Statistical inference is using data analysis to infer properties of an underlying probability distribution

Statistical inference is "the process of using data analysis to infer properties of an underlying probability distribution." It involves making propositions about a population based on sample data, typically through hypothesis testing and parameter estimation.

Statistical inference generalizes logical inference by incorporating probability: rather than demanding certainty, it licenses conclusions that are likely or credible given the data — and it provides quantitative measures of that credibility.

Two main traditions

Frequentist inference "calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution." Parameters are treated as fixed but unknown quantities. The core method is maximum likelihood estimation (MLE). Frequentist methods are viewed as more objective because they rely on repeated sampling rather than prior beliefs, and are standard for large-scale hypothesis testing and regression analysis.

Bayesian inference uses "the available posterior beliefs as the basis for making statistical propositions." Bayes' Theorem updates the probability of a hypothesis as evidence accumulates, yielding a posterior distribution that combines a prior with the likelihood of observed data. Bayesian methods are considered subjective (the prior must be chosen) but automatically provide optimal decisions in a decision-theoretic framework. They are especially useful when data is scarce or when prior knowledge is genuinely available.

A critical terminological divergence in machine learning

Wikipedia's entry on statistical inference flags a notable shift: in machine learning, "the term 'inference' shifts meaning — it refers to 'making a prediction, by evaluating an already trained model.'" Building the model is called training rather than inference, "reversing typical statistical usage."

This divergence is consequential. A statistician doing Bayesian inference is performing inference about parameters. An AI engineer doing "inference" is performing a forward pass through frozen parameters. The word is used by both communities, but they mean opposite ends of the same pipeline. See claim-ai-inference-means-running-a-model for the engineering sense and claim-inference-logical-types for the logical root both senses share.

See also: claim-inference-logical-types, claim-ai-inference-means-running-a-model, claim-inference-word-etymology