Bayesian Inference & Probabilistic Models

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Abstractions about updating probabilistic models with evidence through priors, likelihoods, posterior distributions, and hierarchical structure. They include Bayesian regression and networks, empirical and recursive methods, model reduction and comparison, uncertainty estimation, and Bayesian decision criteria.

23 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bayes classifier — The decision rule that assigns each feature vector to the class with greatest posterior probability, minimizing expected classification loss when the true class distributions and loss function are known.
• Bayesian inference in marketing — The application of prior-to-posterior probabilistic updating to marketing data, consumer models and decisions under uncertainty.
• Bayesian linear regression — A linear conditional model that combines a likelihood for outcomes with prior distributions over coefficients and noise parameters to obtain posterior inference and prediction.
• Bayesian model reduction — A method deriving evidence and posterior parameters for models with altered priors from a previously fitted full Bayesian model.
• Chauvenet's criterion — Flag a single extreme observation when, under a fitted normal-error model, the expected number of sample observations at least as far from the mean is below one half.
• Chinese restaurant process — An exchangeable partition process in which each arriving item joins an existing block in proportion to its size or starts a new block with parameter-controlled probability.
• Dynamic Bayesian network — A Bayesian network template replicated across time slices to represent probabilistic dependencies within and between successive states of a stochastic process.
• Empirical Bayes method — Estimate a shared prior distribution or its hyperparameters from the same ensemble of observations and then perform Bayesian-style shrinkage or posterior inference conditional on that estimate.
• False confidence theorem — Show that a continuous data-dependent additive probability distribution can, for some false assertion, assign arbitrarily high belief with high sampling probability, motivating assertion-wise validity checks.
• Gamma-minimax inference — A robust statistical decision rule that minimizes worst-case risk over a specified class Gamma of plausible prior distributions rather than committing to one prior.
• Generalised likelihood uncertainty estimation — A hydrological uncertainty framework that weights an ensemble of behaviorally acceptable model realizations using chosen likelihood-like measures and thresholds.
• Hidden Markov model — Model an observed sequence as emissions from an unobserved Markov state process, separating state transition dynamics from state-conditioned observation distributions.
• Hierarchical Dirichlet process — A Bayesian nonparametric prior for grouped data in which group-specific discrete distributions share a global random set of mixture components while retaining different group weights.
• Information field theory — A Bayesian statistical field theory for reconstructing continuous fields from incomplete noisy data using field priors and methods adapted from quantum field theory.
• Jeffreys prior — A Bayesian prior measure proportional to the square root of the Fisher-information determinant, constructed to remain invariant under smooth reparameterization.
• Marginal likelihood — The probability density of observed data under a Bayesian model after integrating the likelihood over the prior distribution of its parameters.
• Normal-inverse-gamma distribution — A four-parameter joint distribution in which a variance has an inverse-gamma law and a mean conditional on that variance is normal, conjugate for a normal model with unknown mean and variance.
• Observed information — The negative Hessian of a sample log-likelihood evaluated at a specified parameter value, measuring local realized curvature.
• Posterior probability — The probability distribution for an uncertain hypothesis or parameter after combining a prior distribution with observed-data likelihood through Bayes' rule.
• Probability of direction — Summarize a posterior effect's sign certainty as the larger of Pr(θ>0) and Pr(θ<0), ranging from one-half to one for continuous posteriors while deliberately not measuring magnitude or practical importance.
• Recursive Bayesian estimation — Sequential estimation of a changing hidden state by alternating model-based prediction with Bayesian updating from each new observation.
• Variable-order Bayesian network — A Bayesian-network model whose parent set for a variable can change according to the realized context of preceding variables.
• Widely applicable information criterion — A Bayesian predictive-fit criterion combining log pointwise posterior predictive density with a variance-based effective-complexity penalty.