Metropolis-Hastings Proposals for SAEM

Why SAEM Needs a Conditional Random-Effect Proposal

In a nonlinear mixed-effects model, the distribution of a subject’s random effects conditional on that subject’s observations is usually not available in a form that can be sampled from directly. SAEM can use a Markov chain to generate these latent effects: at each transition, it proposes a new value and accepts or rejects it according to a probability that preserves the required conditional distribution. The proposal can be convenient even when it is not itself a good approximation to that distribution, provided the acceptance probability accounts for the proposal correctly.

Consider subject \(i\), with observations \(y_i\), random effects \(b_i\), population parameters \(\theta\), and random-effect covariance matrix \(\Omega\). During a single SAEM simulation update, treat the observed data and current parameter values as fixed. Assume the subject’s random effects have density \(\phi(b_i;0,\Omega)\), and that the observation density conditional on the random effects is \(p(y_i\mid b_i;\theta)\). The subject-specific conditional distribution of interest is proportional to the product of these two densities.

The target combines the likelihood contribution from the subject’s observations with the random-effects density. Using the likelihood alone would generally target the wrong distribution; using the random-effects density alone would ignore the information in the observations. The proposal need not equal the target, but it must have positive density wherever the target can require movement and the reverse move has positive probability.

Acceptance Probability and Its Justification

From current state \(b_i\), first draw a candidate \(b_i^\star\) from \(q_i(b_i^\star\mid b_i,y_i;\theta,\Omega)\). Then accept that candidate with probability \(\alpha_i(b_i,b_i^\star)\); otherwise retain the current state. The acceptance probability is

The proposal-density ratio is essential when the proposal is not symmetric. Substituting the target density shows that its normalizing constant cancels:

A short balance argument explains why this rule preserves the conditional target. For two distinct states \(b_i\) and \(b_i^\star\), the probability flow under the target density from the first state to the second is proportional to

The expression on the right is unchanged when \(b_i\) and \(b_i^\star\) are exchanged. Thus the flow in either direction is equal. Rejected proposals stay at the current state, supplying the probability of remaining there; together, these moves define a transition that leaves the conditional target invariant. This argument requires the proposal and target densities in the acceptance ratio to be well-defined on the states involved.

For computation, it is usually safer to work with the log of the ratio before applying the minimum. Define

Then \(\alpha_i=\min\{1,\exp(r_i)\}\). If \(r_i\geq 0\), accept with probability one; if \(r_i<0\), accept with probability \(\exp(r_i)\). Drawing \(U\sim\mathrm{Uniform}(0,1)\), accept exactly when \(\log U<r_i\), with automatic acceptance when \(r_i\geq0\).

Worked Proposals in a Nonlinear Observation Model

The examples use a simple hypothetical model to make the likelihood calculations explicit. For one observation \(y\), suppose \(y\mid b\sim N(\exp(b),\sigma^2)\) and \(b\sim N(0,1)\). The observation mean depends nonlinearly on the scalar random effect. The conditional target is proportional to the normal observation density times the standard normal prior density. These examples illustrate proposal arithmetic, not results from a real study.

Choosing and Checking a Proposal

Several proposals are useful, but they make different trade-offs. A symmetric random walk is straightforward because its proposal terms cancel. An independence proposal can aim at a region of high conditional density, but its potentially asymmetric density must be included in the ratio. A prior proposal also has a simple cancellation, leaving the likelihood ratio, but it may propose values poorly suited to the observations. Whenever a suitable conditional sampler is available, drawing directly from the conditional distribution avoids an MH accept-or-reject step; in nonlinear models such a sampler is often unavailable or computationally inconvenient.

Before using an MH update in SAEM, check that the likelihood is evaluated at both the current and proposed random effects, the prior contribution is included, and the proposal densities are in the correct order: reverse move in the numerator, proposed move in the denominator. Verify that any proposal density is positive for moves needed to explore the target’s support. For a multivariate random effect, ensure that likelihoods, prior densities, and proposal densities are evaluated for the full vector, with compatible covariance matrices and dimensions.

Calculate in log space when densities are very small, and check computations using simple limiting cases. For example, a symmetric proposal must have a zero log proposal correction, and a proposal equal to the prior must cancel the prior contribution. A common error is to accept a candidate whenever its likelihood is higher. That rule ignores both the prior and, for an asymmetric proposal, the proposal correction. A candidate with a lower likelihood can still be accepted; conversely, a higher likelihood does not guarantee acceptance when the prior or proposal correction offsets the gain.

An MH transition preserves the conditional target when its acceptance ratio is correct, but a single transition is not generally an independent draw from that target. Repeated proposals may be rejected, and the resulting chain can move slowly if the chosen proposal is poorly suited to the conditional distribution. The acceptance formula ensures the intended invariant distribution; it does not by itself guarantee that a finite run has adequately represented it.