linearmodel: Compute the Posterior Distribution for a Linear Model In lestat: A Package for Learning Statistics

Description

Given a vector of data and a design matrix, describing how these data are thought to relate to some predictors in a linear model, the posterior for the parameters of this linear model is found, using a flat prior.

Usage

 1 linearmodel(data, design)

Arguments

 data A vector with data values. design A design matrix. The number of rows must be equal to the length of the data vector. The number of columns corresponds to the number of explanatory variables.

Details

If y_i is the i'th data value and β_j is the j'th unknown parameter, and if x_{ij} is the value in the i'th row and j'th column of the design matrix, then one assumes that y_i is normally distributed with exptectation

x_{i1}β_1 + x_{i2}β_2 + … + x_{ik}β_k

and logged standard deviation λ. The computed probability distribution is then the posterior for the joint distribution of

(β_1,β_2,…,β_k,λ)

.

Value

If k is the number of columns in the design matrix and if k>1, then the output is a multivariate Normal-ExpGamma distribution representing the posterior for the corresponding k values and the logged scale parameter in the linear model. If k=1, the output is a Normal-ExpGamma distribution representing the posterior.