Inference reasons from observations toward unknown quantities through a chain of assumptions. Keeping each link visible prevents a good optimizer from being mistaken for a good scientific explanation.
The generative chain¶
A physical model maps parameters and state to ideal observables. A measurement model maps those observables to possible recorded data, including calibration, selection, censoring, and noise. Together they define a likelihood: the probability model for data conditional on parameters and assumptions.
A prior represents information or regularization supplied before the current data. Bayes’ rule combines prior and likelihood into a posterior. A posterior predictive distribution propagates posterior uncertainty into possible observations. A nuisance parameter affects the data model without being the main scientific target; ignoring it does not make its effect disappear.
Information has several meanings¶
Retained scientific detail, stored data volume, Shannon information, and information about parameters are related but not interchangeable. Compression can reduce storage while retaining a model-relative sufficient statistic, or it can destroy the exact structure needed to distinguish models.
Once discarded information is absent from the data product or forward map, no optimizer or sampler can reconstruct it without new assumptions. High Fisher information can imply narrow local uncertainty under a model, while the same analysis remains biased under a misspecified model. A precise posterior is not proof of an adequate model.
Identifiability before computation¶
Parameters are identifiable only to the extent that distinct values produce distinguishable predicted data under the experiment. Flux and distance, stellar radius and temperature, or calibration and amplitude can create degeneracies. Priors may regularize those directions, but they do not retroactively add information to the likelihood.
Model checking¶
Posterior predictive checks, residual structure, held-out prediction, and sensitivity to assumptions ask whether model-generated data resemble relevant features of observations. Passing one check supports only the feature and domain tested. Failure can identify useful missing structure rather than merely a bad fit.
Predict¶
Draw the generative chain from parameters and state to observable and recorded data. Mark nuisance quantities, selection effects, assumed independence, and which parameter combinations may be degenerate.
Compute¶
Evaluate the forward and measurement models separately before combining them. Inspect likelihood geometry, parameterization, prior influence, and numerical diagnostics rather than retaining only a posterior summary.
Audit¶
Use simulated recovery, prior predictive checks, posterior predictive checks, held-out observations, residual structure, and alternative parameterizations. Test whether conclusions survive scientifically plausible model changes.
State the warranted claim¶
State conclusions conditional on the model, measurement process, prior, selection, and validated computational procedure. Separate “identified by the likelihood” from “constrained after the prior.”
Misconception check¶
Bayes’ rule does not validate the likelihood. A narrow posterior need not mean the parameter is physically well determined, and optimization cannot recover information the model or data representation discarded.
Continue to Sensitivity, conditioning, and identifiability and What is a model?.