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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

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Markov chain, Markov chain Monte Carlo, monte carlo, stationary distribution, Gibbs sampler, MCMC Algorithms, transition probabilities, Metropolis-Hastings algorithm, probability distribution, state space

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posted: | 1/30/2011 |

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