Markov Chain Monte Carlo Algorithms for Gaussian Processes

Document Sample
Markov Chain Monte Carlo Algorithms for Gaussian Processes Powered By Docstoc
					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