Markov Chain Monte Carlo Simulation Made Simple


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1 Markov Chain Monte Carlo Simulation Made Simple Alastair Smith Department of Politics New York University April2,2003 1
2 Markov Chain Monte Carlo (MCMC) simualtion is a powerful technique to perform numerical integration. It can be used to numerically estimate complex economometric models. In this paper I describe the intuition behind the process, show its flexiblity and applicability. I conclude by demonstrating that these methods are often simpler to implement than many common techniques such as MLE. This paper serves as a brief introduction. I do not intend to derive any results or prove any theorems. I beleive that MCMC offers a powerful estimation tool. This paper is designed to remove the mystery surround the process. Not only it extremely powerful and flexible but it is easy to implement. Given the recent growth in the power of computers I beleive that numerical procedures will be the estimation tools of the future. I outline the underlying logic, show why these techniques work. MCMC techniques are most often used in the Bayesian context. I start by outlining the simple linear model in the Bayesian framework. Although, analytical techniques exist for this model they are complex. In general, more complex model are analytically intractable. Having setup the estimation technique I examine the properties of Markov chains. These properties provide the basis for the estimation procedure. 1 The Bayesian Model While I beleive that the Bayesian approach a superior, consistent approach to statistics than the standard frequentist approach, this debate is volumous and not the topic of this paper. For practical purposes if is usually possible to use diffuse priors that do not influence the posterior results. prior f(θ) likelihood L(Y θ) posterior f(θ Y ) f(θ) L(Y θ) For example in the simple linear model θ = {β,σ 2 } 2 Markov Chains A Markov chain is a stochastic process. It generates a series a observations, X. To illustrate the concept I focus on a discrete time, descrete state space model. At each time period the process generates a sample, X t,from 2
3 the state space. For a simple example, suppose that the state space is the numbers 1, 2,and 3. A Markov chain is simply a string of these numbers. The Markov property is that the probability distribution over the next observation depends only upon the current observation. Let p ij represent the probability that the next observation is j (X t+1 = j), given that the current observation is i (X t = i). A convenient way to present these transition probabilities is through a transition matrix P, P = p 11 p 12 p 13 p 21 p 22 p 23. The elements of the first row represent the probabilities of moving to the different states if the current state is 1. Therefore, p 31 p 32 p 33 p 11 represents the probability that state X t+1 =1ifthecurrentstateisalso 1; p 12 represents the probability that state X t+1 =2if the current state is 1, etc... Suppose that our initial observation is indeed 1 (X 0 =1). The probability distribution for state is given by the first row of P. The next question is what is the probability distribution over the following observation. To illustrate, I consider the more specific question, with what probability does X 2 =3? There are three possible paths by which the second observation could equal 3; they are illustrated in the table below. Thus the probability that X 2 =3 is p 11 p 13 + p 12 p 23 + p 13 p 33. Pathway X 0 X 1 X 2 Probability # p 11 p 13 # p 12 p 23 # p 13 p 33 Thus, for any initial state, we can calculate the probability density over the states for a given number of moves. Obviously, as the number of moves increases these become increasingly difficult to calculate. Yet, matrix notation simplifies the calculation. Suppose, rather than start with a specific state we consider a probability distribution over these states, ν 0 = 0. v1 0 v2 0 v3 0 If we randomly select the initial state from this distribution, then what is the probability distribution of the next state in the chain is given by v (1) 1 v (1) = v (1) 2 = Pv (0) = p 11 p 12 p 13 p 21 p 22 p 23 v (1) p 31 p 32 p 33 3 v0 1 v 0 2 v 0 3 = p 11v p 12 v p 13 v 0 3 p 21 v p 22 v p 23 v 0 3 p 31 v p 32 v p 33 v 0 3 3
4 This idea can be extended, the probability distribution over the states after the second move is simply v 2 = Pv 1 = P 2 v 0. This idea can be generalised; specifically, v (t) = P t v (0). Of particular interest, is the distribution as the chain becomes long. As the chain s length increases then the distribution over the states becomes less and less determined by the starting distribution and more and more determined by the transition probabilities. Indeed, providing the chain satisfies certain regularity conditions, i.e. it does not get stuck in one state, there exists a unique invariant distribution associated with every transition matrix. Let π represent this invariant distribution. So for any starting distribution, π (0), as the chain becomes long then the π (t) tends to π ( lim π (t) = π). t There are two ways to calculate this invariant distribution. The first is analytical. This method exploits the fact that π = Pπ, and solves this system of equations. The second, and of more relevance for this paper, is to similate π by actually running the Markov chain. This involves choosing a starting value and simply running the Markov chain. The initial values in the chain depend strongly upon the starting values. However, as the chain becomes longer then the elements of the chain represent random draws from the probability distribution π Suppose, for example, that the transition matrix is P = We could start by setting X 0 =1and then running the Markov chain. We could estimate the density of each state by examining the frequency of each state. Figure 1 demonstrates that, as the number of iterations becomes large, that the relative frequency of each state converges to its invariant density. We can arbitarily increase the accuracy of these estimates simply by taking more iteration. In this example, I use a discrete state space model; however, these ideas are readily extendable to continuous state space models, where the transition matrix is replaced by a transition kernel (a probability density over the next state that depends only upon the current state). 2.1 Exploiting Markov chains for estimation Most of Markov theory revolves around finding the invariant distribution of Markov chains. MCMC turns the problem arround. Rather than finding 4
5 x1 x2 x N Figure 1: 5
6 the invariant distribution of a specific Markov chain, it starts with a specific invariant distributions and says, can I find a Markov chain that has this invariant distribution. 1 Typically, we already know the distribution of interest: the posterior distribution of the parameters. The key is to find a transition kernel that has this invariant distribution f(θ Y ). In Bayesian estimation we want to find the posterior distribution of the parameters, f(θ Y ). As discussed above this is often analytically intractible. However, suppose we have a Markov chain, P, whose invariant distribution is f(θ Y ). If we run thismarkovprocessthen,asthechainbecomeslong,itselementsrepresent random draws from the posterior distribution f(θ Y ). To illustrate how the process works consider the following algorthym. 1. Choose starting values, θ (0), and length of the chain, n 0 + m. 2. Given the current element in the chain, θ (t), use the Markov process P, to draw the next element θ (t+1). 3. If t>m,thenstoreθ (t+1). 4. If t<m+ n 0, then return to step 2; otherwise calculate and report the descriptive statistics for the elements stored in step 3. This algorthym generates and stores n 0 elements from the chain. These elements represent random samples from the posterior distribution of f (θ Y ). Thus the sample average represents an estimate of the expected value of θ. Other properties of f (θ Y ) can also be estimated by examining the properties of the sample. The accuracy of these estimates depends upon the number of draws, n 0. Accurracy is improved by running the chain longer. Note that the first m iterations of the chain were discarded. The initial elements in the chain are strongly influenced by the starting value (as the figure above demonstrates). If these starting values are drawn from a low density region of the posterior denisty then the chain contains too many draws from this region. 2 1 Each Markov process has a unique invariant distribution. Yet, many Markov chains could have the same invariant distribution. Thus, we are free to use any of these process to simulate the invariant distribution. 2 Another practical problem with running this algorthym is the high autocorrelation between elements in the chain. This reduces the rate a which convergence is acheived. A practical solution is to subsample from elements stored at step 3. 6
7 In summary, if we can find a Markov process with transition kernel P, such that its invariant distribution is f (θ Y ), then we can numerically estimate this posterior distribution by running the Markov chain. Obviously, there are many importance convergence consideration that I have not considered. However, the basic point is that if an appropriate transition kernel can be found, then estimations involves nothing more than running the Markov process. So far I have said nothing about how to find an appropriate transition kernel. It is to this point that I turn next. 3 Transition Kernels Table 1 compares the analytically calculated probability distribution with the numerically simulated values. The accuracy of the simulation can be increased by simply increases the number of iterations of the chain. 3 Most of Markov theory revolves around finding the invariant distribution of Markov chains. MCMC turns the problem arround. Typically, we already know the distribution of interest: the posterior distribution of the parameters. The key is to find a transition kernal that has this invariant distribution. Then to estimate this distribution we simply need to run the Markov chain for a suitably long period. 4 Joint, marginal and conditional distributions In the linear model we want to estimate f(β,σ 2 Y ). Being somewhat informal, this is the probability density of seeing a particular value of β and σ 2. Bayesian have calculated this density. It turns out that, with suitable conjugate priors 4, f(β,σ 2 Y ) is distributed inverse gamma normal. Unfortunately, this is about the most complicated model for which we can work with 3 These is a convenient time to discuss several aspects associated with implementing MCMC methods. First the starting value of the Markov chain affect the initial values of the chain. Over time their effect diminishes. However, if the starting values represent very low density portions of the state space then the choice of starting values affects the results. The usual solution is to discard the early part of the chain. This tends to disregard those draws from the chain that are highly dependent upon the starting values. Convergence criticeria??? literature????? 4 what is a conjugate prior? 7
8 the joint posterior density analytically. For more complex models the joint density is simply intractable. Yet, generally our interest is in the marginal denisty of a particular parameter. In particular case of the simple linear model we typically want to know about β and σ 2, separately. For example, this is all we report from a regression model, the distribution of β. This marginal density is simply the joint density of β and σ 2 intregrated across all possible values of σ 2. The key do using MCMC is to stop thinking in terms of calculating things analytically and imagine how you could simulate a single parameter in a model if you knew all the other parameters. Suppose for example that you knew the marginal distribution of σ 2 and wanted to calculate the marginal distribution of β. In order to estimate the marginal density of β Icould simply, integrate out σ 2 from the joint density. While simply tricky in this problem, it is impossible in more complex econometric models. However, knowing the marginal density of σ 2, I can draw a large number of random draws from this density. For each of these draws, the conditional density of β is simple to calculate (with normal priors, f(β Y,σ 2 ) is also distributed normally). To numerically estimate β I could draw a random sample from this distribution. Algorithm to calculate the marginal density of β given that the density of σ 2 is known. 1. set t=1 2. randonly draw (σ 2 ) (t) from its known posterior marginal distribution 3. calculate the posterior density of β given (σ 2 ) (t) (f(β Y,(σ 2 ) (t) )) 4. randomly draw (β) t from f(β Y,σ 2(t) )) 5. let t=t+1 and go to 2 Suppose this algorthm is repeated T times. Then the T samples of β represent random draws from its marginal density. The algorthm effectively integrates out σ 2. As an analogy, in our 101 econometrics classes we learn how to estimate the means of a variable if we know its variance. We then learn to calculate the variance if we know the mean. Being an order of magnitude harder, the calculation of the joint distribution of the mean and variance is typically 8
9 ommitted. Calculating the posterior density of the mean and the variance togther is much harder than calculating either conditional density. However, providing we can break a model down into a series of simple conditional densities we can estimate the marginal density of a parameter. The algorithm above assumed that the distribution of σ 2 was known and it produced a random sample from the posterior density of β. However,ifthe draws from the algorithm represent random draws from the marginal density of β, then we could simply reverse the logic of the argument, and draw random samples from the conditional density of σ 2 given the current value of β. Given that the β s are random draws from the marginal density for β, then random draws of σ 2 represent random draws from the marginal density of σ 2. Hence the following algorithm simulates the posterior ditributions for β and σ 2. Algorithm to calculate the marginal density of β and σ set t=1 and choose starting values, β (0) and (σ 2 ) (0). 2. calculate the posterior density of β given (σ 2 ) (t) (f(β Y,(σ 2 ) (t) )) 3. Randomly β (t+1) draw from this distribution. 4. calculate the posterior density of (σ 2 ) (t) given (β) (t+1) (f((σ 2 ) (t) Y,(β) (t+1) )) 5. randomly draw (σ 2 ) (t+1) from f((σ 2 ) Y,(β) (t+1) ) 6. let t=t+1 and go to 2 Providing the prior are appropriately choosen then the calculates of f(β Y,(σ 2 ) (t) ) and f((σ 2 ) (t) Y,(β) (t+1) ) are straightforward. The following code shows how simply this algorthym can be implied in STATA. See program OLS_MCMC.do 5 Bayesian Updates for simple models. Suppose we assume that the likelihood function is normal and so is our prior: Likelihood: p(y θ) = 1 2π exp ( 1(y 2 θ)2 ) 9
10 Normal prior: f(θ) = 1 2π exp ( 1 2 (θ µ 0) 2 ). To make life as simple as possible, suppose initially that the variance of both the likelihood and the prior density in one. By Bayes rule the posterior density is proportional to the product of the prior and the likelihood: p(θ y) p(y θ)f(θ). We can show that posterior denisty is also normal. Specifically, p(θ y) exp ( 1 2 (y θ)2 )exp ( 1 2 (θ µ 0) 2 )=exp ( 1 2 [(y θ)2 +(θ µ 0 ) 2 ]) We can expand the terms in the exponential and then collect them (completing the square). 1 2 [(y θ)2 +(θ µ 0 ) 2 ]= θ 2 +( y µ 0 ) θ y µ2 0. We only care about term is θ since everything else is in the nomalizing constant. θ 2 +( y µ 0 ) θ y µ2 0 =(θ b) 2 = θ 2 2θb+b 2 so b = 1 (y + µ 2 0) Hence p(θ y) exp ((θ 1 2 (y + µ 0)) 2 ) so p(θ y) is distributed normal with mean 1 2 (y + µ 0) and variance 1 2.We can now move to a more realistic example. The normal prior is referred to as a conjugate prior since it results in posterior density from the same class of distributions. 5.1 Simple Linear Model Consider the OLS simple linear model, y i = x i β + e i where e i N(0,σ 2 ). Using conjugate prior, β 0 N(β 0,B 0 ) and σ 2 0 GAMMA(υ 0,δ 0 ),wecan derive the posterior conditional densities. The posterior β parameter is normally distributed: β y, σ 2 N( β,b) b where β b = B(B0 1 β 0 + P N i=1 x iy i ) and B =(B0 1 + P N i=1 x0 ix i ) 1,and The posterior σ 2 is inverse gamma distributed: So σ 2 is distributed G( υ 0+N, δ 0+SSE ) 2 2 where SSE = P N i=1 (y i x i β) 2 i.e. sum of squared errors. 5.2 More Complex Models A key advantage of MCMC is that models can be built up in simple stepwise fashion. Suppose from example, that instead of a continuous dependent variable we have binary outcomes. Such data is typically analysed as a probit model. Specifically, z i = x i β + e i where e i N(0, 1), asify i =1 then z i > 0 and if y i =0then z i < 0. The variable z i is referred to a latent variable since we never actually observe it. The standard approach 10
11 to estimating such a model is to integrate out the latent variable and then apply maximium likelihood. A simple MCMC approach utilitizes a data augmentation technique (Tanner and Wong, 198?). If we knew the value of these latent data then we could simulate the β sjustaswedidintheols model above. Although we don t care directly about the latent data we can simulate these data. The probit model tells us the distribution of the latent data. Specific, if y i =1then z i has a left truncated normal distribution with mean xβ and variance 1: x TN [0,+ ] (x i β,1). Similarly, if y i =0 then we know that the corresponding latent variable lies between and 0: x TN [,0] (x i β,1). We now that the tools to implement this model. Let Z refer to the set of latent data (i.e. all the z i s.) Algorithm to calculate the marginal density of β in a probit model. 1. set t=1 and choose starting values, β (0) and (Z) (0). 2. calculate the posterior density of β given Z (t) (f(β Z t )) 3. Randomly β (t+1) draw from this distribution. 4. calculate the posterior density of Z (t) given (β) (t+1) (f(z Y,(β) (t+1) )) 5. randomly draw (Z) (t+1) from f(z Y,(β) (t+1) ) 6. let t=t+1 and go to 2 The key simplification here is that given Z, the posterior distribution of β is independent of the binary observed dependent variable. Now while in this context, the MLE approach provides highly reliable estimates in more complex models, such are multivariate, multinominal, or censored descrete choice models, MLE is less reliable. MCMC provides a powerful tool in these cases, being easy to program and less prone to the convergence failure problems of MLE. The construction of MCMC can be done peicewise. For example, the OLS code above with estimate the probit model with two additions. First, set the variance, σ 2, equal to one. Second, add the set to draw the latent data, Z. This is easily acheived using the following simulation. If z is a truncated normal variable with mean xβ, variance 1 with a range p to q, then the following algorithym readily provides a method to genrate a random sample 11
12 from the distribution of z. If x TN [p,q] (µ, σ 2 ) and u is a uniform random number then x = µ + σφ 1 (Φ((p µ)/σ)+u(φ((q µ)/σ) Φ((p µ)/σ))), represents a random draw of x. 12
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