A Markov-chain Monte Carlo sampler targets the unnormalised density on (a Gamma(2,1)…

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems · Monte Carlo and lattice simulation of physical systems

Error, Chance and the Machine, Ch Markov-chain Monte Carlo

Problem

A Markov-chain Monte Carlo sampler targets the unnormalised density

p(x) \propto x e^{-x}

x>0

(a Gamma(2,1) shape with its normalising constant omitted, since only ratios of

p

are needed). Proposed moves are multiplicative: from the current point

x

the candidate is

y = x e^{Z}

with

Z \sim \mathcal{N}(0,\sigma^2)

, so the proposal density is

q(y\mid x) = \frac{1}{y\sigma\sqrt{2\pi}}\exp\left[-\frac{(\ln y - \ln x)^2}{2\sigma^2}\right], \qquad y>0.

This proposal is asymmetric:

q(y\mid x) \ne q(x\mid y)

in general, because of the leading factor

1/y

. ((a)) Starting from detailed balance

p(x)K(x,y) = p(y)K(y,x)

for the Metropolis-Hastings kernel

K(x,y) = q(y\mid x)\,\alpha(x,y)

(for

y \ne x

), derive the general Metropolis-Hastings acceptance probability

\alpha(x,y) = \min\left(1, \frac{p(y)\,q(x\mid y)}{p(x)\,q(y\mid x)}\right).

((b)) Show that for the multiplicative proposal above the ratio

q(x\mid y)/q(y\mid x)

reduces to

y/x

, independent of

\sigma

, and hence write

\alpha(x,y)

for this target and proposal as a function of

x

and

y

alone. ((c)) Evaluate the acceptance probability for a move proposed from

x=2

y=3

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