Gaussianprior

Gaussian prior is a prior distribution used in Bayesian inference and machine learning, defined as a multivariate normal distribution over parameters or latent variables. If θ is a d-dimensional parameter vector, the prior is p(θ) = N(θ | μ0, Σ0), with mean μ0 and covariance Σ0. In the univariate case, p(θ) ∝ exp(- (θ − μ0)^2 / (2 σ0^2)). The prior encodes beliefs about plausible parameter values before observing data; μ0 expresses central tendency, while Σ0 controls uncertainty and correlations among components. A common special case is an isotropic prior p(θ) ∝ exp(-λ/2 ||θ − μ0||^2) with λ = 1/σ0^2.

In Bayesian linear models with Gaussian likelihood, the Gaussian prior is conjugate, so the posterior p(θ | data) is also Gaussian. This yields closed-form updates for the posterior precision and mean, leading to efficient inference. In practice, a Gaussian prior induces ridge-like regularization in maximum a posteriori estimates, linking Bayesian inference to L2 regularization.

Gaussian priors are also used as function-space priors, including Gaussian processes that define a distribution over functions where any finite collection of inputs has a multivariate Gaussian distribution. Choosing μ0 and Σ0 reflects prior beliefs and can be fixed or learned through hierarchical modeling or evidence maximization.

Limitations include sensitivity to the covariance choice and potential mismatch with the true parameter structure. Gaussian priors are foundational in probabilistic modeling and provide a tractable default when little is known about parameter values.