[TM] Change of Variables in MCMC

This post is about change of variables in Markov chain Monte Carlo (MCMC), which is used quite often when the target distribution is supported on a subset of {\mathbb{R}^n}. For example, the Exponential distribution and the Log-Normal distribution are only supported on positive reals.

Consider a target distribution {\pi(x)} that is supported on a subset {S \subset \mathbb{R}^n}. If we use a random walk proposal {q_X(x' \mid x) = \mathrm{MVN}(x' ; x,\Sigma)}, then we might end up with a proposal {x'} such that {\pi(x') = 0} and, this might cause too few acceptance in the MCMC chain. If we can find a transformation {h:D \to \mathbb{R}^n} that is one-to-one, differentiable and spans {\mathbb{R}^n}, then we can consider a proposal {x' = h^{-1}(y')} where {y' \sim q_Y(\cdot \mid y = h(x))}. This proposal always yields a proposal {x'} such that {\pi(x') > 0.}

Of course, when we employ such a transformation in the proposal kernel, we need to be careful about evaluating the proposal densities. We know that the acceptance probability is {\alpha(x,x') = 1 \wedge \frac{\pi(x') q(x \mid x')}{\pi(x) q(x' \mid x)}}, and it should be no surprise that {q_X(x' \mid x) \not= q_Y(y' \mid y)} unless {h} is the identity map.

Let’s work out the acceptance ratio together carefully. Recall that change of variables proceeds as follows: when {Y \sim f_Y(y) } and we consider the transformation { y = h^{-1}(x)}, the pdf of {X = h^{-1}(Y)} is

\displaystyle f_X(x) = f_Y(h(x))|J_{h}(x)|.

When we apply this to the kernels {q_Y} and {q_X} we get that

\displaystyle q_X(x' \mid x ) = q_Y(h(x') \mid h(x)) \cdot |J_{h}(x')|.

Example 1 {Symmetric proposal on transformed space} If {q_Y(y' \mid y)} is a symmetric proposal, then the acceptance probability becomes

\displaystyle \alpha(x,x') = 1 \wedge \frac{\pi(x') |J_{h}(x)|}{\pi(x)|J_{h}(x')|} .

Here are two common transformations.

Example 2 (Log-transformation for {x} supported on {\mathbb{R}_+})

If {h = \log}, then {|J_{h}(x)| = 1 / x} and acceptance probability is

\displaystyle \alpha(x,x') = 1 \wedge \frac{\pi(x')x'}{\pi(x)x}.

Example 3 (Logit transformation for {x} supported on {(0,1)}) If {h(x) = \log(\frac{x}{1 - x})},then the inverse transformation is {h^{-1}(y) = \frac{exp(y)}{1 + \exp(y)}.} The acceptance probability is

\displaystyle \alpha(x,x') = 1 \wedge \frac{\pi(x')x'(1-x')}{\pi(x)x(1-x)}.

This post is the first one in the category `trivial matters’, where I formally write down some notes to myself about tricks and facts that I repeatedly use but (unfortunately) need to re-derive everytime I use them.

Author: PhyllisWithData

Statistics PhD student at Harvard University.

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