stationary distribution Archives

The Metropolis(-Rosenbluth-Rosenbluth-Teller-Teller)-Hastings algorithm

Consider a collection of random variables described by a joint probability distribution. Often, in any field with probability or statistics, one faces the task of simulating these random variables, which typically depend on each other in some fashion.

A now standard way for simulating or sampling such random variables is to use the Metropolis-Hastings algorithm, undoubtedly the cornerstone of Markov chain Monte Carlo methods. This method creates a discrete-time Markov chain that has a stationary or invariant distribution being the aforementioned distribution.

The algorithm was born out of a 1953 paper by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall Rosenbluth, Augusta H. Teller, and Edward Teller (two husband-wife pairs), who looked at a special case, and a 1970 paper by W.K. Hastings, who generalized the method. It is typically called the Metropolis-Hastings or the Metropolis algorithm. And some have called it the M(RT)²H algorthm.

(The history is a bit complicated, but perhaps we should drop the name Metropolis. The late Arianna (née Wright) Rosenbluth did most of the work. She was also a dab hand at fencing.)

Although the algorithm’s initial adoption and use was slow, taking decades partly due to slower computers, the Metropolis-Hastings algorithm is now a widely used method for simulating collections of random variables. This in turn gives fast ways for exploring, integrating, and optimizing otherwise unwieldy mathematical functions, such as those found in Bayesian statistics, machine learning, statistical physics, and combinatorial optimization. The algorithm serves as the foundation for other random simulation methods, such as the Gibbs sampler, hence it’s been called the workhorse of Marko chain Monte Carlo methods.

There are many books, articles, lecture notes, and websites describing the Metropolis-Hastings algorithm; see the Further reading section below. But I’ll detail the core ideas here. This post is designed to be somewhat self-contained, but it arose from a series of posts, starting with this one and ending with this particularly relevant one.

Constructing a Markov process

Take a collection of $n$ random variables $X_1,\dots,X_n$ with a (joint) probability distribution $\pi(x)=\pi(x_1,\dots,x_n)$. This distribution will either be a (joint) probability mass function or (joint) probability density for discrete or continuous random variables, respectively.

We wish to construct a Markov chain on an abstract mathematical space $\mathbb{X}$. We assume we can write a point $x\in \mathbb{X}$ as $x=(x_1,\dots,x_n)$. More specifically, the space $\mathbb{X}$ is a Cartesian product of spaces $\mathbb{X}_1,\dots,\mathbb{X}_n$ on which the variables are defined.

Which mathematical space is $\mathbb{X}$? That will, of course, depend on the random variables you’re trying to simulate. But it’s usually the lattice $\mathbb{Z}^n$, Euclidean space $\mathbb{R}^n$, or a subset of one of these two spaces.

For this post, we’ll assume that the space $\mathbb{X}$ is discrete, which makes the things simpler. But the mathematics is similar for continuous spaces, with small changes such as replacing probabilities and sums with probability densities and integrals, respectively. We’ll further assume the space is finite, so we can use matrices to describe the Markov transition kernels. But everything covered here will work on infinite spaces such as $\mathbb{R}^n$, which is the most common space used in practice.

Again, our overall aim is to construct a Markov chain with a stationary $\pi$ being the same as the distribution that we want to sample.

Jumper process

There’s a random jumper that wants to jump around the space $\mathbb{X}$. The jumper randomly jumps from one point in this mathematical space $x\in \mathbb{X}$ to another point $y\in \mathbb{X}$ according to the probability $J(x,y)$. If the the state space $\mathbb{X}$ is finite, then $J$ becomes a matrix. The matrix row $J(x,\cdot)$ is a probability mass function for each $x\in \mathbb{X}$, so it sums to one. By definition, this random jumping forms a Markov chain.

The only thing we ask is that, for our jumper, every point $x$ in $\mathbb{X}$ where $\pi(x)>0$ is reachable with positive probability in a single step. This implies the easy-to-achieve condition $J(x,y)>0$ where $\pi(x)>0$ for all points $x,y\in\mathbb{X}$.

Now we have a Markovian jumper on the space $\mathbb{X}$. But this turns out to be too much jumping for our jumper. Furthermore, the jumper is jumping more in certain directions than others. Occasionally the jumper wants to stay put (and have a rest) with the aim of balancing the jump directions.

The jumper still wants to jump sometimes from a point $x\in \mathbb{X}$ to another point $y\in \mathbb{X}$ based on $J(x,y)$. But now at each time step, after choosing the jump direction but before jumping, the jumper flips a biased coin whose success probability $\alpha(x,y)$ depends on the current position $x\in \mathbb{X}$ and the (potential) next position $y\in \mathbb{X}$. For the coin, the acceptance probability, which allows (or not) the jumper to move from $x$ to $y$, is given by

$$\alpha(x,y)=\min[1,\frac{\pi(y)}{\pi(x)}\frac{J(y,x)}{J(x,y)} ]\,,\quad x, y\in \mathbb{X}\,.$$

The function $\alpha(x,y)$ is clearly never negative. The minimum in the above expression ensures that $\alpha(x,y)$ is a valid probability.

Metropolis-Hastings ratio

The ratio in the expression for $\alpha(x,y)$ is sometimes called the Metropolis-Hastings ratio, which we’ll soon see is designed specifically to balance the jump directions. The ratio means that a constant factor in the target distribution $\pi(x)$ will vanish due to cancellation.

More specifically, if we can write the target distribution as $\pi(x)=f(x)/C$, where $C>0$ is some constant and $f(x)$ is a non-negative function, then the ratio term $\pi(y)/\pi(x)=f(y)/f(x)$. This reasoning also applies to a constant factor in the transition kernel $M$.

The constant factor being irrelevant in the target distribution $\pi(x)$ is very useful. It is particularly important for posterior distributions in Bayesian statistics and the Gibbs distributions in statistical physics, as these distributions typically have difficult-to-calculate constants.

Metropolis-Hastings process

The pairing of the original jumper Markov chain with the coin flipping creates a new Markov chain, which we call the Metropolis-Hastings process. What’s remarkable is its stationary distribution will be the target distribution $\pi(x)$.

Transition kernel (matrix)

For the Metropolis-Hastings process, we can readily reason the structure of the transition kernel (matrix) $M$ that describes this Markov chain. First we’ll look at the off-diagonal entries of $M$.

Jumping from $x$ to $y\neq x$

To jump from point $x$ and to another point $y\neq x$, the probability is simply the previous probability $J(x,y)$ multiplied by the probability of that proposed jump being accepted, which is $\alpha(x,y)$, giving

$$ M(x,y) = \alpha(x,y) J(x,y), \quad x\neq y\,.$$

Now we examine the diagonal entries of $M$.

Jumping from $x$ to $x$

There are two different ways for the jumper to remain at point $x$. The first way is that the jumper simply jumps from $x$ to $x$, which happens with probability $J(x,x)$. This proposed jump, so to speak, is accepted with probability one because $\alpha(x,x)=1$. Consequently, we can write this probability as $\alpha(x,x)J(x,x)$ or $J(x,x)$.

The second way consists of all the possible jumps from $x$ to $z$, but then for each of those proposed jumps to be rejected, which happens (for each jump) with probability $[1-\alpha(x,z)]$. (I am using here the dummy or bound variable $z$ instead of $y$ for clarity.) Adding up the probabilities of these events gives the probability of the second way being the sum $\sum_{z\in \mathbb{X}}[1-\alpha(x,z)] J(x,z) \,.$

Consequently, for a single time step, the probability that the jumper starts at point $x$ and remains at $x$ is

$$ M(x,x) = \alpha(x,x)J(x,x)+\sum_{z\in \mathbb{X}}[1-\alpha(x,z)] J(x,z) \,.$$

The transition matrix $M$ should be stochastic, so the rows sum to one, which we see is the case

$$ \sum_{y\in\mathbb{X}}M(x,y)=1\,.$$

Of course, we could have derived the diagonal entry $M(x,x)$ immediately by starting with the above sum, but that approach lacks intuition into what’s happening.

Expression for the transition kernel $M$

$$M(x,y) = \begin{cases}
\alpha(x,y) J(x,y) & \text{if }
\begin{aligned}[t]
x&\neq y
\end{aligned}\\
\alpha(x,x)J(x,x)+\sum_{z\in \mathbb{X}}[1-\alpha(x,z)] J(x,z) & \text{if } x=y
\end{cases}$$

Often the above expression is written as a single line by placing an indicator function or similar in front of the sum for the diagonal entries. (In the continuous case, where the sum is replaced with an integral, a Dirac delta distribution is often used instead.)

Reversibility

A Markov process on $\mathbb{X}$ with kernel (matrix) $K$ is (time) reversible with respect to the distribution $\mu$ if the following holds

$$ \mu(x)K(x,y) = \mu (y) K(y,x)\quad x,y\in\mathbb{X}\,.$$

This reversibility condition is also called the detailed balance equation. If this condition is met, then the Markov process will have a stationary distribution $\mu$. By summing over $x$, we can verify this because we obtain

$$ \sum_{x\in\mathbb{X}}\mu(x)K(x,y) =\mu(y)\sum_{x\in\mathbb{X}} K(y,x)=\mu(y)\,.$$

This is just the balance equation, often written as $\mu=K\mu$, which says that the transition kernel $K$ has a stationary distribution $\mu $.

(Strictly speaking, we don’t necessarily need reversibility, as long as the Markov chain has a stationary distribution. Reversibility just makes the algebra easier.)

The Metropolis-Hastings process is reversible

We can show that the Metropolis-Hastings process with the transition kernel $M$ satisfies the reversibility condition. The proof essentially just requires the swapping of rows and columns. Clearly then we only need to look at the off-diagonal entries, so we assume $x\neq y$.

We further assume that $\pi(y)J(y,x)\geq \pi(x) J(x,y)$. Then $\alpha(x,y)=1$, so $M(x,y)=J(x,y)$. Now we use this last fact in the last line of algebra that follow:

$$\begin{aligned}\pi(y) M(y,x)&=\pi(y)J(y,x) \alpha(y,x)\\ &= \pi(y)J(y,x) \frac{\pi(x)}{\pi(y)}\frac{J(x,y)}{J(y,x)}\\ &= \pi(x)J(x,y)\\&= \pi(x)M(x,y)\,,\end{aligned}$$

which shows that the reversibility condition holds with the last assumption.

But of course we can reverse that assumption, due to symmetry, so $\pi(y)J(y,x)\leq \pi(x) J(x,y)$, then $\alpha(x,y)=[\pi(y)J(y,x)]/[\pi(x) J(x,y)]$ and $\alpha(y,x)=1$, implying this way also works. Then the reversibility condition holds for all cases.

We could have shown this more quickly by observing that

$$\begin{aligned}\pi(y) M(y,x)&=\pi(y)J(y,x)\alpha(y,x)\\ &= \pi(y)J(y,x) \min[1, \frac{\pi(x)}{\pi(y)}\frac{J(x,y)}{J(y,x)}]\\ &= \min[\pi(y)J(y,x) , \pi(x)J(x,y)]\,,\end{aligned}$$

which is symmetric in $x$ and $y$.

In summary, the Metropolis-Hastings process is reversible and has the stationary distribution $\pi(x)$.

Continuous case

As mentioned earlier, the continuous case is similar to the discrete-case with a few modifications. For example, assuming now $\mathbb{X}$ is continuous, then the stationary distribution is described by a probability density $\pi(x)$. The jumper process will be described by transition kernel (function) $j(x,y)$, where $j(x,\cdot)$ is now a probability density for each $x\in \mathbb{X}$.

It follows that the acceptance probability is

$$\alpha(x,y)=\min[1,\frac{\pi(y)}{\pi(x)}\frac{j(y,x)}{j(x,y)} ]\,,\quad x, y\in \mathbb{X}\,.$$

The transition kernel (function) is

$$m(x,y) = \begin{cases}
\alpha(x,y) j(x,y) & \text{if }
\begin{aligned}[t]
x&\neq y
\end{aligned}\\
\alpha(x,x) j(x,x)+\int_{\mathbb{X}}[1-\alpha(x,z)]j(x,z) dz & \text{if } x=y
\end{cases}$$

For more details, there are derivations online such as this one, as well as the sources cited in the Further reading section below.

Libraries

Of course, before writing your own code, I would check out any pre-written functions in your favourite language that already implement the Metropolis-Hastings algorithm.

For example, in MATLAB there’s the function mhsample. Again I would be using this before writing my own code, unless it’s for illustration or educational purposes.

In Python there’s the library pymcmcstat. For those with a machine learning bent, there’s also a TensorFlow function MetropolisHastings.

In R, which is a language designed for statistics, implementations of any Markov chain Monte Carlo methods will be couched in the language and notation of (Bayesian) statistics. For a specific library, I can’t say I know which is the best, but you can check out the MCMCPack library. For R libraries, be wary of using the ones that were published and are (or were?) maintained by a single contributor.

Code

For a couple of simple examples in both one dimension and two dimensions, I’ve implemented the Metropolis-Hastings algorithm in the programming languages R, MATLAB, Python (NumPy) and Julia. The code can be found here.

The first MC in MCMC methods

Markov chains form a fundamentally important class of stochastic processes. It would be hard to over stress their importance in probability, statistics, and, more broadly, science and technology. They are indispensable in random simulations, particularly those based on the Markov chain Monte Carlo methods. In this post, we’ll have a look some Markov chain basics needed for such simulation methods.

This is the first part of a series of posts on Mark chain Monte Carlo methods. This post covers the basics of Markov chains, which is the more involved part. The second part will cover Monte Carlo methods. The third part will combine the ideas from the first two parts. Overall, the three posts will sketch the mechanics of Markov chain Monte Carlo (MCMC) methods, whose importance and applications I detailed in a previous post.

Markov chains vs Markov processes

All Markov chains are Markov processes. Some people use the term Markov chain to refer to discrete-time Markov processes with general state spaces. Other people prefer the term Markov chain for continuous-time Markov processes with countable state spaces.¹

Nevertheless, the first MC in the MCMC suggests the Markov chain Monte Carlo crowd prefers the former sense of Markov chain, given the use of discrete-time Markov processes in their simulations.

Markov the frog

Some writers introduce Markov chains with a mental image of a frog jumping around lily pads scattered over a pond. (Presumably the frog never misses a lily pad.) We assume the frog randomly chooses the next lily pad through some random mechanism. Perhaps the distances between lily pads or their sizes influence the chances that frog will jump between them.

We further assume that the frog is a bit particular, preferring to jump in certain directions more than others. More precisely, the probability of our frog jumping from a lily pad labelled $x$ to another labelled $y$ is $P(x,y)$. But jumping in the opposite direction happens with probability $P(y,x)$, which in general is not equal to $P(x,y)$.

I typically use the term points, but the Markov literature usually says that the Markov chain visits states.

State space

We can interpret a Markov chain, a type of stochastic process, as a collection or sequence of random variables. ²The values of the random variables are points in some mathematical space $\mathbb{X}$. This space can be quite abstract, but in practice it’s usually the lattice $\mathbb{Z}^n$, Euclidean space $\mathbb{R}^n$, or a subset of one of these two spaces. For our frog example, all the lily pads in the pond form the state space.

We’ll only consider countable Markov chains where the number of points in the state space $\mathbb{X}$ is countable. Although the results and theory generally hold for more general state spaces, the accompanying work requires more technical mathematics. For finite and countable state spaces, we can use standard probability and matrix knowledge. But when we use uncountable state spaces such as $\mathbb{R}^n$, we enter the world of measure theory and functional analysis.

I will often write a point $x$ in a (state) space $\mathbb{X}$. But you can say an element $x$ of a set $\mathbb{X}$. Many authors refers to the points or elements as states of the Markov chain. In the frog example, each lily pad is a different state.

Markov property

A discrete-time countable Markov chain is a random process that jumps between points of some countable mathematical space $\mathbb{X}$ such that, when at point $x \in \mathbb{X}$, the next position is chosen according to a probability distribution $P(x,·)$ depending only on $x$.

More specifically, a sequence of random variables $(X_0, X_1, . . .)$ is a discrete-time Markov chain $X$ with a countable state space
$\mathbb{X}$ and kernel $P$ if for all $x,y \in \mathbb{X}$ and all $t \geq 1$ satisfying $\mathbb{P}[X_{t−1}=x_{t-1},\dots,X_0=x_0]>0$, we have

$$ \begin{align}\mathbb{P}[X_{t+1} =y|&X_{t}=x,X_{t−1}=x_{t-1},\dots,X_0=x_0]\\&=\mathbb{P}[X_{t+1} =y|X_t =x]\\&=P(x,y)\,.\end{align}$$

This equation is often called the Markov property.

The Markov property says that the conditional probability of jumping from point $x$ to $y$ remains the same, regardless of which points or states $x_0,x_1,\dots,x_{t-1}$ were previously visited. This is precisely why the kernel $P$ contains all the information needed to describe the future random evolution of the Markov chain.

We have assumed the probabilities given by $P$ are fixed, meaning we have described a homogeneous Markov chain.

Markov kernel

The kernel $P$ is called the Markov (transition) kernel or probability kernel. Assuming a countable state space $\mathbb{X}$, we can reference any probability value of the kernel $P$ with two variables $x,y\in\mathbb{X}$. If we assume a finite state space $\mathbb{X}$, then the kernel $P$ becomes a regular matrix taught in linear algebra. An infinite but countable state space gives an infinite matrix $P$. The rows of the kernel matrix $P$ must add up to one, because each row is a probability measure.

A more general space, such as Euclidean space $\mathbb{R}^n$, results in a more general kernel with respect to a suitable measure. In this setting, $P(x,·)$ is no longer a probability mass function, but a general probability measure.

Initial distribution

At time $t=0$ we describe the random initial configuration of a Markov process with a probability distribution $\mu_0$. For a finite or countable Markov chain, this initial distribution $\mu_0$ corresponds to a probability mass function encoded as a row vector.

Jumping from $x$ to $y$

The probability distribution $\mu_0$ gives the probability of starting in state (or at point) $x\in\mathbb{X}$. After one time step, we can write down the probability distribution $\mu_1$ that gives us the different probabilities of the Markov chain being at different states. At $n=1$, basic matrix algebra and probability rules give us the matrix equation

$$\mu_1=\mu_0 P$$

By induction, after $t$ time steps we have the expression

$$\mu_n=\mu_0 P^n\,.$$

where the superscript $n$ denotes matrix power. We can write the $n$-time step kernel as $P_{(n)}$, which for a finite Markov chain is given by the matrix equation $P_{(n)}=P^n$.

Seeing how $P_{(n)}$ behaves as $n$ approaches infinity forms part of work that studies the convergence and ergodicity properties of Markov chains. I’ll make these concepts clearer below. But first I’ll give some conditions that are typically needed.

Regularity conditions

A Markov chain with a countable state space needs some conditions to ensure convergence and ergodicity.

Regularity conditions

A stationary distribution $\pi$

Aperiodicity

Irreducibility

Postive recurrence

The nature of the state space and the kernel will dictate these conditions. These conditions are also not necessarily logically distinct. For example, on a finite state space, you’ll get positive recurrence for free, because an aperiodic, irreducible Markov chain with a finite state space is always positive recurrent.

We now briefly detail these conditions and in another post I’ll give examples how the conditions can be met.

Stationary distribution $\pi$

It’s possible to encounter a probability distribution $\pi$ where applying the kernel $P$ returns the same distribution $\pi$, meaning

$$ \pi=\pi P\,.$$

This (fixed-point) equation is called the balance equation.

The distribution $\pi$ is called the stationary, invariant or steady-state distribution. A Markov chain does not need to have a stationary distribution. And if a Markov chain does have one, it may not be unique. Its existence and uniqueness will depend on the Markov kernel $P$.

Showing that a unique stationary distribution exists and it is possible to reach it with probability one is the stuff of Markov convergence results. Markov chain Monte Carlo methods hinge upon these results .

Aperiodicity

It is possible for a Markov chain to get trapped in a loop, periodically visiting the same states. The period $d_x$ of a state $x\in \mathbb{x}$ is the greatest common divisor of all $n$ values such that $P(x,x)^n>0$. If the period of a point is $d_x=1$, then we say it’s aperiodic. If every state of a Markov chain is aperiodic, we says it’s an aperiodic Markov chain.

Aperiodicity means there are no loops to trap the Markov chain. This property is typically needed for convergence results.

Irreducibility

A Markov chain with a countable state space $\mathbb{X}$ is irreducible if the Markov chain can go from any point $x\in\mathbb{X}$ to another other point $x\in\mathbb{X}$ with a positive probability in a finite number of time steps. In other words, there exists a natural number $s$ such that $P(x,y)^s>0$ for all $x,y\in\mathbb{X}$.

Irreducibility ensures that a Markov chain will visit all the states in its state space. This property is also needed for convergence results.

Recurrence

When studying Markov processes, a quantity of interest is how much time it takes to return to a state or point. For a point $x\in\mathbb{X}$, we define its first return time as

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

As the name suggests, this random variable is the number of time steps for the Markov process return to state $x$, taking whichever path, conditioned on it starting at $x$.

We call a state $x$ recurrent if the probability of its first return time being finite is one, meaning $\mathbb{P}_x(T_x^+<\infty)=1$. Otherwise the state $x$ is said to be transient.

Positive recurrence

We can classify different types of recurrence based on the expected value of the first return times. A state $x$ is called positive recurrent if the expected value of its first return time is finite, meaning $\mathbb{E}_x(T_x^+)<\infty$. Otherwise state $x$ is null recurrent.

For a countable Markov chain, if all the states in the state space are (positive) recurrent, so $\mathbb{E}_x(T_x^+)<\infty$ for all $x\in\mathbb{X}$, then we say the Markov chain is (positive) recurrent.

Again, the concept of positive recurrence is needed for convergence results.

Ergodicity

We say a countable Markov chain is ergodic if it is irreducible, aperiodic and positive recurrent.³ Ergodicity allows one to find averages by employing a more general form of the law of large numbers, which Monte Carlo methods rely upon. We stress that definitions of ergodicity vary somewhat, but in general it means convergence and laws of large numbers exists.

Tag: stationary distribution

The Metropolis(-Rosenbluth-Rosenbluth-Teller-Teller)-Hastings algorithm

Constructing a Markov process

Jumper process

Metropolis-Hastings ratio

Metropolis-Hastings process

Transition kernel (matrix)

Jumping from \(x\) to \(y\neq x\)

Jumping from \(x\) to \(x\)

Expression for the transition kernel \(M\)

Reversibility

The Metropolis-Hastings process is reversible

Continuous case

Libraries

Code

Further reading

Articles

Historical

Introductory

History

Books

Websites