Bayesian statistics

Bayesian statistics represents uncertainty with probability distributions and updates prior distributions using observed data and Bayes’ theorem.

Bayes’ theorem: A probability identity that relates conditional probabilities and underlies Bayesian updating. It gives the mathematical rule for turning prior probabilities and likelihoods into a posterior.
Probability distribution: A mathematical description assigning probabilities or probability densities to possible outcomes. Bayesian analyses express uncertainty about data and unknown quantities through distributions.
Bayesian network: A directed graphical model representing conditional dependencies among random variables. It applies Bayesian probability to complex systems of interdependent variables.
Frequentist statistics: A statistical framework that interprets probability through long-run frequencies of repeated events. It treats unknown parameters as fixed rather than assigning them prior probability distributions.
Prior sensitivity analysis: An examination of how inferential results change under alternative prior distributions. It tests whether conclusions depend strongly on subjective or weakly supported prior choices.
Prior probability: A probability distribution representing uncertainty about a quantity before incorporating the current data. It encodes assumptions or previous evidence before the observed data enter the analysis.
Conditional probability: The probability of an event given that another event is known to have occurred. Bayesian updating is built around probabilities conditioned on observed evidence.
Kalman filter: An algorithm that estimates a changing hidden state from noisy sequential measurements. Its Bayesian interpretation updates state estimates as each new measurement arrives.
Maximum likelihood estimation: A method that selects parameter values maximizing the likelihood of observed data. It uses the likelihood without combining it with a prior distribution.
Bayesian nonparametrics: Bayesian methods that place probability distributions over models with flexible or unbounded complexity. It relaxes fixed-dimensional assumptions while retaining probabilistic updating.

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