Blocked MCMC for Correlated Mechanistic Parameters

Why Update Correlated Parameters Together?

In QSP calibration, distinct parameter combinations can produce similar predicted biomarker trajectories. For example, a change in binding affinity may be offset by a change in target turnover, leaving much of the model output nearly unchanged. The posterior can then have a narrow, tilted region of high density: moving one parameter alone may leave that region, even though moving several parameters together along it is plausible.

Metropolis–Hastings can update one parameter at a time or propose changes to several parameters jointly. A block update changes a selected group of coordinates in one proposal while holding the remaining coordinates fixed. Its purpose is not simply to update more parameters at once; a well-designed block can move in directions that respect posterior dependence.

The density used in that rule must be the posterior density in the coordinates actually sampled. If parameters are transformed, any required change-of-variables factor belongs in that density. For instance, when sampling a log parameter instead of a positive parameter, the target density must reflect the prior and transformation used in the model.

Acceptance Rule for a Block

Write the proposal density for a block as \(q(\theta'_B\mid\theta_B,\theta_{-B})\). With all coordinates outside the block fixed, the acceptance probability is

The proposal densities in the numerator and denominator account for possible asymmetry. For a symmetric random-walk proposal, such as adding a zero-mean multivariate normal increment with a fixed covariance matrix, the forward and reverse proposal densities are equal. The acceptance probability then reduces to \(\min\{1,\pi(\theta'\mid y)/\pi(\theta\mid y)\}\). In computation, the log form is often safer: for a symmetric proposal, accept when \(\log U < \min\{0,\log\pi(\theta'\mid y)-\log\pi(\theta\mid y)\}\), where \(U\) is drawn uniformly from \((0,1)\).

A block can be updated by itself, leaving the other parameters unchanged for that step. A sequence of such updates, one block after another, is valid when each update uses its correct conditional target and Metropolis–Hastings acceptance rule. Updating blocks does not make the parameters independent; it is a way to construct transitions that can move more effectively through a dependent posterior.

Choosing Blocks From Posterior and Mechanistic Structure

A practical starting point is to examine posterior dependence from preliminary sampling or other posterior-geometry work. Correlations are useful clues, but they are not a complete description of nonlinear dependence. Mechanistic knowledge can also suggest which parameters may compensate for one another. For example, binding and target-turnover rates can jointly influence the duration and magnitude of target engagement.

Blocks should usually be small enough to propose and evaluate efficiently, but large enough to include important dependencies. A single very large block may be difficult to scale and may have low acceptance if its proposal covariance is poorly estimated. Conversely, leaving strongly coupled parameters in separate blocks can require small, cautious moves. These are design trade-offs to assess by examining proposed movement and acceptance, not rules that can be decided from parameter names alone.

Why Joint Proposals Can Help: A Short Derivation

Consider a standardized bivariate normal target with correlation \(\rho\). Its density, apart from a constant, is

Assume a symmetric random-walk proposal. At the current point \((0,0)\), compare a joint proposal to \((a,a)\) with a single-coordinate proposal to \((a,0)\). Because the proposal is symmetric, the acceptance ratio is the target density ratio. At \((a,a)\), the quadratic expression in the exponent is \(2a^2(1-\rho)\); at \((a,0)\), it is \(a^2\). Thus the log target-density changes are

For positive \(\rho\) near one, the joint move follows the direction of dependence and incurs a smaller density penalty than the single-coordinate move of the same size. This calculation explains a possible advantage of blocking. It does not prove that every block proposal will be efficient: proposal scale, orientation, posterior shape, and computational cost still matter.

Computing a Block Acceptance Probability

For a symmetric block random walk, a proposal can be written as \(\theta'_B=\theta_B+\varepsilon\), with \(\varepsilon\) drawn from a multivariate distribution symmetric about zero. The proposal covariance determines the scale and orientation of the proposed changes. The next example shows the acceptance calculation directly from log posterior values.

Diagnostics, Alternatives, and Limitations

A useful block design should produce substantive movement in the parameters it updates, not merely a high acceptance rate. In practice, inspect proposals and accepted states on the sampled parameter scale, and check whether paired parameters move together in the expected directions. A very low acceptance rate may indicate proposals that are too large, poorly oriented, or incompatible with the target’s shape. A very high acceptance rate accompanied by tiny changes can indicate proposals that are too conservative. Neither rate alone establishes that the chain explores the posterior adequately.

Single-parameter updates are a useful alternative when posterior dependence is weak, when parameters are easy to update separately, or when estimating a joint proposal covariance is impractical. They are also easier to implement and can isolate which coordinate causes frequent rejections. Their disadvantage is that, in a strongly correlated posterior, useful movement may require many small coordinate changes.

Joint blocks can address that problem, but introduce their own limitations. As the block grows, the proposal covariance becomes harder to estimate and scale, and a poor proposal can change several parameters in an implausible combination. Very strong nonlinear dependence may not be captured by a fixed covariance matrix. Blocks also do not repair structural non-identifiability or make a poorly specified likelihood informative. They are a proposal design choice, not a substitute for examining the model and the information in the data.

• Check proposal orientation. If accepted proposals repeatedly move away from the observed posterior dependence, reconsider the covariance or the block membership.
• Check parameter scaling. Coordinates with very different scales can make a shared proposal step inappropriate; choose scales in the coordinates being sampled.
• Check block size. If joint proposals are frequently rejected, test whether a smaller block or a better-scaled covariance gives more useful movement.
• Compare movement, not acceptance alone. A proposal can be accepted often but make negligible changes, or be accepted less often while making useful moves.