# Markov Chain Monte Carlo Algorithms for Gaussian Processes

Document Sample

```					Markov Chain Monte Carlo Algorithms for Gaussian Processes

Markov Chain Monte Carlo Algorithms for
Gaussian Processes

Michalis K. Titsias, Neil Lawrence and Magnus Rattray
School of Computer Science
University of Manchester

20 June 2008
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Outline

Gaussian Processes
Sampling algorithms for Gaussian Process Models
Sampling from the prior
Gibbs sampling schemes
Sampling using control variables
Applications
Demonstration on regression/classiﬁcation
Transcriptional regulation
Summary/Future work
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gaussian Processes

A Gaussian process (GP) is a distribution over a real-valued
function f (x). It is deﬁned by
a mean function

µ(x) = E (f (x))
and a covariance or kernel function

k(xn , xm ) = E (f (xn )f (xm ))
E.g. this can be the RBF (or squared exponential) kernel

||xn − xm ||2
k(xn , xm ) = α exp −
2 2
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gaussian Processes

We evaluate a function in a set of inputs (xi )N :
i=1

fi = f (xi )

A Gaussian process reduces to a multivariate Gaussian
distribution over f = (fi )N
i=1

1                    f T K −1 f
p(f) = N(x|0, K ) =                   N        1   exp −
(2π) |K |
2        2               2

where the covariance K is deﬁned by the kernel function
p(f) is a conditional distribution (a precise notation is p(f|X ))
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gaussian Processes for Bayesian learning

Many problems involve inference over unobserved/latent functions
A Gaussian process can place a prior on a latent function
Bayesian inference:
Data y = (yi )N (associated with inputs (xi )N )
i=1                            i=1
Likelihood model p(y|f)
GP prior p(f) for the latent function f
Bayes rule

p(f|y) ∝ p(y|f) × p(f)
Posterior ∝ Likelihood × Prior

For regression, where the likelihood is Gaussian, this
computation is analytically obtained
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gaussian Processes for Bayesian Regression
Data and the GP prior (rbf kernel function)
2.5                                                                                            2

2
1.5

1.5
1
1

0.5
0.5

0                                                                                             0

−0.5
−0.5

−1
−1
−1.5

−1.5
−2

−2.5                                                                                           −2
0   0.1   0.2   0.3   0.4   0.5           0.6         0.7     0.8    0.9    1                 0     0.1     0.2         0.3         0.4       0.5   0.6   0.7   0.8   0.9   1

Posterior GP process
2.5

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0         0.1         0.2     0.3    0.4   0.5          0.6     0.7         0.8         0.9         1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gaussian Processes for non-Gaussian Likelihoods

When the likelihood p(y|f) is non-Gaussian computations are
analytically intractable
Non-Gaussian likelihoods:
Classiﬁcation problems
Spatio-temporal models and geostatistics
Non-linear diﬀerential equations with latent functions
Approximations need to be considered
MCMC is a powerful framework that oﬀers:
Arbitrarily precise approximation in the limit of long runs
General applicability (independent from the functional form of
the likelihood)
Markov Chain Monte Carlo Algorithms for Gaussian Processes

MCMC for Gaussian Processes

The Metropolis-Hastings (MH) algorithm
Initialize f (0)
Form a Markov chain. Use a proposal distribution
Q(f (t+1) |f (t) ) and accept with the MH step

p(y|f (t+1) )p(f (t+1) ) Q(f (t) |f (t+1) )
min 1,
p(y|f (t) )p(f (t) ) Q(f (t+1) |f (t) )
The posterior is highly-correlated and f is high dimensional
How do we choose the proposal Q(f (t+1) |f (t) )?
Markov Chain Monte Carlo Algorithms for Gaussian Processes

MCMC for Gaussian Processes

Use the GP prior as the proposal distribution
Proposal: Q(f (t+1) |f (t) ) = p(f (t+1) )
MH probability

p(y|f (t+1) )
min 1,
p(y|f (t) )
Nice property: The prior samples functions with the
appropriate smoothing requirement
Bad property: We get almost zero acceptance rate. The chain
will get stuck in the same state for thousands of iterations
Markov Chain Monte Carlo Algorithms for Gaussian Processes

MCMC for Gaussian Processes

Use Gibbs sampling
Proposal: Iteratively sample from the conditional posterior
p(fi |f−i , y) where f−i = f \ fi
Nice property: All samples are accepted and the prior
smoothing requirement is satisﬁed
Bad property: The Markov chain will move extremely slowly
for densely sampled functions:
The variance of p(fi |f−i , y) is smaller or equal to the variance
of the conditional prior p(fi |f−i )
But p(fi |f−i ) may already have a tiny variance
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gibbs-like schemes

Gibbs-like algorithm: Instead of p(fi |f−i , y) use the conditional
prior p(fi |f−i ) and accept with the MH step (it has been used
in geostatistics, Diggle and Tawn, 1998)
Gibbs-like algorithm is still ineﬃcient to sample from highly
correlated functions
Block or region sampling:
Cluster the function values f into regions/blocks {fk }M
k=1
Sample each block fk from the conditional GP prior
(t+1) (t)
p(fk     |f−k ), where f−k = f \ fk and accept with the MH step
This scheme can work better
But it does not solve the problem of sampling highly correlated
functions since the variance of the proposal can be very small
in the boundaries between regions
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Gibbs-like schemes

Region sampling with 4 regions (2 of the proposals are shown
below)
2                                                                    2

1.5                                                                  1.5

1                                                                    1

0.5                                                                  0.5

0                                                                    0

−0.5                                                                 −0.5

−1                                                                   −1

−1.5                                                                 −1.5

−2                                                                   −2

−2.5                                                                 −2.5

0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1          0   0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9   1

Note that the variance of the conditional priors is small close
to the boundaries between regions
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables

Let fc be a set of auxiliary function values. We call them
control variables
The control variables provide a low dimensional representation
of f (analogously to the inducing/active variables in sparse GP
models)
Using fc , we can write the posterior

p(f|y) =              p(f|fc , y)p(fc |y)dfc
fc

When fc is highly informative about f, ie. p(f|fc , y)                  p(f|fc ),
we can approximately sample from p(f|y):
Sample the control variables from p(fc |y)
Generate f from the conditional prior p(f|fc )
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables

Idea: Sample the control variables from p(fc |y) and generate
f from the conditional prior p(f|fc )
Make this a MH algorithm: We only need to specify the
(t+1) (t)
proposal q(fc    |fc ), that will mimic sampling from p(fc |y)
The whole proposal is

(t+1)             (t)                 (t+1)         (t+1)    (t)
Q(f (t+1) , fc           |f (t) , fc ) = p(f (t+1) |fc         )q(fc         |fc )
(t+1)
Each (f (t+1) , fc             ) is accepted using the MH step
(t+1)        (t)   (t+1)
p(y|f (t+1) )p(fc           ) q(fc |fc         )
A=                        (t)           (t+1) (t)
p(y|f (t) )p(fc )        q(fc    |fc )
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Speciﬁcation of
(t+1) (t)
q(fc    |fc )
(t+1)      (t)
q(fc   |fc ) must mimic sampling from p(fc |y)
The control points are meant to be almost independent, thus
Gibbs can be eﬃcient
Sample each fci from the conditional posterior p(fci |fc−i , y)
Unfortunately computing p(fci |fc−i , y) is intractable
But we can use the Gibbs-like algorithm: Iterate between
diﬀerent control variables i:
(t+1)                  (t+1)   (t)
Sample fci         from p(fci     |fc−i ) and f (t+1) from
(t+1) (t)
p(f (t+1) |fci    , fc−i ). Accept with the MH step
The proposal for f is the leave-one-out conditional prior

(t)
p(f t+1 |fc−i ) =                                     (t)              (t)
p(f t+1 |fc(t+1) , fc−i )p(fc(t+1) |fc−i )dfc(t+1)
i                 i               i
(t+1)
fci
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
(t)
Data, current f (t) (red line) and current control variables fc              (red
circles)
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
(t+1)    (t)
First control variable: The proposal p(fc1                       |fc−1 ) (green bar)

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6        0.8       1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
(t+1)
First control variable: The proposed fc1                       (diamond in magenta)

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6       0.8     1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
First control variable: The proposed function f (t+1) (blue line)

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
First control variable: Shaded area is the overall eﬀective proposal
(t)
p(f (t+1) |fc−1 )
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Demonstration
Iteration between control variables: Allows f to be drawn with
considerable variance everywhere in the input space.
2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5
0            0.2           0.4    0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: Input control locations

To apply the algorithm, we need to select the number M of
control variables and their input locations Xc
Choose Xc using a PCA-like approach
Knowledge of fc must determine f with small error
−1
Given fc the prediction of f is Kf ,c Kc,c fc
−1
Mininimize the averaged error ||f − Kf ,c Kc,c fc ||2

−1
G (Xc )      =              ||f − Kf ,c Kc,c fc ||2 p(f|fc )p(fc )dfdfc
f,fc
−1
=     Tr(Kf ,f − Kf ,c Kc,c KfT )
,c

Minimize G (Xc ) w.r.t. Xc using gradient-based optimization
Note: G (Xc ) is the total variance of the conditional prior p(f|fc )
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control points: Choice of M

To ﬁnd the number M of control variables
Minimize G (Xc ) by incrementally adding control variables
until G (Xc ) becomes smaller than a certain percentage of the
total variance of p(f) (5% used in all our experiments)
Start the simulation and observe the acceptance rate of the
chain
Keep adding control variables until the acceptance rate
becomes larger than 25% (following standard heuristics
Gelman, Carlin, Stern and Rubin (2004))
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Sampling using control variables: G (Xc ) function

The minimization of G places the control inputs close to the
clusters of the input data in such a way that the kernel function is
taken into account
2                                             2.5

1.5                                              2

1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5                                            −1.5
−2                                             −2

−2.5                                            −2.5
0      0.2     0.4     0.6     0.8      1       0   0.2   0.4   0.6   0.8   1
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Applications: Demonstration on regression
Regression: Compare Gibbs, local region sampling and control
variables in regression (randomly chosen GP functions of
varied input-dimensions: d = 1, . . . , 10, with ﬁxed N = 200
training points)
60                                                                                                            0.25
Gibbs                                        70

number of control variables
region
50                           control                                      60                                   0.2
KL(real||empirical)

40                                                                        50
0.15
30
40
30                                   0.1
20
20
0.05
10                                                                                                 corrCoef
10
control
0                                                                         0                                     0
2   4          6    8             10                                      2   4         6     8      10
dimension                                                                 dimension

Note: The number of control variables increases as the
function values become more independent... this is very
intuitive
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Applications: Classiﬁcation
Classiﬁcation: Wisconsin Breast Cancer (WBC) and the Pima
Indians Diabetes. Hyperparameters ﬁxed to those obtained by
Expectation-Propagation

−254                                                          −30                                        0.7

0.6
−256
−35
0.5
Log likelihood

Log likelihood

−258
0.4
−40
−260                                                                                                     0.3

−45                                        0.2
−262
0.1
−264                                                          −50
200   400       600     800   1000                          200   400       600     800   1000    0
MCMC iterations                                             MCMC iterations                      gibbs contr ep   gibbs contr ep

Figure: Log-likelihood for Gibbs (left) and control (middle) in WBC
dataset. (right) shows the test errors (grey bars) and the average
negative log likelihoods (black bars) on the WBC (left) and PID (right)
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Applications: Transcriptional regulation
Data: Gene expression levels y = (yjt ) of N genes at T times
Goal: We suspect/know that a certain protein regulates ( i.e.
is a transcription factor (TF) ) these genes and we wish to
model this relationship
Model: Use a diﬀerential equation (Barenco et al. [2006];
Rogers et al. [2007]; Lawerence et al. [2007])

dyj (t)
= Bj + Sj g (f (t)) − Dj yj (t)
dt
where
t - time
yj (t) - expression of the jth gene
f (t) - concentration of the transcription factor protein
Dj - decay rate
Bj - basal rate
Sj - Sensitivity
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Transcriptional regulation using Gaussian processes
Solve the equation
t
Bj
yj (t) =        +Aj exp(−Dj t)+Sj exp(−Dj t)                g (f (u)) exp(Dj u)du
Dj                                     0

Apply numerical integration using a very dense grid (ui )P
i=1
and f = (fi (ui ))P
i=1
Pt
Bj
yj (t)          +Aj exp(−Dj t)+Sj exp(−Dj t)                wp g (fp ) exp(Dj up )
Dj
p=1

Assuming Gaussian noise for the observed gene expressions
{yjt }, the ODE deﬁnes the likelihood p(y|f)
Bayesian inference: Assume a GP prior for the transcription
factor f and apply MCMC to infer (f, {Aj , Bj , Dj , Sj }N )
j=1
f is inferred in a continuous manner (P        T)
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Rogers, Khanin and Girolami
(2007)

One transcription factor (lexA) that acts as a repressor. We
consider the Michaelis-Menten kinetic equation

dyj (t)                  1
= Bj + Sj                 − Dj yj (t)
dt              exp(f (t)) + γj

We have 14 genes (5 kinetic parameters each)
Gene expressions are available for T = 6 time slots
TF (f) is discretized using 121 points
MCMC details:
6 control points are used
Running time was 5 hours for 5 × 105 iterations plus burn in
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Predicted gene expressions

dinF Gene                                      dinI Gene                                   lexA Gene
10                                         7.5                                             6

9.5                                         7                                             5.5

9                                         6.5                                             5

8.5                                         6                                             4.5

8                                         5.5                                             4

7.5                                         5                                             3.5

7                                         4.5                                             3

6.5                                         4                                             2.5

6                                         3.5                                             2
0   10   20      30       40   50   60      0      10   20      30       40   50   60     0    10   20      30       40   50   60

recA Gene                                      recN Gene                                   ruvA Gene
4                                          3.5                                           5.5

3.5                                              3                                         5

3                                          2.5                                           4.5

2.5                                              2                                         4

2                                          1.5                                           3.5

1.5                                              1                                         3

1                                          0.5                                           2.5

0.5                                              0                                         2

0                                         −0.5                                           1.5
0   10   20      30       40   50   60       0     10   20       30      40   50   60      0   10   20      30       40   50   60
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Predicted gene expressions

ruvB Gene                                   sbmC Gene                                  sulA Gene
6.5                                            4.5                                         4.5

6                                              4                                           4

3.5
5.5                                            3.5
3
5                                              3
2.5
4.5                                            2.5
2
4                                              2
1.5
3.5                                            1.5
1
3                                              1                                          0.5

2.5                                            0.5                                          0
0      10   20      30       40   50   60      0   10   20      30       40   50   60     0   10   20      30       40   50   60

umuC Gene                                   umuD Gene                                  uvrB Gene
3.5                                            4                                           5

3                                        3.5                                         4.5

2.5                                            3                                           4

2                                        2.5                                         3.5

1.5                                            2                                           3

1                                        1.5                                         2.5

0.5                                            1                                           2

0                                        0.5                                         1.5

−0.5                                            0                                           1
0     10   20       30      40   50   60     0    10   20      30       40   50   60     0   10   20      30       40   50   60
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Predicted gene expressions

yebG Gene                                  yjiW Gene
5.5                                          7

5                                          6.5

4.5                                          6

4                                          5.5

3.5                                          5

3                                          4.5

2.5                                          4

2                                          3.5

1.5                                          3
0   10   20      30       40   50   60     0   10   20      30       40   50   60
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Protein concentration

Inferred protein
7

6

5

4

3

2

1

0
0             10            20               30   40   50   60
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Kinetic parameters

Basal rates                                                                                         Decay rates
7                                                                                                    1.4

6                                                                                                    1.2

5                                                                                                     1

4                                                                                                    0.8

3                                                                                                    0.6

2                                                                                                    0.4

1                                                                                                    0.2

0                                                                                                     0
dinF    dinI    lexA    recA    recN    ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW             dinF    dinI    lexA   recA   recN   ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW

Sensitivities                                                                               Gamma parameters
0.7                                                                                                  0.25

0.6

0.2

0.5

0.15
0.4

0.3
0.1

0.2

0.05

0.1

0                                                                                                      0
dinF    dinI    lexA    recA    recN   ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW              dinF    dinI   lexA   recA   recN   ruvA   ruvB sbmC sulA umuC umuD uvrB yebG     yjiW
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Results in E.coli data: Conﬁdence intervals for the
kinetic parameters
Basal rates                                                                                           Decay rates
20                                                                                               4

18
3.5

16
3
14

2.5
12

10                                                                                               2

8
1.5

6
1
4

0.5
2

0                                                                                               0
dinF   dinI   lexA   recA   recN   ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW            dinF    dinI    lexA    recA    recN   ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW

Sensitivities                                                                               Gamma parameters
10                                                                                           9

9                                                                                           8

8
7

7
6

6
5
5
4
4

3
3

2
2

1                                                                                           1

0                                                                                           0
dinF   dinI   lexA   recA   recN   ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW           dinF    dinI    lexA    recA    recN    ruvA   ruvB sbmC sulA umuC umuD uvrB   yebG   yjiW
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Data used by Barenco et al. [2006]

One transcription factor (p53) that acts as an activator. We
consider the Michaelis-Menten kinetic equation

dyj (t)             exp(f (t))
= Bj + Sj                 − Dj yj (t)
dt              exp(f (t)) + γj

We have 5 genes
Gene expressions are available for T = 7 times and there are 3
replicas of the time series data
TF (f) is discretized using 121 points
MCMC details:
7 control points are used
Running time 4 hours for 5 × 105 iterations plus burn in
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Data used by Barenco et al. [2006]: Predicted gene
expressions for the 1st replica

DDB2 Gene − first Replica                            BIK Gene − first Replica                         TNFRSF10b Gene − first Replica
3.5                                                  5                                                    3.5

3                                                                                                         3
4
2.5
2.5
3
2
2
2                                                    1.5
1.5
1
1
1
0.5

0.5                                                  0
0

0                                                  −1                                                   −0.5
0   2        4        6       8         10   12     0   2        4        6       8          10   12       0   2     4       6       8      10   12

CIp1/p21 Gene − first Replica                       p26 sesn1 Gene − first Replica
5                                                  3.5

4                                                   3

2.5
3
2
2
1.5
1
1

0                                                  0.5

−1                                                   0
0   2        4       6        8         10   12     0   2         4       6       8          10   12
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Data used by Barenco et al. [2006]: Protein
concentrations

Inferred p53 protein                                    Inferred p53 protein                                    Inferred p53 protein
2                                                    2                                                       3

2.5
1.5
1.5
2
1                                                  1.5
1
0.5                                                          1

0.5
0.5
0
0

0                                              −0.5                                                    −0.5
0   2   4         6        8       10   12        0         2   4          6       8       10   12        0         2   4          6       8       10   12

Linear model (Barenco et al. predictions are shown as crosses)

Inferred protein                                        Inferred protein                                        Inferred protein
0.7                                              0.7                                                      3

0.6                                              0.6                                                     2.5

0.5                                              0.5
2
0.4                                              0.4
1.5
0.3                                              0.3
1
0.2                                              0.2

0.1                                              0.1                                                     0.5

0                                                0                                                       0
0    2   4          6           8   10    12     0       2       4          6           8   10    12     0       2       4          6           8   10    12
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Data used by Barenco et al. [2006]: Kinetic parameters

Basal rates                                               Decay rates
1.6                                                       10

9
1.4

8
1.2
7

1
6

0.8                                                        5

4
0.6

3
0.4
2

0.2
1

0                                                         0
DDB2   p26 sesn1      TNFRSF10b    CIp1/p21   BIK         DDB2   p26 sesn1     TNFRSF10b   CIp1/p21   BIK

Sensitivities                                     Gamma parameters
30                                                        0.9

0.8
25
0.7

20                                                        0.6

0.5
15
0.4

10                                                        0.3

0.2
5
0.1

0                                                         0
DDB2   p26 sesn1      TNFRSF10b    CIp1/p21   BIK         DDB2   p26 sesn1     TNFRSF10b   CIp1/p21   BIK

Our results (grey) compared with Barenco et al. [2006] (black).
Note that Barenco et al. use a linear model
Markov Chain Monte Carlo Algorithms for Gaussian Processes

Summary/Future work

Summary:
A new MCMC algorithm for Gaussian processes using control
variables
It can be generally applicable
Future work:
Deal with large systems of ODEs for the transcriptional
regulation application
Consider applications in geostatistics
Use the G (Xc ) function to learn sparse GP models in an
unsupervised fashion without the outputs y being involved

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 14 posted: 1/30/2011 language: English pages: 48