Markov goes to Monte Carlo

Numerical and scientific fields such as statistics, computational physics, and optimization methods heavily rely upon Markov chain Monte Carlo methods. These simulation techniques use the power of Markov chains to sample general probability distributions, which in turn give Monte Carlo methods for estimating integrals and optimal solutions.

This is the third part of a multi-part series of posts. The first part covers Markov chains. The second part covers the basics of Monte Carlo methods. This post combines the ideas from the first two parts. Overall, the posts sketch the mechanics of Markov chain Monte Carlo (MCMC) methods, whose importance and applications I detailed in a previous post.

We will only examine Markov chains with countable state spaces. In this setting, we only need standard probability and matrix knowledge. But the results extend to general state spaces, such as Euclidean space.

By the way, I doubt that Andrey A. Markov ever went to Monte Carlo in the small country of Monaco.1But Herbie did.

Monte Carlo conditions

Sum

In the first post we covered Markov chains with countable state spaces, because the mathematics is more transparent. Consequently, we can use such Markov chains to estimate the sum of a function \(f\) over a countable (possibly infinite) set \(\mathbb{S}\). We can write the sum as

$$\begin{align}S(f)&=\sum_{x\in \mathbb{S}} f(x) \\&= \sum_{x\in\mathbb{S}} \frac{f(x)q(x)}{q(x)}\\& = \mathbb{E}_{q}[ \frac{f(Y)}{q(Y)}]\\& = \mathbb{E}_{q}[ g(Y)]\,,\end{align}$$

where \(g=f/q\) and \(Y\) is a suitable random variable that has a probability mass function \(q\) with support \(\mathbb{S}\).

Sampling the random variable (or point) \(Y\), gives the samples \(y_1,\dots,y_n\), which yields an unbiased Monte Carlo estimate of the sum, namely

$$S_n (f)=\frac{1}{n}\sum_{i=1}^n \frac{f(y_i)}{q(y_i)}.$$

The closer the function \(q\) is to \(|f|\), the better the estimate.

Integral

We can write an integral over some region \(\mathbb{S}\) as

$$\begin{align}\int_{\mathbb{S}} f(x) dx &= \int_{\mathbb{S}} \frac{f(x)}{p(x)}p(x)dx\\ & =\mathbb{E}_p[\frac{f(Y)}{p(Y)}]\,,\end{align}$$

where \(p\) is the probability density of a random variable (or point) \(Y\) with support \(A\). Then by sampling \(Y\), the resulting samples \(y_1, \dots, y_n\in [0,1]\) give the unbiased Monte Carlo estimate

$$I_n (f)=\frac{1}{n}\sum_{i=1}^n \frac{f(y_i)}{p(y_i)}\,.$$

Ergodic sequence

To estimate the sum or integral, we’ll need a sequence of ergodic random variables \(\{Y_i\}_{i\in\mathbb{N}}\), which we can achieve with an ergodic Markov chain with a state space \(\mathbb{S}\).

Markov chain conditions

In the context of Markov chains and, more generally, stochastic processes, we can think of ergodicity as convergence and a slightly more general version of the law of large numbers. A Markov chain with a countable state space needs some conditions to ensure ergodicity.

Regularity conditions for countable a Markov chain

  1. A stationary distribution \(\pi\)
  2. Aperiodicity
  3. Irreducibility
  4. Postive recurrence

In general, most of the needed structure for the Markov chain comes from irreducibility and a stationary distribution existing. Sometimes the above conditions are redundant. For example, an aperiodic, irreducible Markov chain with a finite state space is always positive recurrent. We look at ways on how to satisfy these conditions in the Monte Carlo setting when the underlying state space is countable.

Stationary distribution \(\pi\)

A stationary distribution \(\pi\) satisfies equation

$$ \pi=P\pi.$$

We know exactly what our stationary distribution \(\pi\) needs to be. It is the one used in the Monte Carlo method, namely the distribution of \(Y\). The challenge is constructing a Markov chain with such a stationary distribution. This will of course depend on the Markov (transition) kernel \(P\) of the Markov chain.

We’ll look at the other conditions on the Markov kernel \(P\).

Aperiodicity

We recall that the period \(d_x\) of a state \(x\in \mathbb{X}\) is the greatest common divisor of all \(n\) values such that \(P(x,x)^n>0\). We need every state of the countable Markov chain to be aperiodic, meaning \(d_x=1\) for all \(x\in\mathbb{X}\).

One sufficient way to achieve this requirement is to first have the countable Markov chain to be irreducible, which we’ll cover in the next section. Then if there exists at least one state \(x\in \mathbb{x}\) such that \(P(x,x)>0\), then the Markov chain is aperiodic.2Lemma 1.8.2 in Markov Chains by Norris or Remark 6.2.5 in Probability and Stochastic Processes by Brémaud.

Clearly the last condition is not difficult to satisfy, putting the focus now on the irreducibility requirement.

Irreducibility

The irreducibility property says that it is possible for the Marko chain to get from any point to another point in a finite number of steps. More precisely, for an irreducible Markov chain with countable state space, there must exist for all \(x,y\in\mathbb{X}\) a natural number \(s\) (possibly depending \(x\) and \(y\)) such that \(P(x,y)^s>0\).

We can achieve this requirement by introducing a stronger requirement, which is easier to verify. A countable Markov chain with a transition \(P\) will be irreducible if for all \(x,y\in\mathbb{X}\) the condition \(P(x,y)>0\) holds. We have simply required that the above number \(s=1\) for all \(x,y\in\mathbb{X}\).

Furthermore, clearly \(P(x,x)>0\) for all \(x\in\mathbb{X}\), then our irreducible countable Markov chain is also aperiodic.

Positive recurrence

For a point or state \(x\in\mathbb{X}\), we recall its first return time being defined as

$$ T_x^+=\min\{ t\geq 1: X_t=x\} \,.$$

A state \(x\) is called positive recurrent if the expected value of its first return time is finite, meaning \(\mathbb{E}_x(T_x^+)<\infty\). For a countable Markov chain, if all the states in the state space are positive recurrent, then we say the Markov chain is positive recurrent.

A countable irreducible Markov chain is positive recurrent if (and only if) it has a stationary distribution \(\pi\). Furthermore, that stationary distribution will be unique and nonzero, meaning \(\pi(x)>0\) for all \(x\in\mathbb{X}\).3Theorem 1.7.7 in Markov Chains by Norris or Theorem 6.3.14 in Probability and Stochastic Processes by Brémaud.

In other words, for positive recurrence, we need to prove that the Markov chain has a stationary distribution.

Ergodicity

A countable Markov chain that is aperiodic and irreducible is ergodic. In other words, an ergodic Markov process \(X\) with stationary distribution \(\pi\) is a random sequence \(X_0, X_1,\dots\) such that the following holds almost surely

$$ \frac{1}{t}\sum_{i=0}^{t-1} g(X_i) \rightarrow \sum_{x\in\mathbb{X}}g(x)\pi(x)$$

as \(t\rightarrow\infty\) for all bounded, nonnegative functions \(g\). Consequently, we can use such an ergodic Markov chain to make statistical estimates.

Sampling with a Markov chain

For the Monte Carlo sample \(y_1,\dots,y_n\), we simply use the Markov chain samples \(x_0,x_1,\dots,x_{n-1}\), giving the Monte Carlo estimate. In the discrete case, this is simply

$$S_n (f)=\frac{1}{n}\sum_{i=1}^n g(x_{i-1}).$$

which, due to ergodicity, converges to \(S(f)=\mathbb{E}_{Y}[g(Y)]\).

Summary

In summary, a sufficient way to have a countable ergodic Markov chain is for the transition kernel to be such that \(P(x,y)>0\) for all \(x\in\mathbb{X}\) and there exists a (stationary) distribution \(\pi=P\pi\). For the first requirement, one imagines using a non-negative function such as \(e^{-(x-y)^2}/C\), where \(C>0\) is a suitable normalization constant.

The requirements seem possible, but to bring them all together is still a great challenge for the uninitiated. Fortunately, some great minds almost seven decades ago proposed the first Markov chain that meets all the requirements for a specific stationary distribution, which was used for a Monte Carlo estimate.

The first such Markov chain Monte Carlo method appeared in a paper written by five scientists, including two husband-wife pairs. It became known as the Metropolis or Metropolis-Hastings algorithm, and it will be the subject of a future post.

Further reading

Books

Markov chains

Any stochastic process book will include a couple of chapters on Markov chains. For more details, there are many books dedicated entirely to the subject of Markov chains. For example, introductory books include:

  • Brémaud – Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues;
  • Levin, Peres, and Wilmer – Markov Chains and Mixing Times;
  • Norris – Markov Chains.

Those books cover Markov chains with countable state spaces. If you want to read about discrete-time Markov chains with general state spaces, try the book:

  • Meyn, Tweedie – Markov chains and stochastic stability

All the above books have a section detailing a Markov chain Monte Carlo method, such as the Metropolis-Hastings algorithm and the Gibbs sampler.

Monte Carlo methods

Books dedicated to Monte Carlo approaches include:

For a statistics context, there’s also the good book:

  • Casella and Robert – Monte Carlo Statistical Methods.
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