Bayesian Model Averaging

You've fit five plausible factor models for a stock's expected return, and each gives a slightly different answer for next month's forecast. Picking the single best-fitting one and reporting only its forecast throws away a real fact: you're genuinely not certain which of the five is right, and that uncertainty should show up somewhere in the final number. Bayesian model averaging (BMA) is the technique for combining several models into one forecast that honestly reflects not knowing which model is correct — rather than pretending the winner is the truth.

An analogy: polling several analysts instead of trusting the loudest one

Suppose you ask five analysts to independently forecast a stock's next-quarter earnings. Rather than picking whichever analyst has the best track record and using only their number, a smarter approach is a weighted average of all five forecasts, where each analyst's weight reflects how much their track record and reasoning deserve to be trusted given everything you know. An analyst who's been reliably close gets more weight; one who's been erratic gets less, but even the erratic one still contributes something if their reasoning has occasionally been useful. The blended forecast is more robust than betting on any single analyst, because it doesn't collapse to zero the moment your favorite analyst has an off quarter — the weighting adapts continuously as track records update.

The maths, one symbol at a time

Given a set of candidate models $M_1, \dots, M_K$ and data $D$, Bayesian model averaging computes a forecast for a new quantity $\Delta$ as:

$$p(\Delta \mid D) = \sum_{k=1}^{K} p(\Delta \mid M_k, D)\, p(M_k \mid D)$$

In words: take each model's own forecast, $p(\Delta \mid M_k, D)$, and average them together weighted by $p(M_k \mid D)$ — the posterior probability that model $k$ is the right one, given the data. Each $p(M_k \mid D)$ is computed from the model's marginal likelihood (see marginal likelihood) combined with a prior probability on the model itself, via Bayes' theorem applied at the level of whole models rather than parameters:

$$p(M_k \mid D) \propto p(D \mid M_k)\, p(M_k)$$

A model that explained the data well gets a large weight in the final average; a model that explained it poorly gets a small one, and models can even get weight zero without being formally discarded, just outvoted.

Worked example 1: averaging two return forecasts

Suppose two models for a factor's next-month return: $M_1$ forecasts $+2\%$ with posterior probability $p(M_1 \mid D) = 0.7$, and $M_2$ forecasts $-1\%$ with posterior probability $p(M_2 \mid D) = 0.3$ (these sum to 1 across the two candidates considered). The BMA forecast is:

$$0.7 \times 2\% + 0.3 \times (-1\%) = 1.4\% - 0.3\% = 1.1\%$$

The blended forecast, $1.1\%$, sits between the two individual forecasts, pulled toward the more-trusted model but not entirely dismissing the other. If instead you'd simply picked "the best model" (M1, at 70% posterior weight) and used only its $+2\%$, you'd be implicitly claiming 100% confidence in a model that only actually earned 70% — overstating certainty by exactly the 30% you swept under the rug.

Worked example 2: model-averaged uncertainty is wider, not just a different center

Suppose $M_1$'s forecast has its own 90% interval of $[-1\%, 5\%]$ around $+2\%$, and $M_2$'s has $[-4\%, 2\%]$ around $-1\%$. The BMA distribution isn't a single narrow interval around $1.1\%$ — it's a mixture of the two models' distributions, weighted 70/30, and mixtures of different-centered distributions are always at least as wide as either alone, because part of the spread comes from disagreement between models on top of each model's own uncertainty. A risk manager relying on the BMA interval sees a wider, more honest range, reflecting that "I'm not sure which model is right" is itself a source of risk.

Picture two of these bell curves at different centers, blended with weights 0.7 and 0.3 — the resulting mixture is the shape BMA produces, wider and possibly bimodal compared to either curve alone.

What this means in practice

BMA is used when several structurally different models are all defensible and none clearly dominates — competing factor specifications, competing regime models, or competing priors on a signal's decay rate — and picking just one would overstate confidence. It's especially valuable for risk management, where the width added by genuine model disagreement is often the difference between a plausible-looking VaR number and one that misses a real source of uncertainty.