# Parallel Tempering

## General Chain-Structured Models

There is an important probability distribution used in applications (Liu, 2008):
$\pi(\mathbf x) \propto \exp\Big\{-\sum_{i=1}^dh_i(x_{i-1},x_i)\Big\} \equiv \exp(-H(\mathbf x))$
where
$\mathbf x = (x_0,x_1,\ldots,x_d)$
. This type of model exhibits a so-called "Markovian structure" because
$\pi(x_i\mid \mathbf x_{[-i]}) \propto \exp\{-h(x_{i-1},x_i)-h(x_i,x_{i+1})\}\,.$
We can simulate from
$\pi(\mathbf x)$
by an exact method. Note that
$Z\equiv \sum_{\mathbf x}\exp\{-H(\mathbf x)\} = \sum_{x_d}\Big[\cdots\Big[ \sum_{x_1}\Big\{\sum_{x_0}e^{-h(x_0,x_1)}\Big\}e^{-h_2(x_1,x_2)} \Big] \cdots\Big]\,.$
The simulating procedure is as follows: ## Exact Method for Ising Model

For a one-dimensional Ising Model,
$\pi(\mathbf x)=Z^{-1}\exp\{\beta(x_0x_1+\cdots+x_{d-1}x_d)\}\,,$
where
$x_i\in\{-1,1\}$
. And thus the simulating procedure is much easy. Note that for
$t=1,\ldots,d$
,
$V_t(x) = (e^{-\beta}+e^{\beta})^t$
and
$Z=2(e^{-\beta}+e^{\beta})^d$
.
We can use the following Julia program to simulate from
$\pi(\mathbf x)$
:
function exactMethod(M = 2000; beta = 10, d = 100)
V = ones(d)
V = exp(beta) + exp(-beta)
for t = 2:d
V[t] = V * V[t-1]
end
Z = 2*V[d]
x = ones(Int, d)
x[d] = 2*rand(Bernoulli(1/2))-1
for t = (d-1):1
# p1 = V[t] * exp(x[d] * 1)
# p2 = V[t] * exp(x[d] * (-1))
p1 = exp(x[d])
p2 = exp(-x[d])
x[t] = 2*rand(Bernoulli(p1/(p1+p2)))-1
end
return x
end

## Parallel Tempering for Ising Model

Liu (2008) also introduced the parallel tempering strategy: I wrote the following Julia program to implement this procedure and reproduce the simulation example of Liu (2008):
function parallelTemper(M = 2000, T = [0.1, 0.2, 0.3, 0.4]; d = 100, alpha0 = T)
I = length(T)
# initial
x = 2*rand(Bernoulli(1/2), d, I) .- 1
num1 = zeros(3)
num2 = zeros(3)
res = zeros(M, 4)
for m = 1:M
if rand() < alpha0 # parallel step
#for i = 1:I
# !!! Doesn't work !!!
# y1 = x[:,i]
# sampleX!(x[:,i], T[i])
# y2 = x[:,i]
# println(sum(abs.(y2.-y1)))
#end
sampleX!(x, T)
else
# idx = sample(1:I, 2, replace = false) # not neigbor
idx1 = sample(1:(I-1))
num1[idx1] += 1
idx = [idx1, idx1+1]
# rho = pdfIsing(x[:,idx], T[idx]) * pdfIsing(x[:,idx], T[idx]) / (pdfIsing(x[:,idx], T[idx]) * pdfIsing(x[:,idx], T[idx]))
rho = logpdfIsing(x[:,idx], T[idx]) + logpdfIsing(x[:,idx], T[idx]) - (logpdfIsing(x[:,idx], T[idx]) + logpdfIsing(x[:,idx], T[idx]))
if log(rand()) < rho
# swappin step
num2[idx1] += 1
tmp = copy(x[:, idx])
x[:,idx] .= x[:, idx]
x[:,idx] .= tmp
end
end
res[m, :] = sum(x, dims=1)
end
return res, num2 ./ num1
end
It is important to note that we should use logpdf to avoid too big values, in which the rejection part doesn't work well. Refer to Ising-model.jl for complete source code.
Finally, I reproduce the Figure 10.1 as follows: and the corresponding autocorrelation plots: 