Markov Chain Monte Carlo simulation sounds, admittedly, like a method better left to professional practitioners and the like; but please don’t let the esoteric name fool you. MCMC, as we like to call is, is a powerful yet deceptively simple technique that can be useful in problems ranging throughout science and engineering.

Since this promises to be a practical introduction, let’s skip the usual statistical jargon (and nonsense) about priors, posterior distributions and $P(d|\theta )$. Instead we will focus on a direct and intuitive demonstration.

### Optimization

Let’s start by setting up a problem for us to solve. A frequently encountered class of problems found in science, engineering and even finance, is to minimize (or *optimize*) a value. For example, given that building a bridge is a rather pricey endeavour, civil engineers will attempt to minimize the length of the bridge so to keep the cost down. In a similar vein, a stock broker would love to estimate how high (or low) a stock will go so as to know when to buy and when to sell. These are the sorts of problems that optimization tries to solve.

The second example is actually quite pertinent to our discussion because unlike finding the shortest distance between two shores (where there can be a sure answer), most problems don’t have a guaranteed answer. In these cases we’re searching for a good enoughÂ solution, one where the trade off between time and accuracy is appropriate.

### Markov Chain and CostÂ Functions

A Markov Chain is a special term for a certain class of processes that are essentially ‘memory-less’. What this means is that the evolution of the system does not depend on where the system has been. To illustrate this, consider a coin flip. At each flipping, their is a 50/50 chance for eith side to land upright. This is a Markov process. Now something which would not fulfill this criteria might be the spread of the flu. When trying to predict how many people will suffer the flu in the next week, you better take into accountÂ how many people had the flu right now. With the flu, the past trends are quite helpful in predicting the course of future events, but in the Markov process of a coin flip is doesn’t help at all.

In the case of a coin flip, we’ve already discussed the idea of a costÂ function without even knowing it. When we said the coin had a 50/50 chance of landing heads, we intuitively realized that a heads-up flip and a tails-up flip are equally difficult for the coin. If you were a fair coin, you wouldn’t care which way you land. So a Cost Function is simply a black box (for all we care) that tells us how costlyÂ a certain outcome, such as when deciding how best to avoid traffic on the way to work. For a coin this cost function is flat, or indifferent to either outcome.

This cost functionÂ tells the Markov Chain how to evolve in time by describing the weights associated with each option. Going back to the optimization that we talked about before, the cost function should always point us in the right direction in terms of providing a better and better solution as time goes on. But enough of all this talking, let’s get right into it.

Let’s first setup a domain, or area, to explore. We’ll use a well known test function that should provide the complexity we require.

test = function (x,y) { return(-20*exp(-0.2*sqrt((x^2+y^2)/2))-exp(0.5*(cos(2*pi*x)+cos(2*pi*y)))+22.71828 ) } library(lattice) points = matrix(nrow=61, ncol=61, seq(-3,3,0.1)) ## Test points for plotting filled.contour(x=points[,1], y=points[,1], z =test(points,t(points)), nlevels=20)

The test function is the mathematical surface thatÂ we’ll use to setup our domain. As seen in the contour plot, the function is topographically complex and offers many minimums (but only one global minimum at 0,0).

Here is an example of a cost function. This function compares the misfit between the new solution and the old solution to determine how much an improvement (or digression) the new solution offers.

accept = function(x1, y1, x2, y2) { score1 = test(x1,y1) score2 = test(x2,y2) if (exp(-15*(score2-score1)) > runif(1) ) { ## is score > random number? return(TRUE) ## Accept new solution } return(FALSE) ## Reject new solution }

If the new solution costs less than the old solution then it is guaranteed to be accepted and a poorer solution is unlikely to be accepted (but it can be, and that is important!).

For the MCMC method, we’ll start at a random point in the domain and then take steps based on whether or not the score is good enough (based on the accept function above). We’ll save all the steps we take in the MCMC with a dataframe (i.e. table) so we can plot the evolution over time.

x = runif(1,-1,1) y = runif(1, -1,1) position = data.frame(x,y) position

Now we just have to setup the MCMC algorithm to use our starting point and the acceptance function. The jump length is simply how far from the current position the algorithm will search, and iter is the number of steps to take.

jmp = 0.2 iter = 50000 naccepted = 1 x = runif(1,0,2) x = 2.3 y = runif(1, 0,2) y = 1.98 position = data.frame(x,y) for (i in 1:iter) { n = length(position[,1]) new_x = rnorm(1, position[n,1], jmp) new_y = rnorm(1, position[n,2], jmp) if (accept( position[n,1], position[n,2],new_x, new_y )) { position[n+1,1] = new_x position[n+1,2] = new_y naccepted = naccepted + 1 } } print("Number accepted") naccepted print("Ratio of acceptance:") naccepted/iter

If we let it run, this is what we get:

The first point is at the top right and then as time went on the chain moved down into the local minimum right next to it. And then it climbed out of it eventually and fell into a better minimum and another before finally reaching the global minimum in the center.

Depending on how we choose the parameters, especially the cost function, the behavior and properties of the algorithm can change. Sometimes we’d want to explore the space more fully and therefore have a relatively relaxed cost function and a larger number of steps. Other times we just want a computationally cheap and quick solution for which we would use another combination of parameters.

MCMC methods are especially useful in higher dimensional cases (here $n=2$) such as for my inverse modeling research (see here) where $n=100$ or more.

Thank you for your post, very interesting.

I have the impression that the key point of your algorithme is the accept function.

Could you please explain how you chose this function (why an exponential form ? why -15 ?) and what is the rationale behind this function ?

When you are stuck in a local minimum, this function has to allow to jump into a worse solution, in order to get out from the local minimum and then be able to jump into a better solution. But I don’t get how your accept function manges to do this.

Yeah, spot on. The acceptance function dictates a lot of the overall performance and run-time considerations of the algorithm as a whole. I should mention that

anymetric which will bias the walk towards lower costs will theoretically guarantee the algorithm to converge to the global minimum (given some constraints of the topology of the solution space are valid, as they are in these linear models), yet the timeframe in which that convergence is found can be highly problematic if you need the model to run in bounded time (say a few hours).The choice here for the acceptance function is commonly used generic function which is traditionally seen with a coefficient of -0.5. After running my model a few times, I decided to tweak that coefficient in order to produce a better animation–a change in that coefficient will alter the sensitivity of the acceptance algorithm to non-optimal steps (where cost goes up) but not change the actual functioning of the algorithm structurally. Let me know if anything here isn’t clear and I can do a better job of explaining (and illustrating) what I mean.

The key aspect in allowing the model to step “out” of a local minimum stems from the random number generation in the acceptance function. The random number might be 0.0 which would allow

anysolution to be accepted, or it may be 1.0 which strongly biases the acceptance of less than optimal steps towards rejection. I can put together a diagram that shows how this works in practice if you’d like, just let me know!