Represent uncertainty before the current data.
Priors can encode previous evidence, domain constraints or deliberately weak information.
Side 120
Bayesian Inference
A framework for updating uncertainty when new evidence arrives, with explicit separation between prior assumptions, data-generating models and posterior conclusions.
prior→likelihood→posterior→prediction→decision
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16working concepts
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Bayesian inference is structured belief revision.
The posterior combines what was plausible before the data with how compatible the observed data are with competing parameter values.
Specify how data would arise under the model.
The likelihood is a model of evidence conditional on parameters, not the probability that the model is true.
Update uncertainty after observing data.
The posterior redistributes probability according to both prior structure and evidential fit.
Compare possibilities on a common probability scale.
The denominator ensures probabilities sum correctly and can be computationally difficult in complex models.
Bayesian answers inherit the assumptions of the model.
A posterior can be internally precise while the model is externally wrong.
Describe how observable data could be produced.
Thinking generatively exposes hidden assumptions about noise, dependence and sampling.
Share information across related units.
Multilevel models partially pool estimates rather than forcing complete independence or complete equality.
Represent unobserved structure explicitly.
Latent states can explain observed dependence while introducing identifiability challenges.
Ask what the model predicts before seeing data.
Implausible simulated outcomes reveal priors or likelihoods that encode unrealistic worlds.
Inference should be checked, not merely computed.
Posterior summaries are trustworthy only when the computation converges and the model can reproduce relevant features of the data.
Simulate replicated data from the fitted model.
Systematic mismatch reveals dimensions the model fails to capture.
Vary priors and assumptions.
Conclusions that change sharply under reasonable alternatives should be reported as fragile.
Check numerical stability.
Sampling diagnostics assess whether computational approximations explored the posterior adequately.
Evaluate long-run inferential behavior.
Intervals and probabilities should be interpreted with awareness of model assumptions and repeated-use properties.
Inference and action are separate steps.
A posterior describes uncertainty; a decision additionally requires values, losses and feasible actions.
Move from parameters to future outcomes.
Predictions integrate uncertainty rather than substituting a single best estimate.
State what errors cost.
Different asymmetric losses can rationally produce different actions from the same posterior.
Measure whether more evidence could change action.
Additional data matter when they are likely to alter an important decision.
Retain uncertainty across plausible models.
Averaging can reduce overconfidence when no single model dominates.
Bayes' rule is simple; modeling is not. Most of the real judgment lies in defining the model, choosing priors, checking fit and deciding what the posterior is actually allowed to support.