Tuning Random-Walk Metropolis Proposals

Why Proposal Tuning Matters

In a mechanistic calibration, a posterior distribution may have parameters with very different scales and strong dependence. A random-walk Metropolis proposal that is too small tends to be accepted but makes slow progress through that posterior. A proposal that is too large tends to be rejected, leaving the chain repeatedly at its current value. Tuning aims to make useful movement, not to maximize acceptance.

This tutorial assumes the Metropolis-Hastings update and posterior target have been introduced. Let \(\theta\) denote the vector of parameters on the scale used by the sampler, and let \(\pi(\theta)\) be its posterior density. In QSP calibration, evaluating \(\pi(\theta)\) may require solving an ODE model and evaluating an observation likelihood. Proposal tuning can therefore materially affect computational efficiency, even though it does not change the posterior target when the Metropolis-Hastings correction is applied correctly.

For numerical work, the log acceptance ratio is usually evaluated as \(\log \pi(\theta')-\log \pi(\theta)\). The proposal is accepted if this difference is nonnegative, and otherwise is accepted with probability \(\exp\{\log \pi(\theta')-\log \pi(\theta)\}\). The acceptance rate over a run is the number of accepted transitions divided by the number of attempted transitions, including rejected proposals in the denominator.

What a Tuning Change Can and Cannot Do

Proposal tuning changes how the chain explores the target, not the target itself. If a fixed, symmetric proposal is used with the correct Metropolis-Hastings acceptance probability, detailed balance holds:

Here \(q\) is the proposal density and \(\alpha\) is the acceptance probability. Symmetry gives \(q(\theta'\mid\theta)=q(\theta\mid\theta')\), so the displayed equality follows directly from the Metropolis-Hastings rule. This equality is the detailed-balance condition and implies that \(\pi\) is stationary for the transition. It does not establish that a particular finite chain has explored the posterior adequately.

A practical tuning run therefore separates two questions: does the chosen proposal produce useful movement, and does the resulting chain explore the relevant posterior region? Acceptance rates address the first question only indirectly. Trace plots and effective sample sizes (ESS) provide additional evidence about movement and autocorrelation, but none is a standalone proof of convergence.

Scale, Shape, and Starting Choices

For a one-parameter chain, the proposal standard deviation determines the step size. For several parameters, a covariance matrix controls both the relative step sizes and the directions of proposed movement. If the parameters have different units or orders of magnitude, tuning on raw scales can be inefficient. Sampling transformed or standardized parameters can make a single scale factor more interpretable. For example, positive rate constants are often represented on a log scale; bounded parameters may be represented on an appropriate unconstrained scale. The posterior density on the sampled scale must be the correct transformed density, including any required Jacobian.

A useful multivariate starting form is \(\Sigma_q=c^2S\), where \(S\) describes relative scales and correlations, and \(c\) is an overall multiplier. One may begin with a diagonal \(S\) based on plausible parameter scales, or estimate \(S\) from a pilot run. A common reference for a roughly Gaussian, high-dimensional target is \(c=2.38/\sqrt{d}\), with \(d\) the dimension, when \(S\) approximates the target covariance. This is an asymptotic benchmark under restrictive assumptions, not a universal optimum for nonlinear QSP posteriors. Neither that value nor any single acceptance-rate target replaces diagnostics.

Using Correlation in the Proposal

An isotropic proposal uses \(\Sigma_q=c^2I\), where \(I\) is the identity matrix. It proposes equally scaled changes in every coordinate, which can be inefficient if parameter scales differ. A diagonal proposal permits different marginal step sizes but still proposes changes without accounting for posterior correlation. A covariance-aware proposal can orient steps along correlated directions. Its covariance must remain positive definite so it defines a valid non-degenerate Gaussian proposal.

Pilot estimates can be noisy, especially when the chain has not explored the posterior, when parameters are weakly identified, or when the dimension is large relative to the number of pilot draws. A small ridge term can stabilize an estimated covariance: use \(S_\lambda=S+\lambda I\) for a positive \(\lambda\) chosen relative to the scale of \(S\). Check that the resulting matrix is positive definite and that its implied proposal scales are plausible. Do not assume a covariance estimated from a short, sticky trace is a reliable description of posterior geometry.

Read Acceptance Rates Alongside Traces and ESS

A trace plot displays sampled values in iteration order. A trace that barely moves can indicate overly small steps, overly large steps with frequent rejection, or a chain trapped in only one part of a complicated target. A trace that moves rapidly but repeatedly revisits a narrow range may still be highly autocorrelated. Inspect traces for the parameters that matter, and compare their ranges and movement across chains when multiple chains are available. A visually reassuring trace does not by itself establish convergence.

The effective sample size expresses the approximate information in an autocorrelated chain as the equivalent number of independent draws for estimating a mean. For a stationary chain of length \(N\), a basic expression is \(N_{\mathrm{eff}}\approx N/\tau\), where \(\tau=1+2\sum_{k=1}^{\infty}\rho_k\) and \(\rho_k\) is the lag-\(k\) autocorrelation. In practice, software estimates and truncates this sum because only a finite chain is available. ESS is parameter-specific: a proposal may work reasonably for one parameter and poorly for another.

A Practical Tuning Workflow and Its Limits

Tuning choices have limitations. A single overall scale can fail when parameter scales differ greatly. A diagonal covariance cannot align proposals with strong posterior dependence. A full covariance can be unstable if estimated from inadequate pilot output, and even a well-chosen covariance does not solve separated modes or other difficult posterior geometry. If low acceptance persists after sensible rescaling, check the likelihood calculation, parameter transformations, numerical ODE solver behaviour, and whether the chain is proposing outside the meaningful support. If traces remain sticky or ESS remains low, report that limitation rather than treating a preferred acceptance rate as evidence of adequate sampling.

During an adaptive tuning phase, proposal changes depend on the chain history. Arbitrarily continuing such adaptation during production can invalidate the usual fixed-kernel justification. A simple and transparent approach is to adapt during warmup and then hold the proposal fixed. More elaborate adaptive algorithms require conditions that are not supplied merely by observing a desirable acceptance rate.

Common Mistake

The main practical comparison is between proposal simplicity and its ability to reflect the target’s scales and shape. Isotropic proposals are easiest to set up but can waste computation when scales differ. Diagonal proposals address unequal scales but not correlation. Covariance-aware proposals can improve movement along correlated directions but depend on a credible, stable covariance estimate. These are alternatives within random-walk Metropolis; none removes the need for diagnostics or guarantees adequate exploration.