Advanced Statistical Modelling

Transcription

1 Markov chain Monte Carlo (MCMC) Methods and Their Applications in Bayesian Statistics School of Technology and Business Studies/Statistics Dalarna University Borlänge, Sweden. Feb. 05, 2014.

2 Outlines 1 Necessity of MCMC in Bayesian data analysis. 2 Algorithms for MCMC sampling. 3 Example 1: Metropolis-Hastings algorithm for posterior simulation of a Poisson model. 4 Example 2: Gibbs sampler for posterior simulation with a Gaussian model. 5 Software for Bayesian statistics with MCMC sampling. 6 A short introduction to WinBugs 7 What might go wrong while using MCMC sample.

3 Necessity of MCMC in Bayesian Statistics Let, Y = {y 1, y 2,..., y n } be observed data, L (θ Y ) = f (Y θ) be the likelihood function and π (θ) be the prior distribution. Any Bayesian inference is based on posterior distribution p (θ Y ) = θ L (θ Y ) π (θ) L (θ Y ) π (θ) dθ Often, the computation of the normalizing constant (integral term) is too difficult but still we may want to study the properties of the posterior distribution, e.g. it s mean, quantiles, mode, etc. A solution to this problem is to draw random sample by using un-normalized posterior density and study the properties of p with a randomly drawn sample. Markov chain Monte-Carlo offers us with this opportunity.

4 Markov chain Monte Carlo sampling MCMC sampling is about running a Markov chain and accept its state values along the path as a sample from the target distribution. In doing this, we have to make sure that the probability for the chain to stay at a certain state converges to the probability of observing the state value in the target distribution. This can be done in many different ways but the most popular techniques are: Metropolis-Hastings algorithm and the Gibbs sampler.

5 Metropolis-Hastings algorithm 1 Assume we want to draw a sample of size n from a target density f (x) = Cϕ (x) where f can be a multivariate density. 2 We select a proposal density g (y 1 y 0 ) (also called candidate, jumping and instrumental distribution) such that all the states in f are accessible through g. In application we often use g (y 1 y 0 ) = g (y 1 ). 3 Start with an initial value y 0 (within the sample space of x) such that ϕ (y 0 ) > 0. 4 Generate a candidate value y t, t = 1, 2,,..., from g and calculate the following quantity ( ) ϕ (yt ) g (y t 1 y t ) α = min ϕ (y t 1 ) g (y t y t 1 ), 1 5 Accept y t as the new state with probability α, i.e. draw U t Uniform (0, 1) and accept y t if U t α otherwise set y t = y t 1. 6 Set t = t + 1 and go to step 4 until t n.

6 Example : Metropolis-Hastings algorithm Suppose have a random sample of size n from Poisson (θ) and we want to draw Bayesian inference on the model parameter θ. With a flat prior, the posterior becomes: p (θ x) exp [ nθ] θ n i=1 x i. Suppose an observed sample of size n = 12 gives n i=1 x i = 6 then the analytical solution of the posterior mean is E (θ x) = = However, we will use MCMC sample via Metropolis-Hastings algorithm to draw inference. Let us use log-normal proposal density, with mean=0 and variance=1 in the log-scale.

7 Comparison of the theoretical and MCMC sample density f(θ) Theoretical From MCMC θ

8 Plot of first 100 MCMC sample points Plot of the first 100 MCMC sample θ(t) t

9 Gibbs Sampler In many situations, e.g. multivariate distribution, it is not easy to find a good multivariate proposal distribution. An alternative to Metropolis-Hastings is the Gibbs sampler which works with conditional distribution. The Gibbs sampler technique can be described as follows 1 Assume we want to draw sample from f (x) = f (x 1, x 2 ). 2 Compute f (x 1 x 2 ) and f (x 2 x 1 ). 3 With any initial value x (0) (t) ( ) 2, draw x 1, t = 1, 2, )..., n, from ( f x 1 x 2 = x (0) 2 and x (t) 2 from f x 2 x 1 = x (t) 1. Now, the first sample point. 4 Set x (0) 2 = x (t), t = t + 1 and go to 3 until t n. 2 ( x (t) 1, x (t) 2 ) gives

10 Example: Gibbs Sampler for posterior inference with a Gaussian model Suppose we have n independent observations y 1, y 2,..., y n from N ( µ, σ 2) distribution. Then the likelihood function is given by L ( µ, σ 2 y ) = ( [ 2πσ 2) n/2 exp 1 n ( ) ] yi µ 2 2 σ 2 With Jeffreys s prior p ( µ, σ 2) 1 we have the following posterior σ 2 distribution p ( µ, σ 2 y ) ( [ 2πσ 2) n/2 1 exp 1 n ( ) ] yi µ 2 2 σ 2 Though the marginal posteriors are analytically tractable, in this case, we will make our inference based on MCMC sample from the posterior. i=1 i=1

11 Gaussian Example: Cont. The posterior p ( µ, σ 2 y ) simplifies as p ( µ, σ 2 y ) ( σ 2) n/2 1 exp [ It can be verified that ( n )] 1 2σ 2 (y i ȳ) 2 + n (µ ȳ) 2 i=1 p ( µ σ 2, y ) N ( ȳ, σ 2 /n ) Also that p ( σ 2 µ, y ) ( Inverse Gamma n/2, 1 2 ) n (y i µ) 2 i=1 So, we have got everything ready for running a Gibbs sampler.

12 Gaussian Example (cont.) Suppose we have 100 random sample from N (2, 4). This can be drawn in R by using the command y<-rnorm(100,mean=2,sd=2). Suppose we have an initial value of σ = 0.5 then we draw our first sample on mu by using the R command mu<-rnorm(1,mean=mean(y),sd=sigma/sqrt(100)) Again, we draw our first sample on σ 2 as s2<-rigamma(1,alpha=100/2,beta=b.n) where b.n= 100 (y i mu) 2 i=1 So, mu,s2 constitutes the first pair of observations on the parameters. Now, set σ = s2 and continue drawing sample until the desired number of sample observations are drawn.

13 Gaussian Example (cont.) A comparison of theoretical vs. sample (based on Gibbs sampler, n=10000) densities f(θ) Theoretical From MCMC θ

14 Software for MCMC sampling in R There are several R packages that implements MCMC for certain types of statistical models e.g. MCMCGlmm package is useful for fitting GLMM, truncated response, MCMCpack fits GLMMs and latent index models, glmmbugs package implements GLMM through WinBUGS and OpenBUGS. R2WinBUGS and BRugs packages facilitate fitting a Bayesian model using BUGS running from R. Only MCMCpack has a general purpose function MCMCmetrop1R that takes any (log-)un-normalized density as input and returns MCMC Metropolis-Hastings sample from it.

15 Getting software for running Bugs from R 1 You have to install WinBugs which is freely downloadable from http: // Follow the instructions on the above web page regarding installation. 2 You may fit your models directly in WinBugs, but we will run it from R. 3 To run WinBugs from R, you have to install the R package, R2WinBUGS in your currently running R. 4 WinBugs runs only under Windows. Make sure that you have administrative access to the folder where you installed the WinBugs. 5 Once you have all the necessary software installed, you are ready to fit almost any Bayesian model.

16 What else do we need to run MCMC in R/WinBugs? You need the following documents/data files to run a model in R2WinBUGS via R 1 You need a model file (*.bug) where you programme your model. 2 You need a data file where the data are stored in WinBugs preferred format. Alternatively, you can send data from R as a list object. 3 You need a few lines of R programme to coordinate between R and WinBugs. 4 Further R codes for manipulating the MCMC sample. 5 You may wish to go though WinBugs manual ( before starting working with WinBUGS. 6 You may also visit for instructions on running R2WinBUGS from R.

17 Some basic features of WinBugs A WinBugs programme must contain a model. WinBugs models are written with the same syntax structure as the S language. However, there are some differences. A model must be defined for every data point. An object cannot be redefined or reassigned e.g x<-log(y); x dnorm(0,tau) does not work. Missing data are not allowed. Every object must have a value (from the data; if known) or a distribution (if random). Improper prior is not allowed. Any flat prior should be replaced with a weakly informative prior e.g. flat prior on (, ) may be approximated with N (0, 1e8), flat prior on (0, ) might be approximated with Uniform (0, k) and so on. Only the distributions available in WinBugs can be used. However, it is possible to create a new distribution by using already existing distribution.

18 WinBugs Example: A simple Gaussian model Bugs Codes model{ for(i in 1:n){ y[i] dnorm(mu,tau) } Mu dnorm(0,0.0001) Tau< 1/s2 s2 dunif(.01,100) } R Codes library(r2winbugs) y< rnorm(100,2,2) parms< c( Mu, s2 ) MC1< bugs(data=list(y=y,n=100),inits=null,parameters.to.save=parms, model.file= Gaussian.bug, n.chains=3,n.iter=10000,n.thin=3, bugs.directory= C:/Users/maa/WinBUGS14/, working.directory= C:/ local/bayes,debug=t,codapkg=f)

19 What might go wrong with MCMC sampling MCMC sample values might be very highly correlated, needing too huge size of sample for a reliable computation. Suggestion: Accept every k th sample value. Check autocorrelation by plotting the lag-values of the chain. Bad starting value combined with high correlation might produce useless (un-converged) sample. Suggestion: Discard a certain number of initial sample values, thereafter accept every k th sample value. Use multiple chains to check convergence. Too high rejection rate makes MH sample values highly correlated. Suggestion: Try different proposal distribution. Plot the chain for checking acceptance rate. You may check convergence of chain(s) by using formal diagnostic tools (see e.g. coda library in R).