Summary of Elementary Modeling and Computation#

Part 1 · Stage 3 · 🧮 Multiparameter Models · Lesson 027 of 144 · beginner

◀ Previous · Example: Bayesian analysis of a bioassay experiment (logistic, nonconjugate) · Next · Normal Approximations to the Posterior Distribution ▶ · ↑ Section

Important

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What Part I established#

Everything so far rests on a single move: write a full probability model \(p(\theta, y) = p(\theta) p(y \mid \theta)\), condition on the data, and summarise. The stages built this out — one parameter, then several — while the arithmetic stayed the same. Three ideas do most of the work.

The recurring three#

The posterior is a compromise. Beta–Binomial, Normal–Normal, Poisson–Gamma, the multivariate normal: in each, the posterior mean is a weighted average of prior and data, with weights given by effective sample size or precision. Data eventually win; the prior matters most when they are scarce.
Nuisance parameters are integrated, not fixed. Marginalising honestly is what turns a normal into a \(t\). Plug-in estimates discard the second term of the variance decomposition and understate uncertainty.
Priors are choices, and they are checkable. Informative (pseudo-data), noninformative (invariant, possibly improper), weakly informative (regularising) — each states something, and prior predictive simulation reveals what.

Two computational lessons#

Conjugacy is a convenience. It gives exact, instant, interpretable updates, and it will return in Part III as the engine inside Gibbs samplers. But it exists for algebraic reasons, not scientific ones, and the bioassay showed how quickly a realistic model escapes it.

Draws are enough. Every quantity of interest — a mean, an interval, \(\Pr(\beta > 0 \mid y)\), the LD50, a contrast — is a summary of posterior draws. Functions of draws are draws from the function’s posterior; dropping a column is marginalisation. This is why the simulation-based workflow scales where the algebra cannot.

# the entire Part I workflow, in five lines
import pymc as pm, arviz as az
with pm.Model():
    theta = pm.Beta("theta", 1, 1)                 # prior
    pm.Binomial("y", n=10, p=theta, observed=8)    # likelihood
    idata = pm.sample()                            # condition on data
az.summary(idata)                                  # summarise; then check the fit

What comes next#

Part I answered how to compute a posterior when the model is given. Three questions remain, and they organise the rest of the course: is the model any good? (checking and comparison, Part II); what if the posterior has no formula? (approximation and MCMC, Part III); and how do we build models with structure — groups, predictors, nonlinearity? (hierarchies, regression, nonparametrics, Parts IV–V). The next stage begins the answer by asking what happens to a posterior as data accumulate.

Hint