The connection between Simulated Annealing and MCMC (Part 3)

Check out part 1 and part 2. Let’s start off by writing the code for the Metropolis algorithm and comparing it to Simulated Annealing.

def metropolis(p, qdraw, nsamp, stepsize, initials):

   samples=np.empty((nsamp, 5))
   x_prev =initials['xinit']

   for i in range(nsamp):

       x_star = qdraw(x_prev, stepsize)
       p_star = p(x_star)
       p_prev = p(x_prev)

       # note that p is in logs!
       pdfratio = p_star - p_prev                  # (1)
       threshold = np.random.uniform()
       alpha = min(1, np.exp(pdfratio))

       if threshold < alpha:
           samples[i] = x_star                     # (2)
           x_prev = x_star

           samples[i]= x_prev                      # (3)

    return samples

Here’s the simulated annealing algorithm again but using the same variable names where it makes sense.

def annealing(p, qdraw, nsamp, initials):

    L = initials['L']
    T = initials['T']

    x_prev =initials['xinit']
    for e in range(nsamp):
        for l in range(L):
            x_star = qdraw(X)                              
            p_star = p(x_star)
            p_prev = p(x_prev)   

            pdfratio = p_prev - p_star              # (1)
            threshold = np.random.uniform()                
            alpha = min(1, np.exp(pdfratio/T))      

            if (p_star < p_prev) or (threshold < alpha ):
                x_prev = x_star

               # Let's keep track of the best one
               if p_star < p_prev:
                   x_best = x_star                  # (2)

        # Let's calculate new L and T
        T = T * 0.85                                  
        L = L * 1.1                                   

    return x_best

Let’s start off by writing the code for the Metropolis algorithm and comparing it to Simulated Annealing.

Similarities and differences

  1. In both, we have a proposal function that generates an x_star
  2. In both, we calculate the probability of the target distribution at x_star
  3. Temperature for Metropolis is just 1 (T = 1).
  4. Note that in SA, we move from x_prev to x_star in SA if the new energy, or the value of the target function, is lower. In Metropolis, we move if the probability mass function of the target distribution is higher.
  5. We always take a sample in Metropolis.

It’s pretty remarkable how similar they are.

Why does it work?

The main reason this works is because of “detailed balance” which you may remember from the post on Markov Chains. We said that if $\pi_{i} p_{ij} = p_{ij} \pi_{j}$ then $\pi$ is a stationary distribution. Detailed balance for continuous variables is:

\[f(x)p(x,y) = f(y)p(y,x)\]

Now, if this is true (and proof isn’t too hard), then you know that f is a stationary distribution for the chain. And since it is stationary, each step in the Markov Chain lands us back in f. This means that all the samples that we draw are coming from f

If you squint hard enough, you’ll notice that even in simulated annealing, we’re drawing from a stationary distribution at each temperature.

It may take a little while for the chain to get to that stationary distribution depending on where you start but once you’re there you’re (by definition) stuck in the stationary distribution. But how do you know that you actually made it to your stationary distribution? The truth is that you don’t really. There are signs that you haven’t made it to the stationary distribution so high dose of paranoia is advised.

We’ll talk about burnin, autocorrelation, Gewecke, and Gelman-Rubin test, in a future post.