Implicit assumptions and model expansion: an example#

Part 2 · Stage 6 · 🔍 Model Checking & Comparison · Lesson 049 of 144 · intermediate

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Important

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The assumptions you forgot you made#

Every model states some assumptions explicitly — the likelihood, the prior — and smuggles in others implicitly, in the very act of writing it down. Implicit assumptions are dangerous precisely because they are invisible: nothing in the output flags them, and a posterior predictive check will not test a feature the model was never asked about. Expansion is how you drag them into the open.

Inventory the silences#

Take the eight-schools model of Stage 5, an analysis that passed its checks:

\[y_j \mid \theta_j \sim \mathrm{N}(\theta_j, \sigma_j^2), \qquad \theta_j \mid \mu, \tau \sim \mathrm{N}(\mu, \tau^2) .\]

Four assumptions were never stated. That the school effects are normally distributed around \(\mu\) — no outlying school with a genuinely different programme. That the standard errors \(\sigma_j\) are known exactly, rather than estimated from modest samples. That the schools are exchangeable — nothing about size, funding or curriculum predicts the effect. And that each school’s estimate is unbiased for its own effect, so no selective reporting occurred.

Expand to test#

Each silent assumption becomes a parameter, and the posterior for that parameter tells you whether it mattered. Replace the normal population with a \(t\):

import pymc as pm
with pm.Model():
    mu  = pm.Normal("mu", 0, 5)
    tau = pm.HalfCauchy("tau", 5)
    nu  = pm.Gamma("nu", 2, 0.1)            # was: implicitly infinity (normal)
    eta = pm.StudentT("eta", nu=nu, mu=0, sigma=1, shape=8)
    theta = pm.Deterministic("theta", mu + tau * eta)
    pm.Normal("y", theta, sigma=sigma_j, observed=y)

If the posterior for \(\nu\) piles up at large values, the normal assumption was harmless and you have earned the right to it. If \(\nu\) concentrates near 3, one school is an outlier, and the original analysis over-shrank it. Similarly, letting \(\theta_j\) depend on a school-level covariate tests exchangeability; giving \(\sigma_j\) its own uncertainty tests the known-variance assumption.

The general move#

The pattern is: name the assumption, embed it in a family, fit, inspect the posterior of the embedding parameter. This converts an untestable premise into an estimated quantity. Where the data are informative the expansion settles the question; where they are not, the posterior for the new parameter simply reproduces its prior — which is itself the answer: the data cannot tell, and the conclusion rests on judgement rather than evidence.

Two honest limits#

Not every implicit assumption can be expanded away. The claim that \(\sigma_j\) describes the right sampling process, or that no results were suppressed before publication, lies outside the likelihood — no parameter inside the model can detect it. And each expansion costs identifiability: with \(J = 8\), \(\nu\) will be weakly determined, so the check reports “no evidence against” rather than “assumption verified”. Model expansion widens what the data can speak to; it does not make them omniscient.

Hint