BayesAnsatz

BayesAnsatz is a framework for statistical inference and decision making based on Bayes' theorem. It treats unknown quantities as random variables with probability distributions. The central idea is to combine a prior distribution p(theta) with observed data D via the likelihood p(D|theta) to obtain the posterior distribution p(theta|D) ∝ p(D|theta) p(theta).

The posterior expresses updated beliefs after seeing data, and predictions for new data are obtained from the predictive distribution p(y|D) = ∫ p(y|theta) p(theta|D) dtheta. This allows uncertainty about parameters to be propagated into predictions and decisions.

Implementation requires specifying a model: prior, likelihood, and sometimes hierarchical structure. Conjugate priors yield analytic posteriors in some cases; otherwise numerical methods such as Markov chain Monte Carlo (MCMC), variational inference, and Laplace approximation are used. The framework supports a wide range of data types and complex models, including multilevel and latent-variable structures.

Bayesian model comparison uses Bayes factors or posterior model probabilities to weigh competing models, and Bayesian model averaging can combine predictions across models to account for model uncertainty. The BayesAnsatz is widely used in machine learning, statistics, epidemiology, finance, and cognitive science.

Advantages include coherent uncertainty propagation, principled incorporation of prior information, and automatic regularization through the prior. Disadvantages include sensitivity to priors, dependence on correct model specification, and higher computational cost compared with some frequentist approaches. Variants such as hierarchical Bayes, empirical Bayes, and Bayesian networks expand the range of applications and methods within the BayesAnsatz.