Statistics.FastBayes.Linear

This module gives an implementation of Bayesian linear regression, with the scale of the prior chosen by marginal likelihood.

The inputs for a Bayesian linear model are identical to those of a classical linear model, except that in addition to a design matrix and response, we must also specify a prior distribution on the weights and the noise. This leaves us with an open question of how these should be specified.

In his book Pattern Recognition and Machine Learning, Christopher Bishop provides details for an approach that simplifies the situation significantly, and allows for much faster inference. The structure of the linear model allows us to integrate the posterior over the weights, resulting in the marginal likelihood, expressed as a function of the prior precision and noise precision. This, in turn, can be easily optimized.

Documentation

data Fit

Instances

Show Fit

fit

Arguments

:: ([(Double, Double)] -> (Double, Double))

How to take the limit of the (α,β) sequence. A simple approach is, e.g., fit (!! 1000) x y

-> Matrix Double

The design matrix (each column is a feature)

-> Vector Double

The response vector

-> Fit

fit lim x y fits a Bayesian linear model to a design matrix x and response vector y. This is an iterative algorithm, resulting in a sequence (list) of (α,β) values. Here α is the prior precision, and β is the noise precision. The lim function passed in is used to specify how the limit of this sequence should be computed.

design :: Fit -> Matrix Double

The design matrix used for the fit.

response :: Fit -> Vector Double

The response vector used for the fit

priorPrecision :: Fit -> Double

The precision (inverse variance) of the prior distribution, determined by maximizing the marginal likelihood

noisePrecision :: Fit -> Double

The precision (inverse variance) of the noise

numEffectiveParameters :: Fit -> Double

The number of effective parameters in the model

logEvidence :: Fit -> Double

The log of the evidence, which is useful for model comparison (different features, same response)

mapWeights :: Fit -> Vector Double

The MAP (maximum a posteriori) values for the parameter weights

hessian :: Fit -> Matrix Double

The Hessian (matrix of second derivatives) for the posterior distribution