Bayesian Model Averaging for Human Extrapolation

Why Average Across Translational Models?

A human prediction can depend on structural choices: for example, which cross-species scaling relationship is used or how a translational parameter is mapped. If the available evidence does not clearly favor one defensible structure, selecting a single model can conceal an important source of uncertainty. Bayesian model averaging (BMA) represents uncertainty across a stated set of candidate models by weighting each model’s human predictive distribution according to its posterior model probability.

The aim is to average predictions for a common human quantity, such as clearance in L/h or a biomarker response at a specified time. This does not require averaging parameters that have different meanings under different structures. It does require defining the candidate models, their priors, the observed data, and the target prediction precisely.

Definition: Let $M_1,\ldots,M_K$ be a specified set of candidate cross-species models and let $D$ denote the observed translational data. The posterior model probability for $M_k$ is $w_k=P(M_k\mid D)$, conditional on the stated model space, prior probabilities, and data. The weights are nonnegative and sum to one. For a human predictive quantity $Y_H$, the model-averaged predictive distribution is the weighted mixture of the model-specific posterior predictive distributions: $$ p(y_H\mid D)=\sum_{k=1}^{K}p(y_H\mid D,M_k)P(M_k\mid D). $$

These weights are posterior probabilities of the candidate models conditional on the specified model space, data, and priors. They do not establish that a candidate is literally true or that the candidate set is adequate: structures outside the set receive no weight. This qualification matters especially in extrapolation, where animal data may be compatible with several models but provide limited evidence about human behavior.

Calculating Posterior Model Probabilities

Bayes’ rule gives each model’s posterior probability as its prior probability multiplied by its marginal likelihood, then normalized across the candidate set. The marginal likelihood averages the likelihood over that model’s parameter prior; it is not generally the likelihood evaluated at a best-fitting parameter estimate.

P(M_k\mid D)= \frac{p(D\mid M_k)P(M_k)} {\sum_{j=1}^{K}p(D\mid M_j)P(M_j)}, \qquad p(D\mid M_k)=\int p(D\mid \theta_k,M_k)p(\theta_k\mid M_k)\,d\theta_k.

Here $\theta_k$ denotes the parameters under model $M_k$. The model prior probabilities $P(M_k)$ must sum to one, and the marginal likelihood must be computed under the stated parameter prior. Because it integrates over parameter values rather than rewarding only the best-fitting point, the evidence reflects both fit and the prior distribution of plausible parameter values.

Worked Example: Updating Two Model Probabilities

Suppose two candidate models map animal clearance to human clearance using different scaling structures. For illustration, assign each prior probability $0.50$. Assume their marginal likelihoods for the hypothetical animal data are $p(D\mid M_1)=0.20$ and $p(D\mid M_2)=0.05$, in the same data scale.

The unnormalized posterior weights are $0.20(0.50)=0.10$ for $M_1$ and $0.05(0.50)=0.025$ for $M_2$. Their sum is $0.125$, so

P(M_1\mid D)=\frac{0.10}{0.125}=0.800, \qquad P(M_2\mid D)=\frac{0.025}{0.125}=0.200.

Thus $M_1$ has an 80% posterior probability within this stated two-model set, given these priors and data; $M_2$ has 20%. The result is not an unconditional claim that $M_1$ is true, and it does not account for structures omitted from the candidate set. If the prior model probabilities or parameter priors changed, the posterior weights could change as well.

Combining Predictive Distributions

Within each model, $p(y_H\mid D,M_k)$ should include the uncertainty represented by that model’s parameter posterior and any relevant future individual or observation variability. The BMA mixture then adds uncertainty associated with not knowing which candidate structure to use. For a quantity with a finite mean and variance, the mixture mean is the weighted average of the model-specific predictive means. Its variance has a within-model component and a between-model component.

E(Y_H\mid D)=\sum_{k=1}^{K}w_k\mu_k, \qquad \operatorname{Var}(Y_H\mid D) =\sum_{k=1}^{K}w_k\left[\sigma_k^2+(\mu_k-\bar{\mu})^2\right], \qquad \bar{\mu}=\sum_{k=1}^{K}w_k\mu_k.

In this expression, $\mu_k=E(Y_H\mid D,M_k)$ and $\sigma_k^2=\operatorname{Var}(Y_H\mid D,M_k)$. The $\sigma_k^2$ terms represent predictive uncertainty within each candidate model. The squared deviations $(\mu_k-\bar{\mu})^2$ capture how much the model-specific predictions differ from the average. Ignoring the latter term discards structural uncertainty.

The variance identity follows from conditioning on the model. First, the conditional expectation of $Y_H$ is $\mu_k$ when $M_k$ is specified. The law of total variance therefore gives the average conditional variance plus the variance of the conditional means:

\operatorname{Var}(Y_H\mid D) =E\{\operatorname{Var}(Y_H\mid D,M)\mid D\} +\operatorname{Var}\{E(Y_H\mid D,M)\mid D\}.

The first term is $\sum_k w_k\sigma_k^2$, and the second is $\sum_k w_k(\mu_k-\bar{\mu})^2$. Expanding the expression also gives $\sum_k w_k(\sigma_k^2+\mu_k^2)-\bar{\mu}^{\,2}$. This derivation shows why averaging only the model-specific variances is insufficient when predictions differ across structures.

Worked Example: Mean and Variance of a Human Clearance Prediction

Suppose two hypothetical models give posterior predictive distributions for human clearance $CL_H$: $M_1$ predicts a mean of $4$ L/h and variance $1$ (L/h)$^2$, while $M_2$ predicts a mean of $6$ L/h and variance $4$ (L/h)$^2$. Suppose the posterior model weights are $w_1=0.70$ and $w_2=0.30$.

The model-averaged mean is

\bar{\mu}=0.70(4)+0.30(6)=2.8+1.8=4.60\ \text{L/h}.

The average within-model variance is $0.70(1)+0.30(4)=1.90$ (L/h)$^2$. The between-model variance is $0.70(4-4.60)^2+0.30(6-4.60)^2=0.252+0.588=0.840$ (L/h)$^2$. Therefore,

\operatorname{Var}(CL_H\mid D)=1.90+0.840=2.74\ \text{(L/h)}^2, \qquad SD(CL_H\mid D)=\sqrt{2.74}=1.655\ \text{L/h}.

The between-model term is part of the prediction uncertainty even though each model’s individual predictive variance is already included. Reporting only the within-model variance of $1.90$ (L/h)$^2$ would understate uncertainty in this mixture.

Prediction Intervals Belong to the Mixture

A model-averaged prediction interval is obtained from quantiles of the mixture distribution, not by averaging the endpoints of the model-specific intervals. For an equal-tailed 95% interval, the lower and upper endpoints are the 2.5th and 97.5th percentiles of $p(y_H\mid D)$. Except in special cases, the mixture is not itself normal, even when every model-specific predictive distribution is normal. It may be asymmetric or have separated regions of high probability.

Worked Example: Finding a Mixture Prediction Interval

Continue with the hypothetical clearance example, and suppose the model-specific predictive distributions are normal: $CL_H\mid D,M_1\sim N(4,1)$ and $CL_H\mid D,M_2\sim N(6,4)$, where the second argument is the variance in (L/h)$^2$. The mixture has weights $0.70$ and $0.30$. Its cumulative distribution function is

F(x)=0.70\Phi\left(\frac{x-4}{1}\right) +0.30\Phi\left(\frac{x-6}{2}\right),

where $\Phi$ is the standard normal cumulative distribution function. Solving $F(x)=0.025$ and $F(x)=0.975$ numerically gives approximate quantiles of $2.05$ L/h and $8.77$ L/h. Thus the equal-tailed 95% model-averaged prediction interval is approximately $[2.05,8.77]$ L/h. It describes the central 95% of the specified mixture’s predictive distribution, conditional on the data, priors, candidate models, and their predictive assumptions.

For comparison, a normal distribution with the mixture mean and variance would give a moment-matched interval of approximately $4.60\pm1.96(1.655)$, or $[1.36,7.84]$ L/h. That is not the mixture’s equal-tailed interval: matching two moments does not make the mixture normal. Nor should one average the two model-specific interval endpoints and call the result a mixture interval.

Use the Mixture for Decisions and Check Its Construction

BMA can also answer predictive questions directly. For a threshold, calculate the weighted average of the model-specific predictive probabilities. This preserves meaningful differences in model predictions rather than reducing them to an average parameter or a single selected structure.

Worked Example: Probability of Clearance Exceeding a Threshold

Consider another hypothetical prediction with $P(M_1\mid D)=0.70$ and $P(M_2\mid D)=0.30$. Suppose the model-specific predictions are $CL_H\mid D,M_1\sim N(6,1)$ and $CL_H\mid D,M_2\sim N(9,2.25)$, with clearance in L/h. For the threshold $8$ L/h, standardize under each model:

P(CL_H>8\mid D,M_1)=1-\Phi\left(\frac{8-6}{1}\right) =1-\Phi(2)\approx0.0228,

P(CL_H>8\mid D,M_2)=1-\Phi\left(\frac{8-9}{1.5}\right) =\Phi(0.667)\approx0.7475.

The model-averaged probability is $0.70(0.0228)+0.30(0.7475)=0.01596+0.22425=0.2402$. Under the specified mixture, the predictive probability that human clearance exceeds $8$ L/h is approximately 24.0%. It is not the probability that either model is true; it is a prediction probability obtained by averaging the model-specific event probabilities using the posterior model weights.

Before interpreting an average, check that every candidate predicts the same quantity on the same scale, at the same time point and under the same dosing or covariate conditions. Model-specific parameter names need not correspond, so averaging parameters across structures can be meaningless even when averaging their predictions is sensible. Check that the likelihoods and parameter priors used for marginal likelihoods are proper and comparable, and that posterior predictive calculations include the uncertainty intended for the target quantity.

Several warning signs deserve attention. A weight near one may reflect genuinely strong relative evidence within the stated model space, but it can also be sensitive to prior model probabilities or parameter priors. Similar weights do not imply similar human predictions; inspect the model-specific predictive distributions. Conversely, similar predictions may produce a stable average even when model weights vary. Numerical instability in marginal likelihood estimates or predictive quantiles can also distort results, so assess integration or simulation precision. These checks identify fragility; they do not demonstrate that the candidate set spans all plausible biology.

Approach	What it reports	Main limitation
Choose the highest-posterior model	The prediction under one candidate, often the model with the largest $P(M_k\mid D)$	Discards predictive contributions and uncertainty from the other candidates
Bayesian model averaging	A predictive mixture weighted by posterior model probabilities	Depends on the specified model space and priors; cannot protect against omitted structures
Equal-weight predictive averaging	A mixture assigning the same weight to each model	Does not use posterior model probabilities unless equal weights are deliberately justified

BMA is most useful when candidate structures are scientifically defensible and uncertainty across them is relevant to the human prediction. It does not repair a poor likelihood, a misspecified candidate model, weakly informed extrapolation, or a missing structure. A posterior model probability is properly read as conditional on the specified model space, priors, and data; it is neither an unconditional guarantee of literal truth nor evidence that omitted models are impossible.

Key takeaway: Average model-specific predictive distributions using posterior model probabilities. The resulting mixture carries both within-model predictive uncertainty and between-model differences, while its interpretation remains conditional on the candidate set, priors, and observed data.

Check Your Understanding

Work through each question, then compare with the solutions below.

Two models have prior probabilities $0.60$ and $0.40$, and marginal likelihoods $0.10$ and $0.30$. What are their posterior model probabilities?
In one sentence, interpret a posterior model probability of $0.90$ while stating its conditioning and one limitation.
Two models have weights $0.50$ each, predictive means $10$ and $14$ mg/L, and predictive variances $4$ and $9$ (mg/L)$^2$. Find the model-averaged mean and variance.
Why is an average of model-specific 95% interval endpoints generally not a model-averaged 95% prediction interval?

Solutions. (1) The unnormalized weights are $0.10(0.60)=0.060$ and $0.30(0.40)=0.120$; dividing by their sum, $0.180$, gives $P(M_1\mid D)=0.3333$ and $P(M_2\mid D)=0.6667$. (2) It is a 90% posterior probability for that candidate conditional on the stated model space, data, and priors; it does not guarantee unconditional or literal truth, and the candidate set may omit plausible structures. (3) The mean is $0.50(10)+0.50(14)=12$ mg/L. The variance is $0.50[4+(10-12)^2]+0.50[9+(14-12)^2]=0.50(8)+0.50(13)=10.5$ (mg/L)$^2$. (4) Interval endpoints are quantiles, and quantiles of a weighted mixture are found from its combined cumulative distribution; they are not generally weighted averages of the component quantiles. The mixture’s shape, including any asymmetry or separated modes, determines its interval.