918  0.057
6.86699E-19     9.128E-19     0.248     0.752     0.248     0         0        0.752     0       0      0.077
7.81198E-19     9.128E-19     0.144     0.856     0         0         0.144    0         0       0.856  0.131
7.108E-19       9.128E-19     0.221     0.779     0         0         0.221    0         0       0.779  0.128
1.03238E-19     9.128E-19     0.887     0.113     0.887     0         0        0.113     0       0      0.221
1.85132E-20     9.128E-19     0.98      0.02      0.98      0         0        0.02      0       0      0.884
8.6007E-21      9.128E-19     0.991     0.009     0.991     0         0        0.009     0       0      0.975
2.10835E-20     9.128E-19     0.977     0.023     0         0.977     0        0         0.023   0      0.973
5.92856E-21     9.128E-19     0.994     0.006     0.994     0         0        0.006     0       0      0.974
5.90993E-21     9.128E-19     0.994     0.006     0.994     0         0        0.006     0       0      0.989
2.09291E-20     9.128E-19     0.977     0.023     0.977     0         0        0.023     0       0      0.974
1.30407E-19     9.128E-19     0.857     0.143     0         0         0.857    0         0       0.143  0.855
3.49253E-20     9.128E-19     0.962     0.038     0.962     0         0        0.038     0       0      0.853
3.59335E-20     9.128E-19     0.961     0.039     0         0.961     0        0         0.039   0      0.944
1.02468E-20     9.128E-19     0.989     0.011     0.989     0         0        0.011     0       0      0.957
1.40098E-20      9.128E-19    0.985    0.015           0.985     0        0       0.015   0        0          0.978
6.71211E-20      9.128E-19    0.926    0.074           0.926     0        0       0.074   0        0          0.924
4.50433E-19      9.128E-19    0.507    0.493           0         0.507    0       0       0.493    0          0.505
8.33366E-19      9.128E-19    0.087    0.913           0         0        0.087   0       0        0.913      0.085
8.83349E-19      9.128E-19    0.032    0.968           0         0        0.032   0       0        0.968      0.026
8.64687E-19      9.128E-19    0.053    0.947           0         0.053    0       0       0.947    0          0.022
8.71949E-19      9.128E-19    0.045    0.955           0         0        0.045   0       0        0.955      0.028
7.796E-19        9.128E-19    0.146    0.854           0         0.146    0       0       0.854    0          0.04
7.07773E-19      9.128E-19    0.225    0.775           0         0.225    0       0       0.775    0          0.13
TOTAL:    14.679 18.321            9.931     3.212    1.537   1.069   7.788    9.463      12.855
Add up                   Add up expected # of times we ate 1, 2, or 3 ice       Add expected # of trans
expected # of            cream cones in each state. These six numbers           from C to C, from H to H
days we were             total 33, the total number of days on which we ate     (These total 32, the tota
in state C (or           ice cream. We'll use them to get the relative          transitions between da
H). (Note that           probabilities of such days, e.g. p(1 | C).             use them to get relative
these total 33.)                                                                transitions, e.g. p(C|H).

ream Consumption                                                                  Transitions that Best Explain Consumption
Ice Creams               3.5
p(H)
3

2.5

2

1.5

1

0.5

0
22 23 24 25 26 27 28 29 30 31 32 33                        1    3     5    7    9    11   13    15   17   19   21   23    25
Diary Day

e show the probability that the state at the end of the day is                 This graph shows the model's reconstruction of weather changes. Blu
n't bother to show it. How did the model figure out that day                   spikes show days when cold weather probably hit, and yellow spikes
I ate only a medium amount of ice cream?                                      show days when hot weather probably hit. Notice that hot weather
might have arrived on either day 27 or day 28 (or neither). Why is the
e cream graph. The smoothing effect is because weather                         blue spike at day 14 so much bigger than the blue spike at day 11?
anges are more likely. What happens if you remove the                          What happens if you remove the inertia as discussed at left?
0.45? How about putting in "anti-inertia"?
Now for the next iteration. The emission and transition probabilities below are reestimated from the
transition and emission in the data. Double click the probabilities to see how they were computed fr

Terminology note: The computation at left is the expectation step of the Expectation-Maximizatio
(given the observed data)?" The computation below is the maximization step - "What parameters
unsmoothed count ratios, known as maximum likelihood estimates (MLE), since these maximize
with a (2/3, 1/3)-weighted coin. 3 heads + 0 tails are most probable with a (3/3, 0/3) coin, which alw
p(…|C)                p(…|H)                p(…|START)
p(1|…) 0.6765                   0.0584
p(2|…) 0.2188                   0.4251
p(3|…) 0.1047                   0.5165
p(C|…) 0.8757                    0.0925                 0.1291
p(H|…) 0.109                     0.8652                 0.8709
p(STOP|…) 0.0153                     0.0423                 0
Prob given all ice cream data that
today we transitioned from (e.g.)
state H to state C, i.e., that                  1 ITERATION
yesterday was hot and today cold.
This is tricky. We must find the
total p(paths passing through H at                 The new probabilities above are plugged right back into the copy below of the spreadsh
the end of yesterday and C at the                  we do it all over again. Scroll down for a graph of the results. Scroll all the way right fo
end of today), then divide by p(all                iterations.
paths)=p(ice cream data).
p(HC) p(CH) p(HH)                    a(C)                   a(H)                   b(C)
#N/A    #N/A        #N/A                0.028240369            0.370225776            4.7147E-17
0.003   0.109       0.868               0.006174279            0.167039523            1.10994E-16
0.005   0.017       0.972               0.002183699            0.074995746            3.12448E-16
0.021   0.005       0.969               0.001936713            0.027682104            6.77874E-16
0.005   0.02        0.968               0.000445628            0.01247985             2.00668E-15
0.024   0.005       0.963               0.000338044            0.004610318            4.76751E-15
0.009   0.02        0.959               7.56375E-05            0.002079329            1.98482E-14
0.052   0.005       0.927               5.65894E-05            0.000768212            7.00839E-14
0.038   0.018       0.893               2.63963E-05            0.000285144            2.78362E-13
0.025   0.031       0.886               5.18125E-06            0.000128914            2.57419E-12
0.171   0.005       0.747               1.1138E-05             6.54325E-06            4.24142E-12
0.013   0.117       0.739               1.0844E-06             3.55099E-06            3.34487E-11
0.093   0.016       0.762               1.33798E-07            1.64793E-06            3.40726E-10
0.666   0.001       0.113               1.82408E-07            8.40737E-08            5.74402E-10
0.095   0.003       0.018               1.13328E-07            5.40599E-09            9.68176E-10
0.016   0.005       0.005               6.74785E-08            9.93805E-10            1.62864E-09
0.004   0.018       0.005               1.2951E-08             3.49092E-09            8.26777E-09
0.019   0.003       0.004               7.89117E-09            2.58668E-10            1.39376E-08
0.005   0.005       0.002               4.69124E-09            6.32529E-11            2.34844E-08
0.003   0.019       0.004               2.78324E-09            3.30319E-11            3.93461E-08
0.002   0.122       0.021               2.55462E-10            1.71409E-10            3.96875E-07
0.109   0.004       0.034               1.62075E-10            1.02812E-11            6.67585E-07
0.017   0.018       0.021               3.12663E-11            1.1288E-11             3.38716E-06
0.032   0.004       0.008               1.92299E-11            7.68919E-13            5.7083E-06
0.007     0.011    0.005   1.14407E-11            1.61139E-13            9.58471E-06
0.003     0.061    0.012   6.78804E-12            8.09037E-14            1.53829E-05
0.001     0.421    0.072   1.30242E-12            3.44159E-13            3.20013E-05
0.002     0.421    0.492   1.22728E-13            2.27105E-13            7.57126E-05
0.006     0.061    0.907   1.345E-14              1.08398E-13            0.000217073
0.031     0.01     0.937   4.77187E-15            4.04882E-14            0.000487523
0.017     0.025    0.931   8.29551E-16            1.83623E-14            0.001672779
0.106     0.005    0.849   5.30703E-16            6.79149E-15            0.004892091
0.095     0.016    0.759   2.39189E-16            2.52227E-15            0.015298833
1.695     1.599    15.85
from C to C, from H to H, etc.
(These total 32, the total # of
transitions between days.) We'll
use them to get relative probs of
transitions, e.g. p(C|H).
Ice Creams                                                 Weather States that Best Explain Ice Cream
p(C -> C)
3.5
p(H -> C)
p(C -> H)           3
p(H -> H)
2.5

2

1.5

1

0.5

0
1   2   3   4   5   6   7     8    9   10   11   12   13   14   15
25   27   29    31     33

1 Iteration

ion of weather changes. Blue           Take a little time to examine the two graphs at left!
bably hit, and yellow spikes            You might also want to compare them with the graphs above.
Notice that hot weather
y 28 (or neither). Why is the
the blue spike at day 11?
s discussed at left?
es below are reestimated from the number of times the computation at left thought it probably saw each
o see how they were computed from the totals of columns at the left (using simple unsmoothed ratios).

p of the Expectation-Maximization algorithm - "How many times do we expect that various hidden things happened
mization step - "What parameters would make those expected counts most probable?" (We just use simple
tes (MLE), since these maximize the probability of the counts. E.g., counts of 2 heads and 1 tail are most probable
le with a (3/3, 0/3) coin, which always comes up heads.)

How much ice cream did I actually eat on days that the previous model thought were in state C (or H)?
For example, p(1 | H) is estimated as the expected number of (H, 1) days divided by the expected number of H days.

When the previous model thought the day was in state C (or H), what state did it think the next day was in?
For example, p(C | H) is estimated as the expected number of H  C transitions divided by the expected number of H days.

to the copy below of the spreadsheet at left, and
results. Scroll all the way right for the result of 10

a(C)*b(C)
b(H)                 a(C)*b(C)              a(H)*b(H)     +a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2)
2.94639E-16          1.33145E-18            1.09083E-16   1.104E-16    0.012   0.988   0       0.012
6.56904E-16          6.85305E-19            1.09729E-16   1.104E-16    0.006   0.994   0       0
1.46318E-15          6.82293E-19            1.09732E-16   1.104E-16    0.006   0.994   0       0
3.94122E-15          1.31285E-18            1.09101E-16   1.104E-16    0.012   0.988   0       0.012
8.77575E-15          8.94232E-19            1.0952E-16    1.104E-16    0.008   0.992   0       0
2.35998E-14          1.61163E-18            1.08803E-16   1.104E-16    0.015   0.985   0       0.015
5.23789E-14          1.50126E-18            1.08913E-16   1.104E-16    0.014   0.986   0       0
1.38566E-13          3.966E-18              1.06448E-16   1.104E-16    0.036   0.964   0       0.036
3.61454E-13          7.34771E-18            1.03067E-16   1.104E-16    0.067   0.933   0       0.067
7.53037E-13          1.33376E-17            9.70767E-17   1.104E-16    0.121   0.879   0       0
9.65471E-12          4.7241E-17             6.31732E-17   1.104E-16    0.428   0.572   0.428   0
2.08794E-11          3.62718E-17            7.41424E-17   1.104E-16    0.329   0.671   0       0
3.93377E-11          4.55884E-17            6.48259E-17   1.104E-16    0.413   0.587   0       0
6.70678E-11          1.04776E-16            5.63864E-18   1.104E-16    0.949   0.051   0.949   0
1.28148E-10          1.09721E-16            6.92769E-19   1.104E-16    0.994   0.006   0.994   0
5.19095E-10          1.09898E-16            5.15879E-19   1.104E-16    0.995   0.005   0.995   0
9.56361E-10          1.07076E-16            3.33858E-18   1.104E-16    0.97    0.03    0       0.97
1.66403E-09          1.09984E-16            4.30432E-19   1.104E-16    0.996   0.004   0.996   0
3.84531E-09          1.10171E-16            2.43227E-19   1.104E-16    0.998   0.002   0.998   0
2.73798E-08          1.0951E-16             9.04404E-19   1.104E-16    0.992   0.008   0.992   0
5.26667E-08          1.01387E-16            9.02756E-18   1.104E-16    0.918   0.082   0       0
2.15518E-07          1.08198E-16            2.21578E-18   1.104E-16    0.98    0.02    0.98    0
3.99565E-07          1.05904E-16            4.51028E-18   1.104E-16    0.959   0.041   0       0.959
8.37487E-07          1.0977E-16             6.43959E-19   1.104E-16    0.994   0.006   0.994   0
4.70481E-06             1.09656E-16          7.58127E-19          1.104E-16        0.993     0.007    0.993        0
7.40977E-05             1.04419E-16          5.99478E-18          1.104E-16        0.946     0.054    0.946        0
0.000199718             4.16793E-17          6.87349E-17          1.104E-16        0.377     0.623    0            0.377
0.000445267             9.29202E-18          1.01122E-16          1.104E-16        0.084     0.916    0            0
0.000991665             2.91963E-18          1.07495E-16          1.104E-16        0.026     0.974    0            0
0.002669611             2.3264E-18           1.08088E-16          1.104E-16        0.021     0.979    0            0.021
0.005937508             1.38766E-18          1.09027E-16          1.104E-16        0.013     0.987    0            0
0.015875458             2.59625E-18          1.07818E-16          1.104E-16        0.024     0.976    0            0.024
0.042325004             3.65931E-18          1.06755E-16          1.104E-16        0.033     0.967    0            0.033
TOTAL:        14.727    18.273   10.264       2.525

es that Best Explain Ice Cream Consumption                                                                                       Transitions that Best E
Ice Creams                  3.5
p(H)
3

2.5

2

1.5

1

0.5

0
15   16   17   18   19    20   21   22   23   24   25   26   27   28   29     30    31   32   33           1    3     5     7
Diary Day

1 Iteration
As we become less sure that days 11-1
weather (blue spike) starts to shift from
get blue spikes at days 12 or 13.
p(…|C)
p(1|…)     0.697
ected number of H days.                                                             p(2|…)     0.1715
p(3|…)     0.1316
xt day was in?                                                                      p(C|…)     0.904
he expected number of H days.                                                       p(H|…)     0.0937
p(STOP|…)      0.0023

2 ITERATIONS

p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)                a(C)
0       0       0.988   0       #N/A     #N/A     #N/A     #N/A                  0.002067573
0.006   0       0       0.994   0.003    0.004    0.009    0.984                 0.004869021
0.006   0       0       0.994   0.002    0.005    0.005    0.989                 0.002575784
0       0       0.988   0       0.003    0.009    0.004    0.984                 0.001525443
0.008   0       0       0.992   0.003    0.005    0.009    0.983                 0.000531436
0       0       0.985   0       0.004    0.011    0.004    0.981                 0.000280196
0.014   0       0       0.986   0.006    0.008    0.009    0.977                 9.48404E-05
0       0       0.964   0       0.009    0.027    0.004    0.96                  4.94759E-05
0       0       0.933   0       0.027    0.039    0.009    0.925                 2.17617E-05
0.121   0       0       0.879   0.056    0.064    0.01     0.869                 6.97567E-06
0       0.572   0       0       0.118    0.31     0.003    0.569                 1.44817E-05
0.329   0       0       0.671   0.309    0.019    0.119    0.553                 1.78949E-06
0.413   0       0       0.587   0.307    0.106    0.022    0.565                 2.48552E-07
0       0.051   0       0       0.412    0.537    0.001    0.051                 2.42696E-07
0       0.006   0       0       0.948    0.046    0.001    0.005                 1.55986E-07
0       0.005   0       0       0.99     0.005    0.003    0.001                 9.84406E-08
0       0       0.03    0       0.968    0.002    0.027    0.003                 1.52679E-08
0       0.004   0       0       0.969    0.028    0.001    0.003                 9.86366E-09
0       0.002   0       0       0.994    0.003    0.002    0.0                   6.22656E-09
0       0.008   0       0       0.99     0.001    0.007    0.001                 3.92566E-09
0.918   0       0       0.082   0.917    0.001    0.075    0.007                 4.67187E-10
0       0.02    0       0       0.915    0.065    0.003    0.017                 3.04697E-10
0       0       0.041   0       0.953    0.006    0.027    0.014                 4.73419E-11
0       0.006   0       0       0.958    0.037    0.002    0.004                 3.07214E-11
0         0.007     0            0        0.989    0.004     0.005        0.002    1.93979E-11
0         0.054     0            0        0.944    0.001     0.049        0.005    1.22299E-11
0         0         0.623        0        0.377    0.0       0.569        0.054    1.89673E-12
0.084     0         0            0.916    0.082    0.002     0.296        0.62     2.31287E-13
0.026     0         0            0.974    0.022    0.004     0.062        0.912    3.08523E-14
0         0         0.979        0        0.011    0.01      0.015        0.964    6.80372E-15
0.013     0         0            0.987    0.007    0.006     0.014        0.973    1.44899E-15
0         0         0.976        0        0.007    0.016     0.006        0.971    5.88634E-16
0         0         0.967        0        0.014    0.019     0.009        0.957    2.39027E-16
1.938     0.736     8.475        9.062    13.314   1.401     1.38         15.905

Ice Creams
Transitions that Best Explain Consumption                         p(C -> C)
p(H -> C)
p(C -> H)
p(H -> H)

7     9    11   13    15     17     19    21   23   25   27    29     31     33
Diary Day

e become less sure that days 11-13 were hot, the onset of cold
her (blue spike) starts to shift from day 14 to day 11. We do not
lue spikes at days 12 or 13.
p(…|H)        p(…|START)
0.0403
0.4638
0.4959
0.0767       0.0121
0.8704       0.9879
0.0529       0

RATIONS

a(C)*b(C)
a(H)          b(C)          b(H)          a(C)*b(C)     a(H)*b(H)     +a(H)*b(H)
0.458206432   7.88212E-17   5.27741E-16   1.62969E-19   2.41814E-16   2.42E-16
0.197892172   1.86665E-16   1.21818E-15   9.08875E-19   2.41068E-16   2.42E-16
0.085651325   4.70947E-16   2.81098E-15   1.21306E-18   2.40764E-16   2.42E-16
0.03468868    1.09573E-15   6.9275E-15    1.67147E-18   2.40306E-16   2.42E-16
0.015045121   2.96842E-15   1.59786E-14   1.57752E-18   2.404E-16     2.42E-16
0.006096689   8.12571E-15   3.93165E-14   2.27679E-18   2.397E-16     2.42E-16
0.002644808   3.30275E-14   9.03071E-14   3.13234E-18   2.38845E-16   2.42E-16
0.001071809   1.51731E-13   2.18761E-13   7.50702E-18   2.3447E-16    2.42E-16
0.000434831   8.34542E-13   5.1472E-13    1.8161E-17    2.23816E-16   2.42E-16
0.000188717   6.6107E-12    1.03787E-12   4.61141E-17   1.95863E-16   2.42E-16
6.6381E-06    1.04095E-11   1.37434E-11   1.50747E-16   9.12299E-17   2.42E-16
3.53858E-06   7.57672E-11   3.00664E-11   1.35585E-16   1.06392E-16   2.42E-16
1.61069E-06   6.15402E-10   5.52669E-11   1.5296E-16    8.90176E-17   2.42E-16
5.73687E-08   9.76169E-10   8.8283E-11    2.36913E-16   5.06468E-18   2.42E-16
2.92545E-09   1.54832E-09   1.57814E-10   2.41516E-16   4.61675E-19   2.42E-16
6.90907E-10   2.45275E-09   7.62653E-10   2.4145E-16    5.26923E-19   2.42E-16
4.55763E-09   1.54349E-08   1.38647E-09   2.35658E-16   6.31904E-18   2.42E-16
2.17273E-10   2.44832E-08   2.22359E-09   2.41494E-16   4.83125E-19   2.42E-16
4.48202E-11   3.88317E-08   4.228E-09     2.41788E-16   1.895E-19     2.42E-16
2.50583E-11   6.14682E-08   2.69062E-08   2.41303E-16   6.74223E-19   2.42E-16
1.93273E-10   4.96965E-07   5.07142E-08   2.32176E-16   9.80171E-18   2.42E-16
8.53376E-12   7.87251E-07   2.46537E-07   2.39873E-16   2.10389E-18   2.42E-16
1.66887E-11   4.95285E-06   4.49389E-07   2.34477E-16   7.49972E-18   2.42E-16
7.6328E-13    7.85557E-06   8.42815E-07   2.41334E-16   6.43304E-19   2.42E-16
1.4263E-13    1.2437E-05    5.08302E-06   2.41252E-16   7.24991E-19   2.42E-16
7.81704E-14   1.9044E-05    0.00011603    2.32907E-16   9.07011E-18   2.42E-16
5.63132E-13   4.26539E-05   0.000286033   8.0903E-17    1.61074E-16   2.42E-16
3.31246E-13   0.000100604   0.000660261   2.32685E-17   2.18709E-16   2.42E-16
1.5374E-13    0.000250427   0.001523681   7.72627E-18   2.34251E-16   2.42E-16
6.34046E-14   0.000562372   0.003756052   3.82622E-18   2.38151E-16   2.42E-16
2.76864E-14   0.001340274   0.008669792   1.94204E-18   2.40035E-16   2.42E-16
1.12397E-14   0.002648668   0.021390005   1.5591E-18    2.40418E-16   2.42E-16
4.56297E-15   0.002250354   0.052912705   5.37895E-19   2.41439E-16   2.42E-16
TOTAL:
p(STOP|…)

p(C)   p(H)   p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)
0.001   0.999   0       0.001   0       0       0.999   0       #N/A     #N/A     #N/A     #N/A
0.004   0.996   0       0       0.004   0       0       0.996   0.0      0.004    0.0      0.996
0.005   0.995   0       0       0.005   0       0       0.995   0.001    0.004    0.003    0.992
0.007   0.993   0       0.007   0       0       0.993   0       0.002    0.005    0.003    0.99
0.007   0.993   0       0       0.007   0       0       0.993   0.002    0.004    0.005    0.989
0.009   0.991   0       0.009   0       0       0.991   0       0.003    0.007    0.004    0.987
0.013   0.987   0       0       0.013   0       0       0.987   0.005    0.008    0.005    0.982
0.031   0.969   0       0.031   0       0       0.969   0       0.009    0.022    0.004    0.965
0.075   0.925   0       0.075   0       0       0.925   0       0.026    0.049    0.005    0.92
0.191   0.809   0       0       0.191   0       0       0.809   0.071    0.12     0.004    0.805
0.623   0.377   0.623   0       0       0.377   0       0       0.189    0.434    0.001    0.376
0.56    0.44    0       0       0.56    0       0       0.44    0.539    0.021    0.084    0.356
0.632   0.368   0       0       0.632   0       0       0.368   0.541    0.091    0.019    0.349
0.979   0.021   0.979   0       0       0.021   0       0       0.632    0.347    0.0      0.021
0.998   0.002   0.998   0       0       0.002   0       0       0.978    0.02     0.001    0.001
0.998   0.002   0.998   0       0       0.002   0       0       0.996    0.002    0.002    0.0
0.974   0.026   0       0.974   0       0       0.026   0       0.973    0.001    0.025    0.002
0.998   0.002   0.998   0       0       0.002   0       0       0.973    0.025    0.001    0.001
0.999   0.001   0.999   0       0       0.001   0       0       0.997    0.002    0.001    0.0
0.997   0.003   0.997   0       0       0.003   0       0       0.997    0.001    0.003    0.0
0.959   0.041   0       0       0.959   0       0       0.041   0.959    0.001    0.038    0.002
0.991   0.009   0.991   0       0       0.009   0       0       0.958    0.034    0.002    0.007
0.969   0.031   0       0.969   0       0       0.031   0       0.967    0.002    0.025    0.006
0.997   0.003   0.997   0       0       0.003   0       0       0.968    0.029    0.001    0.002
0.997    0.003    0.997    0       0       0.003   0       0       0.995   0.002   0.002   0.001
0.963    0.037    0.963    0       0       0.037   0       0       0.962   0.001   0.035   0.002
0.334    0.666    0        0.334   0       0       0.666   0       0.334   0.0     0.628   0.037
0.096    0.904    0        0       0.096   0       0       0.904   0.094   0.002   0.241   0.663
0.032    0.968    0        0       0.032   0       0       0.968   0.028   0.003   0.068   0.9
0.016    0.984    0        0.016   0       0       0.984   0       0.011   0.005   0.021   0.963
0.008    0.992    0        0       0.008   0       0       0.992   0.004   0.004   0.011   0.981
0.006    0.994    0        0.006   0       0       0.994   0       0.002   0.004   0.006   0.988
0.002    0.998    0        0.002   0       0       0.998   0       0.001   0.001   0.006   0.992
15.472   17.528   10.541   2.425   2.507   0.459   8.575   8.493   14.22   1.252   1.25    15.278
p(…|C)        p(…|H)        p(…|START)
p(1|…)     0.6813        0.0262
p(2|…)     0.1567        0.4892
p(3|…)     0.162         0.4846
p(C|…)     0.9191        0.0714       6.7E-04
p(H|…)     0.0808        0.8717       0.9993
p(STOP|…)      1.4E-04       0.0569       0

3 ITERATIONS

a(C)          a(H)          b(C)          b(H)          a(C)*b(C)
0.000105548   0.488906078   9.57895E-17   6.72484E-16   1.011E-20
0.005672567   0.206501666   2.26284E-16   1.58598E-15   1.284E-18
0.003234006   0.08744147    5.36202E-16   3.7403E-15    1.734E-18
0.001444412   0.037417107   1.32515E-15   8.73605E-15   1.914E-18
0.000648017   0.015860282   3.486E-15     2.05881E-14   2.259E-18
0.000270838   0.006789207   1.10323E-14   4.79885E-14   2.988E-18
0.000118884   0.002878137   4.45354E-14   1.12398E-13   5.295E-18
4.93341E-05   0.001232076   2.38584E-13   2.57307E-13   1.177E-17
2.08946E-05   0.000527367   1.50168E-12   5.63962E-13   3.138E-17
9.21325E-06   0.00022356    9.8041E-12    1.06667E-12   9.033E-17
1.66448E-05   5.12736E-06   1.56132E-11   1.34405E-11   2.599E-16
2.53788E-06   2.81729E-06   9.71838E-11   2.91597E-11   2.466E-16
4.10509E-07   1.28929E-06   6.39091E-10   5.15314E-11   2.624E-16
3.1975E-07    3.03268E-08   1.02045E-09   8.25759E-11   3.263E-16
2.01676E-07   1.3701E-09    1.62932E-09   1.44855E-10   3.286E-16
1.26339E-07   4.58431E-10   2.59954E-09   8.0478E-10    3.284E-16
1.82021E-08   5.18967E-09   1.76575E-08   1.42377E-09   3.214E-16
1.16491E-08   1.57122E-10   2.81939E-08   2.28196E-09   3.284E-16
7.30131E-09   2.82614E-11   4.50163E-08   4.02309E-09   3.287E-16
4.57284E-09   1.61091E-11   7.18195E-08   2.31565E-08   3.284E-16
6.81122E-10   1.85838E-10   4.71285E-07   4.1915E-08    3.21E-16
4.35501E-10   5.68852E-12   7.51924E-07   2.33452E-07   3.275E-16
6.27903E-11   1.96412E-11   5.10703E-06   4.13406E-07   3.207E-16
4.02695E-11   5.81741E-13   8.15423E-06   7.31046E-07   3.284E-16
2.52416E-11   9.85779E-14   1.3009E-05    4.29618E-06   3.284E-16
1.5809E-11    5.57113E-14   2.02874E-05   0.000144837   3.207E-16
2.27764E-12   6.48686E-13   4.79848E-05   0.000338376   1.093E-16
3.46665E-13   3.63157E-13   0.000112413   0.000798063   3.897E-17
5.58234E-14   1.66958E-13   0.000259993   0.001882382   1.451E-17
9.90893E-15   7.34055E-14   0.000597946   0.004398399   5.925E-18
2.32486E-15   3.1392E-14    0.001286777   0.010378452   2.992E-18
6.86183E-16   1.3479E-14    0.002270956   0.024277355   1.558E-18
2.49684E-16   5.7752E-15    0.000143671   0.056925415   3.587E-20
a(C)*b(C)
a(H)*b(H) +a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)
3.29E-16 3.288E-16     0.0     1.0     0       0.0     0       0       1.0     0       #N/A
3.28E-16 3.288E-16     0.004   0.996   0       0       0.004   0       0       0.996   0.0
3.27E-16 3.288E-16     0.005   0.995   0       0       0.005   0       0       0.995   0.001
3.27E-16 3.288E-16     0.006   0.994   0       0.006   0       0       0.994   0       0.002
3.27E-16 3.288E-16     0.007   0.993   0       0       0.007   0       0       0.993   0.002
3.26E-16 3.288E-16     0.009   0.991   0       0.009   0       0       0.991   0       0.003
3.23E-16 3.288E-16     0.016   0.984   0       0       0.016   0       0       0.984   0.005
3.17E-16 3.288E-16     0.036   0.964   0       0.036   0       0       0.964   0       0.012
2.97E-16 3.288E-16     0.095   0.905   0       0.095   0       0       0.905   0       0.032
2.38E-16 3.288E-16     0.275   0.725   0       0       0.275   0       0       0.725   0.093
6.89E-17 3.288E-16     0.79    0.21    0.79    0       0       0.21    0       0       0.274
8.22E-17 3.288E-16     0.75    0.25    0       0       0.75    0       0       0.25    0.733
6.64E-17 3.288E-16     0.798   0.202   0       0       0.798   0       0       0.202   0.735
2.5E-18   3.288E-16    0.992   0.008   0.992   0       0       0.008   0       0       0.798
1.98E-19 3.288E-16     0.999   0.001   0.999   0       0       0.001   0       0       0.992
3.69E-19 3.288E-16     0.999   0.001   0.999   0       0       0.001   0       0       0.998
7.39E-18 3.288E-16     0.978   0.022   0       0.978   0       0       0.022   0       0.977
3.59E-19 3.288E-16     0.999   0.001   0.999   0       0       0.001   0       0       0.977
1.14E-19 3.288E-16     1.0     0.0     1.0     0       0       0.0     0       0       0.999
3.73E-19 3.288E-16     0.999   0.001   0.999   0       0       0.001   0       0       0.999
7.79E-18 3.288E-16     0.976   0.024   0       0       0.976   0       0       0.024   0.976
1.33E-18 3.288E-16     0.996   0.004   0.996   0       0       0.004   0       0       0.975
8.12E-18 3.288E-16     0.975   0.025   0       0.975   0       0       0.025   0       0.974
4.25E-19 3.288E-16     0.999   0.001   0.999   0       0       0.001   0       0       0.975
4.24E-19   3.288E-16   0.999    0.001    0.999    0       0       0.001   0       0       0.998
8.07E-18   3.288E-16   0.975    0.025    0.975    0       0       0.025   0       0       0.975
2.19E-16   3.288E-16   0.332    0.668    0        0.332   0       0       0.668   0       0.332
2.9E-16    3.288E-16   0.119    0.881    0        0       0.119   0       0       0.881   0.116
3.14E-16   3.288E-16   0.044    0.956    0        0       0.044   0       0       0.956   0.041
3.23E-16   3.288E-16   0.018    0.982    0        0.018   0       0       0.982   0       0.015
3.26E-16   3.288E-16   0.009    0.991    0        0       0.009   0       0       0.991   0.006
3.27E-16   3.288E-16   0.005    0.995    0        0.005   0       0       0.995   0       0.002
3.29E-16   3.288E-16   0.0      1.0      0        0.0     0       0       1.0     0       0.0
TOTAL:   16.205   16.795   10.747   2.454   3.003   0.253   8.546   7.997   15.018
p(…|C)      p(…|H)      p(…|START)
p(1|…)     0.6632      0.015
p(2|…)     0.1515      0.5088
p(3|…)     0.1853      0.4761
p(C|…)     0.9268      0.0707     3.1E-05
p(H|…)     0.0732      0.8698     1.0
p(STOP|…)      6.7E-06     0.0595     0

4 ITERATIONS

p(HC)   p(CH)   p(HH)            a(C)        a(H)        b(C)        b(H)        a(C)*b(C)   a(H)*b(H)
#N/A     #N/A     #N/A              4.657E-06   0.5087985   1.07E-16    7.597E-16   4.985E-22   3.87E-16
0.004    0.0      0.996             0.0066625   0.2107217   2.524E-16   1.826E-15   1.682E-18   3.85E-16
0.004    0.003    0.992             0.0039032   0.0875038   5.784E-16   4.391E-15   2.258E-18   3.84E-16
0.004    0.003    0.991             0.0014842   0.0388723   1.497E-15   9.886E-15   2.221E-18   3.84E-16
0.005    0.004    0.99              0.0007639   0.0161509   3.893E-15   2.375E-14   2.974E-18   3.84E-16
0.006    0.004    0.987             0.00028     0.0071764   1.358E-14   5.333E-14   3.803E-18   3.83E-16
0.011    0.004    0.98              0.0001421   0.0029819   5.327E-14   1.271E-13   7.567E-18   3.79E-16
0.023    0.004    0.961             5.185E-05   0.001325    3.052E-13   2.798E-13   1.583E-17   3.71E-16
0.063    0.003    0.901             2.146E-05   0.0005883   2.02E-12    5.833E-13   4.334E-17   3.43E-16
0.182    0.003    0.723             1.139E-05   0.0002444   1.155E-11   1.043E-12   1.315E-16   2.55E-16
0.516    0.001    0.209             1.845E-05   3.211E-06   1.877E-11   1.253E-11   3.463E-16   4.02E-17
0.018    0.058    0.192             3.211E-06   1.973E-06   1.038E-10   2.698E-11   3.333E-16   5.32E-17
0.063    0.016    0.186             5.774E-07   9.291E-07   5.949E-10   4.633E-11   3.435E-16   4.31E-17
0.195    0.0      0.007             3.984E-07   1.279E-08   9.677E-10   7.547E-11   3.856E-16   9.66E-19
0.007    0.0      0.0               2.455E-07   6.063E-10   1.574E-09   1.305E-10   3.864E-16   7.91E-20
0.001    0.001    0.0               1.509E-07   2.784E-10   2.56E-09    8.07E-10    3.863E-16   2.25E-19
0.0      0.022    0.001             2.119E-08   5.746E-09   1.787E-08   1.391E-09   3.785E-16   8E-18
0.022    0.0      0.001             1.329E-08   9.853E-11   2.906E-08   2.266E-09   3.863E-16   2.23E-19
0.001    0.0      0.0               8.175E-09   1.593E-11   4.728E-08   3.875E-09   3.865E-16   6.17E-20
0.0      0.001    0.0               5.025E-09   9.213E-12   7.688E-08   2.088E-08   3.863E-16   1.92E-19
0.0      0.023    0.001             8.632E-10   1.79E-10    4.402E-07   3.649E-08   3.8E-16     6.53E-18
0.021    0.001    0.003             5.39E-10    3.294E-12   7.158E-07   2.258E-07   3.858E-16   7.44E-19
0.001    0.022    0.003             7.569E-11   2.154E-11   4.996E-06   3.895E-07   3.781E-16   8.39E-18
0.024    0.0      0.001             4.753E-11   3.652E-13   8.127E-06   6.606E-07   3.863E-16   2.41E-19
0.001   0.001   0.0      2.923E-11   5.714E-14   1.322E-05   3.16E-06    3.863E-16   1.81E-19
0.0     0.024   0.001    1.797E-11   3.295E-14   2.12E-05    0.0001656   3.811E-16   5.46E-18
0.0     0.643   0.024    2.523E-12   6.841E-13   5.211E-05   0.0003729   1.315E-16   2.55E-16
0.003   0.216   0.665    4.422E-13   3.713E-13   0.0001214   0.0008964   5.37E-17    3.33E-16
0.003   0.078   0.878    8.082E-14   1.692E-13   0.0002693   0.002156    2.176E-17   3.65E-16
0.003   0.03    0.952    1.315E-14   7.789E-14   0.0006295   0.0048562   8.28E-18    3.78E-16
0.003   0.012   0.979    3.279E-15   3.272E-14   0.0012931   0.0116848   4.24E-18    3.82E-16
0.002   0.007   0.988    8.103E-16   1.46E-14    0.002219    0.0263482   1.798E-18   3.85E-16
0.0     0.005   0.995    2.7E-16     6.493E-15   6.733E-06   0.0595337   1.818E-21   3.87E-16
1.187   1.187   14.609
a(C)*b(C)
+a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)
3.865E-16    0.0     1.0     0       0.0     0       0       1.0     0       #N/A     #N/A     #N/A
3.865E-16    0.004   0.996   0       0       0.004   0       0       0.996   0.0      0.004    0.0
3.865E-16    0.006   0.994   0       0       0.006   0       0       0.994   0.002    0.004    0.003
3.865E-16    0.006   0.994   0       0.006   0       0       0.994   0       0.002    0.004    0.004
3.865E-16    0.008   0.992   0       0       0.008   0       0       0.992   0.003    0.005    0.003
3.865E-16    0.01    0.99    0       0.01    0       0       0.99    0       0.004    0.006    0.004
3.865E-16    0.02    0.98    0       0       0.02    0       0       0.98    0.007    0.013    0.003
3.865E-16    0.041   0.959   0       0.041   0       0       0.959   0       0.016    0.025    0.004
3.865E-16    0.112   0.888   0       0.112   0       0       0.888   0       0.038    0.074    0.003
3.865E-16    0.34    0.66    0       0       0.34    0       0       0.66    0.11     0.23     0.002
3.865E-16    0.896   0.104   0.896   0       0       0.104   0       0       0.34     0.556    0.0
3.865E-16    0.862   0.138   0       0       0.862   0       0       0.138   0.851    0.011    0.045
3.865E-16    0.889   0.111   0       0       0.889   0       0       0.111   0.849    0.04     0.013
3.865E-16    0.998   0.002   0.998   0       0       0.002   0       0       0.888    0.109    0.0
3.865E-16    1.0     0.0     1.0     0       0       0.0     0       0       0.997    0.002    0.0
3.865E-16    0.999   0.001   0.999   0       0       0.001   0       0       0.999    0.0      0.001
3.865E-16    0.979   0.021   0       0.979   0       0       0.021   0       0.979    0.0      0.02
3.865E-16    0.999   0.001   0.999   0       0       0.001   0       0       0.979    0.02     0.0
3.865E-16    1.0     0.0     1.0     0       0       0.0     0       0       0.999    0.001    0.0
3.865E-16    1.0     0.0     1.0     0       0       0.0     0       0       0.999    0.0      0.0
3.865E-16    0.983   0.017   0       0       0.983   0       0       0.017   0.983    0.0      0.017
3.865E-16    0.998   0.002   0.998   0       0       0.002   0       0       0.983    0.016    0.001
3.865E-16    0.978   0.022   0       0.978   0       0       0.022   0       0.978    0.0      0.02
3.865E-16    0.999   0.001   0.999   0       0       0.001   0       0       0.978    0.021    0.0
3.865E-16   1.0      0.0      1.0      0       0       0.0     0       0       0.999    0.001   0.0
3.865E-16   0.986    0.014    0.986    0       0       0.014   0       0       0.986    0.0     0.014
3.865E-16   0.34     0.66     0        0.34    0       0       0.66    0       0.34     0.0     0.646
3.865E-16   0.139    0.861    0        0       0.139   0       0       0.861   0.136    0.003   0.204
3.865E-16   0.056    0.944    0        0       0.056   0       0       0.944   0.053    0.003   0.086
3.865E-16   0.021    0.979    0        0.021   0       0       0.979   0       0.018    0.003   0.038
3.865E-16   0.011    0.989    0        0       0.011   0       0       0.989   0.008    0.003   0.014
3.865E-16   0.005    0.995    0        0.005   0       0       0.995   0       0.003    0.002   0.008
3.865E-16   0.0      1.0      0        0.0     0       0       1.0     0       0.0      0.0     0.005
TOTAL:   16.685   16.315   10.874   2.492   3.318   0.126   8.508   7.682   15.526   1.158   1.158
p(…|C)    p(…|H)      p(…|START)
p(1|…)     0.6518    0.0077
p(2|…)     0.1494    0.5214
p(3|…)     0.1989    0.4708
p(C|…)     0.9306    0.071      1.3E-06
p(H|…)     0.0694    0.8677     1.0
p(STOP|…)      2.8E-07   0.0613     0

5 ITERATIONS

a(C)*b(C)
p(HH)            a(C)       a(H)       b(C)       b(H)       a(C)*b(C) a(H)*b(H) +a(H)*b(H)   p(C)
#N/A              1.93E-07   0.521443   1.14E-16   8.06E-16   2.19E-23 4.2E-16    4.201E-16    0.0
0.996             0.007361   0.213043   2.68E-16   1.96E-15   1.97E-18 4.18E-16 4.201E-16      0.005
0.992             0.00437    0.087282   6.05E-16   4.78E-15   2.64E-18 4.17E-16 4.201E-16      0.006
0.991             0.001533   0.039651   1.61E-15   1.05E-14   2.46E-18 4.18E-16 4.201E-16      0.006
0.989             0.000843   0.01625    4.15E-15   2.56E-14   3.5E-18   4.17E-16 4.201E-16     0.008
0.986             0.00029    0.007383   1.52E-14   5.63E-14   4.41E-18 4.16E-16 4.201E-16      0.01
0.977             0.000158   0.003026   5.83E-14   1.36E-13   9.2E-18   4.11E-16 4.201E-16     0.022
0.955             5.4E-05    0.001375   3.43E-13   2.92E-13   1.86E-17 4.02E-16 4.201E-16      0.044
0.885             2.21E-05   0.000624   2.32E-12   5.91E-13   5.12E-17 3.69E-16 4.201E-16      0.122
0.658             1.29E-05   0.000256   1.23E-11   1.02E-12   1.59E-16 2.61E-16 4.201E-16      0.379
0.104             1.97E-05   1.72E-06   2.03E-11   1.2E-11    4E-16     2.05E-17 4.201E-16     0.951
0.093             3.66E-06   1.34E-06   1.05E-10   2.56E-11   3.86E-16 3.44E-17 4.201E-16      0.918
0.098             6.96E-07   6.69E-07   5.62E-10   4.33E-11   3.91E-16 2.9E-17    4.201E-16    0.931
0.002             4.53E-07   4.84E-09   9.26E-10   7.14E-11   4.2E-16   3.46E-19 4.201E-16     0.999
0.0               2.75E-07   2.75E-10   1.53E-09   1.22E-10   4.2E-16   3.35E-20 4.201E-16     1.0
0.0               1.67E-07   1.49E-10   2.52E-09   8.07E-10   4.2E-16   1.2E-19   4.201E-16    1.0
0.0               2.32E-08   6.11E-09   1.77E-08   1.37E-09   4.12E-16 8.36E-18 4.201E-16      0.98
0.0               1.44E-08   5.33E-11   2.93E-08   2.26E-09   4.2E-16   1.2E-19   4.201E-16    1.0
0.0               8.71E-09   8.04E-12   4.82E-08   3.81E-09   4.2E-16   3.06E-20 4.201E-16     1.0
0.0               5.28E-09   4.71E-12   7.95E-08   1.98E-08   4.2E-16   9.33E-20 4.201E-16     1.0
0.0               9.78E-10   1.75E-10   4.24E-07   3.38E-08   4.14E-16 5.9E-18    4.201E-16    0.986
0.001             6.01E-10   1.69E-12   6.98E-07   2.24E-07   4.2E-16   3.79E-19 4.201E-16     0.999
0.001             8.36E-11   2.25E-11   4.92E-06   3.8E-07    4.12E-16 8.55E-18 4.201E-16      0.98
0.0               5.17E-11   1.95E-13   8.12E-06   6.34E-07   4.2E-16   1.24E-19 4.201E-16     1.0
0.0      3.14E-11   2.9E-14    1.34E-05   2.21E-06   4.2E-16    6.4E-20    4.201E-16   1.0
0.0      1.9E-11    1.7E-14    2.19E-05   0.000179   4.17E-16   3.04E-18   4.201E-16   0.993
0.014    2.65E-12   6.97E-13   5.5E-05    0.000394   1.46E-16   2.74E-16   4.201E-16   0.347
0.657    5E-13      3.71E-13   0.000128   0.00096    6.39E-17   3.56E-16   4.201E-16   0.152
0.858    9.77E-14   1.68E-13   0.000277   0.002339   2.71E-17   3.93E-16   4.201E-16   0.065
0.941    1.54E-14   7.95E-14   0.000654   0.005155   1E-17      4.1E-16    4.201E-16   0.024
0.975    3.97E-15   3.3E-14    0.001312   0.012571   5.2E-18    4.15E-16   4.201E-16   0.012
0.987    9.01E-16   1.51E-14   0.002219   0.027732   2E-18      4.18E-16   4.201E-16   0.005
0.995    2.85E-16   6.85E-15   2.82E-07   0.061291   8.04E-23   4.2E-16    4.201E-16   0.0
14.157                                                                        TOTAL:   16.942
p(1|…)
p(2|…)
p(3|…)
p(C|…)
p(H|…)
p(STOP|…)

6

p(H)   p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)
1.0     0       0.0     0       0       1.0     0       #N/A     #N/A     #N/A     #N/A
0.995   0       0       0.005   0       0       0.995   0.0      0.005    0.0      0.995
0.994   0       0       0.006   0       0       0.994   0.002    0.004    0.003    0.991
0.994   0       0.006   0       0       0.994   0       0.002    0.004    0.004    0.99
0.992   0       0       0.008   0       0       0.992   0.003    0.006    0.003    0.989
0.99    0       0.01    0       0       0.99    0       0.004    0.006    0.004    0.985
0.978   0       0       0.022   0       0       0.978   0.007    0.014    0.003    0.975
0.956   0       0.044   0       0       0.956   0       0.018    0.026    0.004    0.952
0.878   0       0.122   0       0       0.878   0       0.041    0.08     0.003    0.875
0.621   0       0       0.379   0       0       0.621   0.12     0.259    0.002    0.619
0.049   0.951   0       0       0.049   0       0       0.379    0.573    0.0      0.049
0.082   0       0       0.918   0       0       0.082   0.912    0.006    0.039    0.043
0.069   0       0       0.931   0       0       0.069   0.906    0.025    0.012    0.057
0.001   0.999   0       0       0.001   0       0       0.931    0.068    0.0      0.001
0.0     1.0     0       0       0.0     0       0       0.999    0.001    0.0      0.0
0.0     1.0     0       0       0.0     0       0       1.0      0.0      0.0      0.0
0.02    0       0.98    0       0       0.02    0       0.98     0.0      0.02     0.0
0.0     1.0     0       0       0.0     0       0       0.98     0.02     0.0      0.0
0.0     1.0     0       0       0.0     0       0       1.0      0.0      0.0      0.0
0.0     1.0     0       0       0.0     0       0       1.0      0.0      0.0      0.0
0.014   0       0       0.986   0       0       0.014   0.986    0.0      0.014    0.0
0.001   0.999   0       0       0.001   0       0       0.986    0.013    0.0      0.001
0.02    0       0.98    0       0       0.02    0       0.979    0.0      0.02     0.001
0.0     1.0     0       0       0.0     0       0       0.98     0.02     0.0      0.0
0.0      1.0      0       0       0.0     0       0       1.0      0.0     0.0     0.0
0.007    0.993    0       0       0.007   0       0       0.993    0.0     0.007   0.0
0.653    0        0.347   0       0       0.653   0       0.347    0.0     0.646   0.007
0.848    0        0       0.152   0       0       0.848   0.149    0.003   0.198   0.65
0.935    0        0       0.065   0       0       0.935   0.061    0.003   0.091   0.845
0.976    0        0.024   0       0       0.976   0       0.021    0.003   0.043   0.933
0.988    0        0       0.012   0       0       0.988   0.009    0.004   0.015   0.973
0.995    0        0.005   0       0       0.995   0       0.003    0.002   0.009   0.986
1.0      0        0.0     0       0       1.0     0       0.0      0.0     0.005   0.995
16.058   10.941   2.517   3.484   0.059   8.483   7.516   15.796   1.146   1.146   13.912
p(…|C)    p(…|H)      p(…|START)
0.6458    0.0037
0.1486    0.5283
0.2056    0.4681
0.9323    0.0714     5.2E-08
0.0677    0.8663     1.0
1.1E-08   0.0623     0

ITERATIONS

a(C)*b(C)
a(C)       a(H)       b(C)       b(H)       a(C)*b(C) a(H)*b(H) +a(H)*b(H)   p(C)   p(H)   p(C,1)
7.75E-09   0.528253   1.17E-16   8.27E-16   9.09E-25 4.37E-16 4.371E-16      0.0     1.0     0
0.007754   0.214208   2.76E-16   2.03E-15   2.14E-18 4.35E-16 4.371E-16      0.005   0.995   0
0.004631   0.087107   6.19E-16   4.99E-15   2.87E-18 4.34E-16 4.371E-16      0.007   0.993   0
0.001565   0.04003    1.67E-15   1.09E-14   2.61E-18 4.34E-16 4.371E-16      0.006   0.994   0
0.000888   0.016282   4.3E-15    2.66E-14   3.81E-18 4.33E-16 4.371E-16      0.009   0.991   0
0.000296   0.007483   1.61E-14   5.78E-14   4.76E-18 4.32E-16 4.371E-16      0.011   0.989   0
0.000167   0.003044   6.08E-14   1.4E-13    1.01E-17 4.27E-16 4.371E-16      0.023   0.977   0
5.54E-05   0.001399   3.62E-13   2.98E-13   2E-17     4.17E-16 4.371E-16     0.046   0.954   0
2.25E-05   0.000642   2.46E-12   5.94E-13   5.54E-17 3.82E-16 4.371E-16      0.127   0.873   0
1.37E-05   0.000261   1.27E-11   1.01E-12   1.74E-16 2.63E-16 4.371E-16      0.398   0.602   0
2.03E-05   8.38E-07   2.1E-11    1.17E-11   4.27E-16 9.78E-18 4.371E-16      0.978   0.022   0.978
3.91E-06   9.83E-07   1.06E-10   2.5E-11    4.13E-16 2.45E-17 4.371E-16      0.944   0.056   0
7.63E-07   5.22E-07   5.44E-10   4.19E-11   4.15E-16 2.19E-17 4.371E-16      0.95    0.05    0
4.84E-07   1.86E-09   9.03E-10   6.95E-11   4.37E-16 1.29E-19 4.371E-16      1.0     0.0     1.0
2.91E-07   1.27E-10   1.5E-09    1.17E-10   4.37E-16 1.49E-20 4.371E-16      1.0     0.0     1.0
1.75E-07   7.31E-11   2.49E-09   8.08E-10   4.37E-16 5.91E-20 4.371E-16      1.0     0.0     1.0
2.43E-08   6.3E-09    1.76E-08   1.36E-09   4.29E-16 8.55E-18 4.371E-16      0.98    0.02    0
1.49E-08   2.62E-11   2.93E-08   2.25E-09   4.37E-16 5.91E-20 4.371E-16      1.0     0.0     1.0
8.98E-09   3.81E-12   4.86E-08   3.79E-09   4.37E-16 1.44E-20 4.371E-16      1.0     0.0     1.0
5.41E-09   2.25E-12   8.08E-08   1.93E-08   4.37E-16 4.35E-20 4.371E-16      1.0     0.0     1.0
1.04E-09   1.72E-10   4.16E-07   3.26E-08   4.32E-16 5.61E-18 4.371E-16      0.987   0.013   0
6.32E-10   8.09E-13   6.91E-07   2.24E-07   4.37E-16 1.81E-19 4.371E-16      1.0     0.0     1.0
8.76E-11   2.3E-11    4.89E-06   3.76E-07   4.28E-16 8.64E-18 4.371E-16      0.98    0.02    0
5.38E-11   9.53E-14   8.12E-06   6.27E-07   4.37E-16 5.97E-20 4.371E-16      1.0     0.0     1.0
3.24E-11   1.37E-14   1.35E-05   1.62E-06   4.37E-16   2.23E-20   4.371E-16   1.0      0.0      1.0
1.95E-11   8.13E-15   2.23E-05   0.000186   4.36E-16   1.51E-18   4.371E-16   0.997    0.003    0.997
2.7E-12    7.01E-13   5.67E-05   0.000405   1.53E-16   2.84E-16   4.371E-16   0.351    0.649    0
5.29E-13   3.7E-13    0.000131   0.000994   6.95E-17   3.68E-16   4.371E-16   0.159    0.841    0
1.07E-13   1.67E-13   0.000283   0.002441   3.02E-17   4.07E-16   4.371E-16   0.069    0.931    0
1.66E-14   8.01E-14   0.000668   0.005318   1.11E-17   4.26E-16   4.371E-16   0.025    0.975    0
4.35E-15   3.3E-14    0.001327   0.013067   5.77E-18   4.31E-16   4.371E-16   0.013    0.987    0
9.53E-16   1.53E-14   0.002226   0.0285     2.12E-18   4.35E-16   4.371E-16   0.005    0.995    0
2.94E-16   7.02E-15   1.13E-08   0.062275   3.32E-24   4.37E-16   4.371E-16   0.0      1.0      0
TOTAL:   17.068   15.932   10.973
p(…|C)
p(1|…)     0.6429
p(2|…)     0.1483
p(3|…)     0.2088
p(C|…)     0.9331
p(H|…)     0.0669
p(STOP|…)      4.4E-10

7 ITERATIONS

p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)            a(C)
0.0     0       0       1.0     0       #N/A     #N/A     #N/A     #N/A              3.09E-10
0       0.005   0       0       0.995   0.0      0.005    0.0      0.995             0.00795
0       0.007   0       0       0.993   0.002    0.004    0.003    0.991             0.004761
0.006   0       0       0.994   0       0.002    0.004    0.004    0.99              0.001583
0       0.009   0       0       0.991   0.003    0.006    0.003    0.988             0.00091
0.011   0       0       0.989   0       0.005    0.006    0.004    0.985             0.000299
0       0.023   0       0       0.977   0.008    0.015    0.003    0.974             0.000171
0.046   0       0       0.954   0       0.019    0.027    0.004    0.95              5.6E-05
0.127   0       0       0.873   0       0.043    0.084    0.003    0.871             2.27E-05
0       0.398   0       0       0.602   0.125    0.273    0.002    0.6               1.42E-05
0       0       0.022   0       0       0.398    0.579    0.0      0.022             2.06E-05
0       0.944   0       0       0.056   0.941    0.003    0.037    0.019             4.03E-06
0       0.95    0       0       0.05    0.932    0.018    0.012    0.038             7.97E-07
0       0       0.0     0       0       0.95     0.05     0.0      0.0               4.99E-07
0       0       0.0     0       0       1.0      0.0      0.0      0.0               2.99E-07
0       0       0.0     0       0       1.0      0.0      0.0      0.0               1.79E-07
0.98    0       0       0.02    0       0.98     0.0      0.019    0.0               2.48E-08
0       0       0.0     0       0       0.98     0.019    0.0      0.0               1.52E-08
0       0       0.0     0       0       1.0      0.0      0.0      0.0               9.11E-09
0       0       0.0     0       0       1.0      0.0      0.0      0.0               5.47E-09
0       0.987   0       0       0.013   0.987    0.0      0.013    0.0               1.07E-09
0       0       0.0     0       0       0.987    0.013    0.0      0.0               6.47E-10
0.98    0       0       0.02    0       0.98     0.0      0.019    0.0               8.95E-11
0       0       0.0     0       0       0.98     0.02     0.0      0.0               5.48E-11
0       0       0.0     0       0       1.0      0.0     0.0     0.0      3.29E-11
0       0       0.003   0       0       0.997    0.0     0.003   0.0      1.97E-11
0.351   0       0       0.649   0       0.351    0.0     0.646   0.003    2.73E-12
0       0.159   0       0       0.841   0.156    0.003   0.195   0.646    5.42E-13
0       0.069   0       0       0.931   0.066    0.004   0.093   0.838    1.11E-13
0.025   0       0       0.975   0       0.023    0.003   0.046   0.928    1.71E-14
0       0.013   0       0       0.987   0.01     0.004   0.016   0.971    4.54E-15
0.005   0       0       0.995   0       0.003    0.002   0.01    0.985    9.79E-16
0.0     0       0       1.0     0       0.0      0.0     0.005   0.995    2.98E-16
2.531   3.564   0.027   8.469   7.436   15.927   1.141   1.141   13.791
p(…|H)     p(…|START)
0.0017
0.5316
0.4667
0.0716    2.1E-09
0.8656    1.0
0.0628    0

ITERATIONS

a(C)*b(C)
a(H)       b(C)       b(H)       a(C)*b(C) a(H)*b(H) +a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2)
0.531571   1.19E-16   8.37E-16   3.67E-26 4.45E-16 4.452E-16      0.0     1.0     0       0.0
0.214756   2.81E-16   2.06E-15   2.23E-18 4.43E-16 4.452E-16      0.005   0.995   0       0
0.08701    6.26E-16   5.08E-15   2.98E-18 4.42E-16 4.452E-16      0.007   0.993   0       0
0.040206   1.7E-15    1.1E-14    2.68E-18 4.43E-16 4.452E-16      0.006   0.994   0       0.006
0.016293   4.37E-15   2.71E-14   3.97E-18 4.41E-16 4.452E-16      0.009   0.991   0       0
0.007529   1.65E-14   5.85E-14   4.94E-18 4.4E-16    4.452E-16    0.011   0.989   0       0.011
0.003051   6.2E-14    1.42E-13   1.06E-17 4.35E-16 4.452E-16      0.024   0.976   0       0
0.00141    3.71E-13   3.01E-13   2.08E-17 4.24E-16 4.452E-16      0.047   0.953   0       0.047
0.000651   2.53E-12   5.96E-13   5.75E-17 3.88E-16 4.452E-16      0.129   0.871   0       0.129
0.000264   1.28E-11   1E-12      1.81E-16 2.64E-16 4.452E-16      0.408   0.592   0       0
3.91E-07   2.14E-11   1.15E-11   4.41E-16 4.51E-18 4.452E-16      0.99    0.01    0.99    0
8.02E-07   1.06E-10   2.47E-11   4.25E-16 1.98E-17 4.452E-16      0.956   0.044   0       0
4.5E-07    5.36E-10   4.12E-11   4.27E-16 1.85E-17 4.452E-16      0.958   0.042   0       0
7.54E-10   8.93E-10   6.87E-11   4.45E-16 5.18E-20 4.452E-16      1.0     0.0     1.0     0
5.79E-11   1.49E-09   1.15E-10   4.45E-16 6.69E-21 4.452E-16      1.0     0.0     1.0     0
3.42E-11   2.48E-09   8.09E-10   4.45E-16 2.77E-20 4.452E-16      1.0     0.0     1.0     0
6.39E-09   1.76E-08   1.35E-09   4.37E-16 8.65E-18 4.452E-16      0.981   0.019   0       0.981
1.23E-11   2.93E-08   2.25E-09   4.45E-16 2.77E-20 4.452E-16      1.0     0.0     1.0     0
1.75E-12   4.88E-08   3.78E-09   4.45E-16 6.61E-21 4.452E-16      1.0     0.0     1.0     0
1.04E-12   8.14E-08   1.91E-08   4.45E-16 1.99E-20 4.452E-16      1.0     0.0     1.0     0
1.71E-10   4.13E-07   3.2E-08    4.4E-16   5.47E-18 4.452E-16     0.988   0.012   0       0
3.74E-13   6.88E-07   2.24E-07   4.45E-16 8.39E-20 4.452E-16      1.0     0.0     1.0     0
2.32E-11   4.87E-06   3.75E-07   4.36E-16 8.69E-18 4.452E-16      0.98    0.02    0       0.98
4.44E-14   8.13E-06   6.26E-07   4.45E-16 2.78E-20 4.452E-16      1.0     0.0     1.0     0
6.31E-15   1.35E-05   1.32E-06   4.45E-16   8.31E-21   4.452E-16   1.0      0.0      1.0      0
3.76E-15   2.25E-05   0.000189   4.44E-16   7.11E-19   4.452E-16   0.998    0.002    0.998    0
7.03E-13   5.75E-05   0.00041    1.57E-16   2.88E-16   4.452E-16   0.353    0.647    0        0.353
3.69E-13   0.000133   0.001011   7.23E-17   3.73E-16   4.452E-16   0.162    0.838    0        0
1.66E-13   0.000285   0.002491   3.17E-17   4.13E-16   4.452E-16   0.071    0.929    0        0
8.03E-14   0.000676   0.005398   1.16E-17   4.34E-16   4.452E-16   0.026    0.974    0        0.026
3.3E-14    0.001335   0.013313   6.06E-18   4.39E-16   4.452E-16   0.014    0.986    0        0
1.53E-14   0.002231   0.02888    2.18E-18   4.43E-16   4.452E-16   0.005    0.995    0        0.005
7.09E-15   4.45E-10   0.062765   1.33E-25   4.45E-16   4.452E-16   0.0      1.0      0        0.0
TOTAL:   17.126   15.874   10.988   2.537
p(…|C)    p(…|H)
p(1|…)     0.6416    7.8E-04
p(2|…)     0.1482    0.5331
p(3|…)     0.2103    0.4661
p(C|…)     0.9335    0.0717
p(H|…)     0.0665    0.8653
p(STOP|…)      1.7E-11   0.063

8 ITERATIONS

p(C,3) p(H,1) p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)            a(C)       a(H)
0       0       1.0     0       #N/A     #N/A     #N/A     #N/A              1.22E-11   0.533113
0.005   0       0       0.995   0.0      0.005    0.0      0.995             0.008043   0.215007
0.007   0       0       0.993   0.002    0.005    0.003    0.99              0.004822   0.086963
0       0       0.994   0       0.003    0.004    0.004    0.99              0.001591   0.040285
0.009   0       0       0.991   0.003    0.006    0.003    0.988             0.00092    0.016297
0       0       0.989   0       0.005    0.006    0.004    0.985             0.0003     0.00755
0.024   0       0       0.976   0.008    0.016    0.003    0.973             0.000173   0.003054
0       0       0.953   0       0.02     0.027    0.004    0.949             5.64E-05   0.001415
0       0       0.871   0       0.044    0.085    0.003    0.868             2.28E-05   0.000655
0.408   0       0       0.592   0.127    0.28     0.002    0.591             1.44E-05   0.000265
0       0.01    0       0       0.408    0.582    0.0      0.01              2.08E-05   1.78E-07
0.956   0       0       0.044   0.954    0.001    0.036    0.009             4.08E-06   7.16E-07
0.958   0       0       0.042   0.944    0.014    0.012    0.03              8.12E-07   4.15E-07
0       0.0     0       0       0.958    0.042    0.0      0.0               5.06E-07   3.21E-10
0       0.0     0       0       1.0      0.0      0.0      0.0               3.03E-07   2.63E-11
0       0.0     0       0       1.0      0.0      0.0      0.0               1.81E-07   1.56E-11
0       0       0.019   0       0.981    0.0      0.019    0.0               2.51E-08   6.44E-09
0       0.0     0       0       0.981    0.019    0.0      0.0               1.53E-08   5.61E-12
0       0.0     0       0       1.0      0.0      0.0      0.0               9.17E-09   7.94E-13
0       0.0     0       0       1.0      0.0      0.0      0.0               5.49E-09   4.74E-13
0.988   0       0       0.012   0.988    0.0      0.012    0.0               1.08E-09   1.71E-10
0       0.0     0       0       0.988    0.012    0.0      0.0               6.54E-10   1.7E-13
0       0       0.02    0       0.98     0.0      0.019    0.0               9.04E-11   2.33E-11
0       0.0     0       0       0.98     0.019    0.0      0.0               5.52E-11   2.03E-14
0       0.0     0       0       1.0      0.0     0.0     0.0      3.31E-11   2.86E-15
0       0.002   0       0       0.998    0.0     0.002   0.0      1.98E-11   1.71E-15
0       0       0.647   0       0.353    0.0     0.646   0.002    2.74E-12   7.03E-13
0.162   0       0       0.838   0.159    0.003   0.193   0.644    5.48E-13   3.69E-13
0.071   0       0       0.929   0.068    0.004   0.095   0.834    1.13E-13   1.66E-13
0       0       0.974   0       0.023    0.003   0.048   0.926    1.74E-14   8.04E-14
0.014   0       0       0.986   0.01     0.004   0.016   0.97     4.63E-15   3.3E-14
0       0       0.995   0       0.003    0.002   0.01    0.985    9.91E-16   1.54E-14
0       0       1.0     0       0.0      0.0     0.005   0.995    3E-16      7.13E-15
3.601   0.012   8.463   7.399   15.987   1.139   1.139   13.735
p(…|START)

8.2E-11
1.0
0

RATIONS

a(C)*b(C)
b(C)       b(H)       a(C)*b(C) a(H)*b(H) +a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2) p(C,3) p(H,1)
1.2E-16    8.42E-16   1.46E-27 4.49E-16 4.489E-16      0.0     1.0     0       0.0     0       0
2.82E-16   2.08E-15   2.27E-18 4.47E-16 4.489E-16      0.005   0.995   0       0       0.005   0
6.29E-16   5.13E-15   3.03E-18 4.46E-16 4.489E-16      0.007   0.993   0       0       0.007   0
1.71E-15   1.11E-14   2.72E-18 4.46E-16 4.489E-16      0.006   0.994   0       0.006   0       0
4.4E-15    2.73E-14   4.05E-18 4.45E-16 4.489E-16      0.009   0.991   0       0       0.009   0
1.67E-14   5.88E-14   5.03E-18 4.44E-16 4.489E-16      0.011   0.989   0       0.011   0       0
6.26E-14   1.43E-13   1.08E-17 4.38E-16 4.489E-16      0.024   0.976   0       0       0.024   0
3.75E-13   3.02E-13   2.11E-17 4.28E-16 4.489E-16      0.047   0.953   0       0.047   0       0
2.56E-12   5.96E-13   5.84E-17 3.9E-16    4.489E-16    0.13    0.87    0       0.13    0       0
1.29E-11   9.97E-13   1.85E-16 2.64E-16 4.489E-16      0.412   0.588   0       0       0.412   0
2.15E-11   1.15E-11   4.47E-16 2.05E-18 4.489E-16      0.995   0.005   0.995   0       0       0.005
1.06E-10   2.45E-11   4.31E-16 1.76E-17 4.489E-16      0.961   0.039   0       0       0.961   0
5.32E-10   4.09E-11   4.32E-16 1.7E-17    4.489E-16    0.962   0.038   0       0       0.962   0
8.88E-10   6.83E-11   4.49E-16 2.19E-20 4.489E-16      1.0     0.0     1.0     0       0       0.0
1.48E-09   1.14E-10   4.49E-16 3.01E-21 4.489E-16      1.0     0.0     1.0     0       0       0.0
2.48E-09   8.1E-10    4.49E-16 1.27E-20 4.489E-16      1.0     0.0     1.0     0       0       0.0
1.76E-08   1.35E-09   4.4E-16   8.69E-18 4.489E-16     0.981   0.019   0       0.981   0       0
2.93E-08   2.25E-09   4.49E-16 1.27E-20 4.489E-16      1.0     0.0     1.0     0       0       0.0
4.89E-08   3.77E-09   4.49E-16 3E-21      4.489E-16    1.0     0.0     1.0     0       0       0.0
8.17E-08   1.9E-08    4.49E-16 9.01E-21 4.489E-16      1.0     0.0     1.0     0       0       0.0
4.11E-07   3.18E-08   4.43E-16 5.41E-18 4.489E-16      0.988   0.012   0       0       0.988   0
6.87E-07   2.25E-07   4.49E-16 3.82E-20 4.489E-16      1.0     0.0     1.0     0       0       0.0
4.87E-06   3.75E-07   4.4E-16   8.71E-18 4.489E-16     0.981   0.019   0       0.981   0       0
8.13E-06   6.26E-07   4.49E-16 1.27E-20 4.489E-16      1.0     0.0     1.0     0       0       0.0
1.36E-05   1.17E-06   4.49E-16   3.35E-21   4.489E-16   1.0      0.0      1.0      0       0       0.0
2.26E-05   0.000191   4.49E-16   3.26E-19   4.489E-16   0.999    0.001    0.999    0       0       0.001
5.79E-05   0.000413   1.59E-16   2.9E-16    4.489E-16   0.353    0.647    0        0.353   0       0
0.000134   0.001018   7.36E-17   3.75E-16   4.489E-16   0.164    0.836    0        0       0.164   0
0.000287   0.002515   3.24E-17   4.16E-16   4.489E-16   0.072    0.928    0        0       0.072   0
0.000679   0.005436   1.18E-17   4.37E-16   4.489E-16   0.026    0.974    0        0.026   0       0
0.001339   0.013428   6.2E-18    4.43E-16   4.489E-16   0.014    0.986    0        0       0.014   0
0.002233   0.029059   2.21E-18   4.47E-16   4.489E-16   0.005    0.995    0        0.005   0       0
1.74E-11   0.062996   5.23E-27   4.49E-16   4.489E-16   0.0      1.0      0        0.0     0       0
TOTAL:   17.153   15.847   10.994   2.54    3.618   0.006
p(…|C)    p(…|H)    p(…|START)
p(1|…)     0.641     3.5E-04
p(2|…)     0.1481    0.5338
p(3|…)     0.2109    0.4658
p(C|…)     0.9337    0.0718    3.3E-12
p(H|…)     0.0663    0.8651    1.0
p(STOP|…)      6.8E-13   0.0631    0

9 ITERATIONS

p(H,2) p(H,3) p(CC)   p(HC)   p(CH)   p(HH)            a(C)       a(H)       b(C)       b(H)
1.0     0       #N/A     #N/A     #N/A     #N/A              4.83E-13   0.533815   1.2E-16    8.44E-16
0       0.995   0.0      0.005    0.0      0.995             0.008085   0.215121   2.83E-16   2.08E-15
0       0.993   0.002    0.005    0.003    0.99              0.00485    0.086941   6.31E-16   5.15E-15
0.994   0       0.003    0.004    0.004    0.99              0.001595   0.040321   1.72E-15   1.11E-14
0       0.991   0.003    0.006    0.003    0.988             0.000925   0.016298   4.41E-15   2.74E-14
0.989   0       0.005    0.006    0.004    0.985             0.000301   0.007559   1.68E-14   5.89E-14
0       0.976   0.008    0.016    0.003    0.973             0.000174   0.003056   6.28E-14   1.44E-13
0.953   0       0.02     0.027    0.004    0.949             5.65E-05   0.001417   3.77E-13   3.03E-13
0.87    0       0.044    0.086    0.003    0.867             2.29E-05   0.000656   2.57E-12   5.97E-13
0       0.588   0.129    0.283    0.002    0.587             1.45E-05   0.000265   1.29E-11   9.96E-13
0       0       0.412    0.584    0.0      0.005             2.09E-05   8.07E-08   2.16E-11   1.15E-11
0       0.039   0.96     0.001    0.035    0.004             4.11E-06   6.77E-07   1.06E-10   2.45E-11
0       0.038   0.949    0.013    0.012    0.026             8.19E-07   4E-07      5.3E-10    4.08E-11
0       0       0.962    0.038    0.0      0.0               5.09E-07   1.4E-10    8.86E-10   6.82E-11
0       0       1.0      0.0      0.0      0.0               3.04E-07   1.19E-11   1.48E-09   1.14E-10
0       0       1.0      0.0      0.0      0.0               1.82E-07   7.08E-12   2.47E-09   8.1E-10
0.019   0       0.981    0.0      0.019    0.0               2.52E-08   6.46E-09   1.75E-08   1.35E-09
0       0       0.981    0.019    0.0      0.0               1.54E-08   2.54E-12   2.93E-08   2.26E-09
0       0       1.0      0.0      0.0      0.0               9.2E-09    3.58E-13   4.9E-08    3.77E-09
0       0       1.0      0.0      0.0      0.0               5.51E-09   2.14E-13   8.18E-08   1.9E-08
0       0.012   0.988    0.0      0.012    0.0               1.08E-09   1.7E-10    4.11E-07   3.16E-08
0       0       0.988    0.012    0.0      0.0               6.57E-10   7.68E-14   6.86E-07   2.25E-07
0.019   0       0.981    0.0      0.019    0.0               9.08E-11   2.33E-11   4.87E-06   3.74E-07
0       0       0.981    0.019    0.0      0.0               5.54E-11   9.17E-15   8.13E-06   6.26E-07
0       0       1.0      0.0     0.0     0.0      3.32E-11   1.29E-15   1.36E-05   1.1E-06
0       0       0.999    0.0     0.001   0.0      1.98E-11   7.71E-16   2.27E-05   0.000192
0.647   0       0.353    0.0     0.646   0.001    2.74E-12   7.03E-13   5.81E-05   0.000414
0       0.836   0.161    0.003   0.193   0.643    5.51E-13   3.68E-13   0.000135   0.001022
0       0.928   0.069    0.004   0.095   0.833    1.14E-13   1.65E-13   0.000287   0.002525
0.974   0       0.024    0.003   0.049   0.925    1.75E-14   8.04E-14   0.000681   0.005453
0       0.986   0.01     0.004   0.016   0.97     4.67E-15   3.3E-14    0.001341   0.013481
0.995   0       0.003    0.002   0.011   0.984    9.97E-16   1.54E-14   0.002235   0.02914
1.0     0       0.0      0.0     0.005   0.995    3.01E-16   7.14E-15   6.79E-13   0.063102
8.46    7.382   16.015   1.138   1.138   13.709
a(C)*b(C)
a(C)*b(C) a(H)*b(H) +a(H)*b(H)   p(C)   p(H)   p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3)
5.81E-29 4.51E-16 4.506E-16      0.0     1.0     0       0.0     0       0       1.0     0
2.29E-18 4.48E-16 4.506E-16      0.005   0.995   0       0       0.005   0       0       0.995
3.06E-18 4.48E-16 4.506E-16      0.007   0.993   0       0       0.007   0       0       0.993
2.74E-18 4.48E-16 4.506E-16      0.006   0.994   0       0.006   0       0       0.994   0
4.08E-18 4.47E-16 4.506E-16      0.009   0.991   0       0       0.009   0       0       0.991
5.07E-18 4.46E-16 4.506E-16      0.011   0.989   0       0.011   0       0       0.989   0
1.09E-17 4.4E-16    4.506E-16    0.024   0.976   0       0       0.024   0       0       0.976
2.13E-17 4.29E-16 4.506E-16      0.047   0.953   0       0.047   0       0       0.953   0
5.89E-17 3.92E-16 4.506E-16      0.131   0.869   0       0.131   0       0       0.869   0
1.86E-16 2.64E-16 4.506E-16      0.414   0.586   0       0       0.414   0       0       0.586
4.5E-16   9.25E-19 4.506E-16     0.998   0.002   0.998   0       0       0.002   0       0
4.34E-16 1.66E-17 4.506E-16      0.963   0.037   0       0       0.963   0       0       0.037
4.34E-16 1.63E-17 4.506E-16      0.964   0.036   0       0       0.964   0       0       0.036
4.51E-16 9.56E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.51E-16 1.35E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.51E-16 5.73E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.42E-16 8.71E-18 4.506E-16      0.981   0.019   0       0.981   0       0       0.019   0
4.51E-16 5.73E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.51E-16 1.35E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.51E-16 4.06E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.45E-16 5.39E-18 4.506E-16      0.988   0.012   0       0       0.988   0       0       0.012
4.51E-16 1.73E-20 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.42E-16 8.72E-18 4.506E-16      0.981   0.019   0       0.981   0       0       0.019   0
4.51E-16 5.74E-21 4.506E-16      1.0     0.0     1.0     0       0       0.0     0       0
4.51E-16   1.42E-21   4.506E-16   1.0      0.0      1.0      0       0       0.0     0       0
4.5E-16    1.48E-19   4.506E-16   1.0      0.0      1.0      0       0       0.0     0       0
1.59E-16   2.91E-16   4.506E-16   0.354    0.646    0        0.354   0       0       0.646   0
7.42E-17   3.76E-16   4.506E-16   0.165    0.835    0        0       0.165   0       0       0.835
3.28E-17   4.18E-16   4.506E-16   0.073    0.927    0        0       0.073   0       0       0.927
1.19E-17   4.39E-16   4.506E-16   0.026    0.974    0        0.026   0       0       0.974   0
6.27E-18   4.44E-16   4.506E-16   0.014    0.986    0        0       0.014   0       0       0.986
2.23E-18   4.48E-16   4.506E-16   0.005    0.995    0        0.005   0       0       0.995   0
2.05E-28   4.51E-16   4.506E-16   0.0      1.0      0        0.0     0       0       1.0     0
TOTAL:   17.165   15.835   10.998   2.542   3.625   0.002   8.458   7.375
Forward-Back
p(ice cream
perplexity per obs (inc

We generally say a model is go
(equivalently, low perplexity) to
guarantee that the model will a
seen ... but the forward-backw
p(…|C)      p(…|H)    p(…|START)                 at least improve the probability
p(1|…)     0.6407      1.6E-04
p(2|…)     0.1481      0.5341                             The numbers above and the gr
p(3|…)     0.2112      0.4657                             the 11 iterations we've just per
p(C|…)     0.9337      0.0718    1.3E-13                  (local) maximum of the function
p(H|…)     0.0663      0.865     1.0                      data). Thus, we see p(ice crea
p(STOP|…)      2.6E-14     0.0632    0                        column a(C)*b(C) +a(H)*b(H),

10 ITERATIONS
Scroll to the bottom to see how the graphs
have smoothed out since the start! (If you
want additional iterations, scroll all the way
right for instructions.)

p(CC)   p(HC)   p(CH)   p(HH)            a(C)        a(H)       b(C)        b(H)          a(C)*b(C)
#N/A     #N/A     #N/A     #N/A              1.908E-14   0.534132   1.204E-16   8.45E-16      2.29707E-30
0.0      0.005    0.0      0.995             0.0081047   0.215172   2.837E-16   2.09E-15      2.29915E-18
0.002    0.005    0.003    0.99              0.0048632   0.086931   6.314E-16   5.16E-15      3.07048E-18
0.003    0.004    0.004    0.99              0.0015973   0.040337   1.719E-15   1.11E-14      2.74525E-18
0.003    0.006    0.003    0.988             0.0009271   0.016299   4.42E-15    2.74E-14      4.09762E-18
0.005    0.006    0.004    0.984             0.0003016   0.007563   1.686E-14   5.9E-14       5.08459E-18
0.008    0.016    0.003    0.973             0.0001742   0.003056   6.294E-14   1.44E-13      1.09659E-17
0.02     0.027    0.004    0.949             5.661E-05   0.001418   3.775E-13   3.03E-13      2.13709E-17
0.045    0.086    0.003    0.867             2.291E-05   0.000657   2.578E-12   5.97E-13      5.90652E-17
0.129    0.285    0.002    0.585             1.449E-05   0.000265   1.291E-11   9.95E-13      1.87155E-16
0.414    0.584    0.0      0.002             2.089E-05   3.64E-08   2.159E-11   1.14E-11      4.50941E-16
0.963    0.0      0.035    0.002             4.12E-06    6.59E-07   1.056E-10   2.44E-11      4.35241E-16
0.952    0.012    0.011    0.025             8.225E-07   3.93E-07   5.293E-10   4.07E-11      4.35357E-16
0.964    0.036    0.0      0.0               5.102E-07   6.22E-11   8.847E-10   6.81E-11      4.51353E-16
1.0      0.0      0.0      0.0               3.052E-07   5.34E-12   1.479E-09   1.14E-10      4.51357E-16
1.0      0.0      0.0      0.0               1.826E-07   3.19E-12   2.472E-09   8.1E-10       4.51355E-16
0.981    0.0      0.019    0.0               2.525E-08   6.46E-09   1.753E-08   1.35E-09      4.42634E-16
0.981    0.019    0.0      0.0               1.54E-08    1.15E-12   2.931E-08   2.26E-09      4.51355E-16
1.0      0.0      0.0      0.0               9.213E-09   1.61E-13   4.899E-08   3.77E-09      4.51357E-16
1.0      0.0      0.0      0.0               5.512E-09   9.63E-14   8.189E-08   1.9E-08       4.51355E-16
0.988    0.0      0.012    0.0               1.087E-09   1.7E-10    4.103E-07   3.16E-08      4.4598E-16
0.988    0.012    0.0      0.0               6.581E-10   3.46E-14   6.858E-07   2.25E-07      4.5135E-16
0.981    0.0      0.019    0.0               9.1E-11     2.33E-11   4.864E-06   3.74E-07      4.4263E-16
0.981    0.019    0.0      0.0               5.551E-11   4.13E-15   8.131E-06   6.26E-07      4.51355E-16
1.0      0.0     0.0     0.0      3.321E-11   5.81E-16   1.359E-05   1.07E-06   4.51357E-16
1.0      0.0     0.0     0.0      1.987E-11   3.47E-16   2.271E-05   0.000192   4.51291E-16
0.354    0.0     0.646   0.0      2.747E-12   7.03E-13   5.818E-05   0.000414   1.59827E-16
0.161    0.003   0.192   0.643    5.524E-13   3.68E-13   0.0001348   0.001024   7.44697E-17
0.069    0.004   0.095   0.832    1.145E-13   1.65E-13   0.0002875   0.00253    3.29327E-17
0.024    0.003   0.049   0.925    1.76E-14    8.05E-14   0.0006815   0.005461   1.19915E-17
0.01     0.004   0.016   0.97     4.691E-15   3.3E-14    0.001342    0.013504   6.29541E-18
0.003    0.002   0.011   0.984    9.993E-16   1.54E-14   0.0022356   0.029177   2.23391E-18
0.0      0.0     0.005   0.995    3.019E-16   7.15E-15   2.648E-14   0.06315    7.99443E-30
16.027   1.138   1.138   13.698

Weather States that Best Explain Ice Cream Consum

3.5

3

2.5

2

1.5

1

0.5

0
1    3   5      7    9      11   13    15      17       19
Diary Day
10 Iterations
Forward-Backward Iteration         0      1      2      3               4      5      6      7      8
p(ice cream observations)    9E-19 1E-16 2E-16 3E-16              4E-16 4E-16 4E-16 4E-16 4E-16
perplexity per obs (including STOP)    3.393 2.9466 2.8794 2.8535            2.84 2.8331 2.8297 2.8282 2.8275

generally say a model is good to the degree that it assigns high probability                                            Every Iteration of Forward-Backward Is
uivalently, low perplexity) to test data. Obviously no modeling technique can                                             Guaranteed to Increase P(Observations)
rantee that the model will assign high probability to test data it has never
n ... but the forward-backward algorithm is guaranteed at every iteration to                                             4
east improve the probability of the observed training data!
3.5

Perplexity per day of # of cones
numbers above and the graph at right show that this is indeed the case for                                              3
11 iterations we've just performed. The parameters are converging on a
al) maximum of the function that maps the 15 parameters to p(ice cream                                                  2.5
a). Thus, we see p(ice cream data) improve on each iteration; it's given by the                                           2
umn a(C)*b(C) +a(H)*b(H), which sums probs of all paths that emit that data.)
1.5

1
Thus, we have gotten a model that better predicts my pattern of           0.5
ice cream consumption in terms of a hidden weather variable
0
that is either C or H and tends to stay the same from day to day.
Analogously, a Hidden Markov Model tagger tries to explain                    0 1 2 3 4 5 6 7 8 9
words in terms of a hidden tag variable that varies predictably                  Iterations of Forward-Backward
over time.
a(C)*b(C)
a(H)*b(H)          +a(H)*b(H) p(C) p(H) p(C,1) p(C,2) p(C,3) p(H,1) p(H,2) p(H,3) p(CC)
4.51357E-16        4.514E-16 0.0         1.0     0        0.0     0     0      1.0         0          #N/A
4.49058E-16        4.514E-16 0.005       0.995   0        0       0.005 0      0           0.995      0.0
4.48287E-16        4.514E-16 0.007       0.993   0        0       0.007 0      0           0.993      0.002
4.48612E-16        4.514E-16 0.006       0.994   0        0.006   0     0      0.994       0          0.003
4.4726E-16         4.514E-16 0.009       0.991   0        0       0.009 0      0           0.991      0.003
4.46273E-16        4.514E-16 0.011       0.989   0        0.011   0     0      0.989       0          0.005
4.40391E-16        4.514E-16 0.024       0.976   0        0       0.024 0      0           0.976      0.008
4.29986E-16        4.514E-16 0.047       0.953   0        0.047   0     0      0.953       0          0.02
3.92292E-16        4.514E-16 0.131       0.869   0        0.131   0     0      0.869       0          0.045
2.64203E-16        4.514E-16 0.415       0.585   0        0       0.415 0      0           0.585      0.129
4.16222E-19        4.514E-16 0.999       0.001   0.999    0       0     0.001  0           0          0.415
1.61161E-17        4.514E-16 0.964       0.036   0        0       0.964 0      0           0.036      0.964
1.60004E-17        4.514E-16 0.965       0.035   0        0       0.965 0      0           0.035      0.953
4.2327E-21         4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          0.965
6.08122E-22        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          1.0
2.58356E-21        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          1.0
8.72321E-18        4.514E-16 0.981       0.019   0        0.981   0     0      0.019       0          0.981
2.58356E-21        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          0.981
6.07576E-22        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          1.0
1.82524E-21        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          1.0
5.37701E-18        4.514E-16 0.988       0.012   0        0       0.988 0      0           0.012      0.988
7.76678E-21        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          0.988
8.72709E-18        4.514E-16 0.981       0.019   0        0.981   0     0      0.019       0          0.981
2.58445E-21        4.514E-16 1.0         0.0     1.0      0       0     0.0    0           0          0.981
6.22283E-22    4.514E-16   1.0          0.0     1.0       0           0           0.0     0          0         1.0
6.66729E-20    4.514E-16   1.0          0.0     1.0       0           0           0.0     0          0         1.0
2.9153E-16     4.514E-16   0.354        0.646   0         0.354       0           0       0.646      0         0.354
3.76888E-16    4.514E-16   0.165        0.835   0         0           0.165       0       0          0.835     0.162
4.18425E-16    4.514E-16   0.073        0.927   0         0           0.073       0       0          0.927     0.069
4.39366E-16    4.514E-16   0.027        0.973   0         0.027       0           0       0.973      0         0.024
4.45062E-16    4.514E-16   0.014        0.986   0         0           0.014       0       0          0.986     0.01
4.49123E-16    4.514E-16   0.005        0.995   0         0.005       0           0       0.995      0         0.003
4.51357E-16    4.514E-16   0.0          1.0     0         0.0         0           0       1.0        0         0.0
TOTAL:   17.17        15.83   10.999    2.543       3.629       0.001   8.457      7.371     16.033

lain Ice Cream Consumption                                                    Transitions that Best Explain Consumption
Ice Creams
p(H)                 3.5

3

2.5

2

1.5

1

0.5

0
1   3       5    7      9     11    13   15    17   19    21   23
19     21   23     25   27   29        31     33
Diary Day

10 Iterations
Below are the final parameters after 11 iterations of the forward-
9     10              backward algorithm. If the graph at left needs > 11 iterations to
5E-16 5E-16               converge, you can plug these parameters back into the start of the
2.8272 2.8271              model and run it again. To do this, click and drag to highlight the
entire table below. Use Edit / Copy to copy it to the clipboard; and
Backward Is
then use Edit / PasteSpecial / Values to paste its values (not the
(Observations)                               formulas!) over the table of initial probabilities at the far left of this
worksheet. To repeat, just paste again (no need to return here!), so
it's easy to run several 11's of iterations.
p(…|C)     p(…|H)         p(…|START)
p(1|…) 0.6406         7.1E-05
If the iteration-0       p(2|…) 0.1481         0.5343
perplexity dwarfs        p(3|…) 0.2113         0.4657
the others, delete       p(C|…) 0.9338         0.0719         5.1E-15
its cell and the         p(H|…) 0.0662         0.865          1.0
graph will rescale p(STOP|…) 1.0E-15           0.0632         0
to focus on the
difference         Below are some other interesting initial parameters for you to try; again, you can
among iterations paste them over the parameter table in the upper left corner of this worksheet.
1-10.              Also feel free to change the ice cream data being modeled.

9   10                         p(…|C)                p(…|H)      p(…|START)
Backward                              p(1|…) 0.7                    0.1
p(2|…) 0.2                    0.2
p(3|…) 0.1                    0.7
p(HC) p(CH) p(HH)     p(C|…) 0.45                   0.45        0.5       No weather inertia here: if today
#N/A   #N/A   #N/A       p(H|…) 0.45                   0.45        0.5       is cold, tomorrow is equally likely
0.005  0.0    0.995  p(STOP|…) 0.1                     0.1         0         to be hot or cold.
0.005  0.003  0.99
0.004  0.004  0.99             p(…|C)                p(…|H)      p(…|START)
0.006  0.003  0.988      p(1|…) 0.4                    0.3                Only a slight preference to eat
0.006  0.004  0.984      p(2|…) 0.3                    0.3                more ice cream on hot days.
0.016  0.003  0.973      p(3|…) 0.3                    0.4
0.027  0.004  0.949      p(C|…) 0.8                    0.1         0.5
0.086  0.003  0.866      p(H|…) 0.1                    0.8         0.5
0.285  0.002  0.584  p(STOP|…) 0.1                     0.1         0
0.584  0.0    0.001
0.0    0.035  0.001            p(…|C)                p(…|H)      p(…|START)
0.012  0.011  0.024      p(1|…) 0.3                    0.3                No preference at all for more ice
0.035  0.0    0.0        p(2|…) 0.4                    0.4                cream on hot days.
0.0    0.0    0.0        p(3|…) 0.3                    0.3
0.0    0.0    0.0        p(C|…) 0.45                   0.45        0.5       No weather inertia either. In
0.0    0.019  0.0        p(H|…) 0.45                   0.45        0.5       short, the model is completely
0.019  0.0    0.0    p(STOP|…) 0.1                     0.1         0         symmetric.
0.0    0.0    0.0
0.0    0.0    0.0              p(…|C)                p(…|H)      p(…|START)
0.0    0.012  0.0        p(1|…) 0.3                    0.4                The same as above, but now we
0.012  0.0    0.0        p(2|…) 0.3                    0.3                have broken the symmetry by
0.0    0.019  0.0        p(3|…) 0.4                    0.3                giving C days a little push to eat
0.019  0.0    0.0        p(C|…) 0.45                   0.45        0.5    more ice cream. With a lot of
iterations, this is enough to make
C days hot and H days cold (the
reverse of before).
The same as above, but now we
have broken the symmetry by
giving C days a little push to eat
more ice cream. With a lot of
0.0      0.0      0.0              p(H|…)    0.45     0.45         0.5     iterations, this is enough to make
0.0      0.0      0.0          p(STOP|…)     0.1      0.1          0       C days hot and H days cold (the
0.0      0.646    0.0                                                      reverse of before).
0.003    0.192    0.643                     p(…|C)   p(…|H)    p(…|START)
0.004    0.096    0.831            p(1|…)     0.3      0.3              As discussed earlier, the forward-backward algorith
0.003    0.049    0.924            p(2|…)     0.3      0.4              red parameters so as to locally maximize p(observ
0.004    0.016    0.97             p(3|…)     0.4      0.3              the observed data are the words or the ice cream c
0.002    0.011    0.984            p(C|…)     0.45     0.45      0.5    figures. There may be multiple local maxima: we fo
0.0      0.005    0.995            p(H|…)     0.45     0.45      0.5    namely a parameter set where C represents cold d
1.137    1.137    13.692       p(STOP|…)      0.1      0.1       0      one where C represents hot days. By symmetry the
good values of p(observed data); which one the alg
depends entirely on the initial parameters. But the
parameters to the left, which favor weather "anti-ine
third, totally different local maximum where p(obser
unfortunately lower. Try it out. What kind of days d
now? What kind of regularity in the ice cream data
Ice Creams
estimated model exploit to predict p(observed data
onsumption                                                                          enlarge the model so that it could find both this reg
p(C -> C)
hot-cold regularity?
p(H -> C)
p(C -> H)
p(H -> H)

1

23     25    27   29   31      33
6
11
16

21

26
Series4
31
Series1
0-10 Iterations
to locally maximize p(observed data), where
e the words or the ice cream consumption
e multiple local maxima: we found two above,
set where C represents cold days and another
nts hot days. By symmetry these give equally
erved data); which one the algorithm picks
he initial parameters. But the initial
, which favor weather "anti-inertia," lead to a
ocal maximum where p(observed data) is
Try it out. What kind of days does C represent
egularity in the ice cream data does the
oit to predict p(observed data)? How could we
that it could find both this regularity and the

0.8-1.0
0.6-0.8
0.4-0.6
0.2-0.4
0-0.2

1.0
0.8
0.6       p(H)
0.4
0.2
0
Series10

Series7

Series4

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 10 posted: 5/11/2011 language: English pages: 48
Description: Historical Weather Records document sample