INTRO TO THE METROPOLIS ALGORITHM

Size: px

Start display at page:

Download "INTRO TO THE METROPOLIS ALGORITHM"

Kelly Gaines
5 years ago
Views:

1 INTRO TO THE METROPOLIS ALGORITHM A famous reliability experiment was performed where n = 23 ball bearings were tested and the number of revolutions were recorded. The observations in ballbearing2.dat are the number of revolutions (recorded in 0 6 units). A commonly used distribution for failure times is the Weibull distribution which has density given by f Y (y) = ( y { ( y ) ) where > 0 is called a shape parameter and > 0 is called a scale parameter. The R function dweibull(y,shape=kappa,scale=lambda) will evaluate () for you with the option log=true which will evaluate () on the log scale. Experts know that ball bearings should average about (0 6 revolutions) and that a revolution less than 0 is nearly impossible. For this assignment, we wish to fit the above Weibull distribution to the ball bearing data using Bayesian techniques. To do this we will follow the steps of a Bayesian analysis outlined in previous research assignments. Specifically, we will () choose appropriate prior distributions, (2) relate our data to the parameters and (3) combine the information in our prior with the information in our data to get a posterior distribution. Step : Choose prior distributions. The two parameters of the Weibull distribution are and which we know are both positive numbers. So, for this analysis lets use Gamma prior distributions of G(shape = a, rate = b ) and G(shape = a, rate = b ) where a, b, a and b are prior numbers that we get to choose according to our understanding of the problem. When initially thinking about and, we don t really have any idea what they should be so choosing a, b, a and b is rather difficult. But, we do know something about ball bearings so we ll try and use that to our advantage. Namely, the problem states that revolutions should be about 60 and a revolution less than 0 is impossible. Setting the expected value of the Weibull distribution equal to 60 we know Γ( + k ) 60 60/Γ( + k ) where Γ( ) is the mathematical gamma function defined by Γ(z) = 0 x z e x dx. To help us choose a, b, a and b, Figure shows 5 different Weibull density plots for 5 different combinations of and (these 5 combinations were chosen using trial and error). Both the green and blue curves seem to give Weibull densities that describe the information given in the problem so my a priori estimate of (, ) will be (3.25, 66.94) (the green curve). I am quite uncertain about this choice though so I ll pick a prior standard deviation of 3 for and a prior standard deviation of 0 for. I can now solve for a and b by solving the system of equations: E() = a /b = 3.25 Var() = a /b 2 = 3 2 a =.7 b = Similar math yields a = and b = Step 2: Link data to our parameters. Luckily, this was already done for us in the problem. Namely we will assume that Y i is the number of revolutions of the i th ball bearing and Y i iid W(, ) where W( ) is ()

2 0.06 wdens k.l k= l= 60 k= 0 l= k= 3.25 l= k= 5.5 l= k= 7.75 l= x Figure : Choices of (, ) for the ball bearing prior. the Weibull density given in Equation () above. Under this assumption, Pr(Data, ) = N ) = prod(dweibull(y,shape=kappa,scale=lambda)) (3) where the second line gives the R code to calculate the probability of our data (also known as the likelihood). Step 3: Update our priors to the posterior. Because we have two parameters ( and ) we will want to use a Gibbs sampler to sample values from the posterior distribution (because calculating it by hand would be impossible). As we learned earlier, the general Gibbs algorithm to do this is for(it in :n.it){ ## Step. Sample kappa from its posterior assuming we know lambda ## Step 2. Sample lambda from its posterior assuming we know kappa ## Step 3. Save (kappa,lambda) as a draw Let s start from Step in the Gibbs algorithm. From Bayes theorem, we know that post( Data, ) = Pr(Data, )Prior() [ N = ) ] Γ(a )b a a exp { b = prod(dweibull(y,shape=kappa,scale=lambda)) dgamma(kappa,shape=a.kappa,rate=b.kappa) (2) 2

3 but looking at this equation we immediately run into a problem - the posterior distribution is nonconjugate meaning that the form of the posterior for is something we don t recognize (and we don t have a canned R function to sample it for us). The main issue with the above Gibbs sampler algorithm is that any prior we choose for will not have a complete conditional distribution that we recognize (note: if we choose a gamma prior for we would get a gamma complete conditional). So, we need a technique to still sample from its complete conditional distribution without relying on a function in R that will draw it for us. The tool we will be using is called the Metropolis algorithm. So, how do we draw? The answer is the Metropolis algorithm. The Metropolis algorithm is a method (technically it is a Markov chain if you have heard of those) that allows you to sample from any distribution (even ones we don t recognize as is the case here). The Metropolis algorithm uses three steps to sample from any distribution: () propose a value that could be from the distribution, (2) check to see how well the proposed value matches the distribution and then (3) decide whether to keep the value as a draw from the distribution or throw it out because it doesn t match the distribution we are trying to sample from. More technically, to use the Metropolis algorithm, to sample we would do the following: 0. Choose a starting value for. Let s just use = 3.25 from our prior distribution to start.. Propose a new value for from N (, s 2 ) where is the current value of the in the algorithm and s is a standard deviation that you get to pick. The value of s is called the proposal standard deviation and is actually very important but I am going to let you mess around with using different values of s to see what affect it has on the result. In, R code, this would be kappa.star <- rnorm(,kappa,sd=s). 2. If > 0, check to see how well the proposed matches the distribution by calculating: ( Pr(Data, ) Prior( ) ) r = log Pr(Data, ) Prior() = log(pr(data, )) + log(prior( )) log(pr(data, )) log(prior()) = sum(dweibull(y,shape=kappa.star,scale=lambda,log=true)) + dgamma(kappa.star,shape=a.kappa,rate=b.kappa,log=true) sum(dweibull(y,shape=kappa,scale=lambda,log=true)) dgamma(kappa,shape=a.kappa,rate=b.kappa,log=true) were the log scale makes the prod become a sum. Note a few things in this step. First, if < 0 we do not need to do this step because we already know that has to be bigger than zero. So, if we propose < 0 we can automatically throw it out. Second, the numerator in r is just evaluating the posterior for at the value of to see how well matches. The ratio r is essentially comparing the posterior under and to see which is better (we ll keep the better value of ). And, third, this ratio is evaluated on the log-scale for computational reasons. If we try to evaluate it on the original scale we will frequently run into errors because the computer will round r either to 0 or. 3. With probability min(, exp{r), keep as a draw of otherwise just keep the original. Using these steps we would get a draw of that we use as Step in our Gibbs algorithm from above. Now that we have (finally) figured out how to do step in our Gibbs algorithm we can continue on to 3

4 Step 2. From Bayes theorem, we know that, post( Data, ) = Pr(Data, )Prior() [ N = ) ] Γ(a )b a a exp { b = prod(dweibull(y,shape=kappa,scale=lambda)) dgamma(lambda,shape=a.lambda,rate=b.lambda) but immediately run into the same problem as we had with. Namely, the posterior for is non-conjugate meaning that the form of the posterior for is something we don t recognize (and we don t have a canned R function to sample it for us). Luckily, we have already solved this problem and can rely on the Metropolis algorithm to do this. At this point we now have all the tools to code an algorithm that will sample (, ) from their posterior distribution. For this assignment, use the following outline to code up your own Metropolis within Gibbs algorithm to obtain 5000 values of (, ) from their posterior distribution. ## Prior Distribution a.kappa <-.7 b.kappa < a.lambda < b.lambda < ## Starting Values for kappa and lambda kappa < #from prior lambda < #from prior ## Proposal standard deviations s.kappa <- s.lambda <- ## Number of Iterations n.it < ## Place holders to keep values as algorithm progresses post.draws <- data.frame(kappa=rep(na,n.it),lambda=rep(na,n.it)) ## Begin the algorithm for(it in :n.it){ ## Draw kappa using the 3 steps of the Metropolis algorithm kappa.star <- rnorm(,kappa,sd=s.kappa) if(kappa.star>0){ r <- Metropolis.prob <- min(,exp(r)) if(rbinom(,,metropolis.prob)==){ kappa <- kappa.star 4

5 ## Draw lambda using the 3 steps of the Metropolis algorithm ## Keep (kappa,lambda) as a draw post.draws$kappa[it] <- kappa post.draws$lambda[it] <- lambda 5

Markov chain Monte Carlo methods

Markov chain Monte Carlo methods (supplementary material) see also the applet http://www.lbreyer.com/classic.html February 9 6 Independent Hastings Metropolis Sampler Outline Independent Hastings Metropolis