%% Set Hyperparameters of Normal-Inverse-Gamma Conjugate Prior Model
% Consider the multiple linear regression model that predicts the US real gross national product (GNPR) using a linear combination of industrial production index (IPI), total employment (E), and real wages (WR).
% \texttt{GNPR}_t=\beta_0+\beta_1\texttt{IPI}_t+\beta_2\texttt{E}_t+\beta_3\texttt{WR}_t+\varepsilon_t.
% For all t, \varepsilon_t is a series of independent Gaussian disturbances with a mean of 0 and variance \sigma^2.
% Suppose that the regression coefficients \beta=[\beta_0,...,\beta_3]^\prime and the disturbance variance \sigma^2 are random variables. Specifically,
% \beta|\sigma^2 \sim N_4\left({\rm M},\sigma^2 V\right). \rm M is a 4-by-1 vector of means, and V is a scaled 4-by-4 positive definite covariance matrix. Suppose you have prior knowledge that  {\rm M} = {\left[ {\begin{array}{*{20}{c}} { - 20}&4&{0.1}&2 \end{array}} \right]^\prime } and V is the identity matrix.
% \sigma^2 \sim IG(A,B). A and B are the shape and scale, respectively, of an inverse gamma distribution.
% These assumptions and the data likelihood imply a normal-inverse-gamma conjugate model.
% Create a normal-inverse-gamma conjugate prior model for the linear regression parameters. Specify the number of predictors p.

p = 3;
Mdl = bayeslm(p,'ModelType','conjugate','Mu',[-20; 4; 0.1; 2],'V',eye(4))

% Mdl is a conjugateblm Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance. bayeslm displays a summary of the prior distributions at the command line. For example, the elements of Positive represent the prior probability that the corresponding parameter is positive.
% By default, bayeslm sets the shape and scale to 3 and 1, respectively. Suppose you have prior knowledge that the shape and scale are 5 and 2.
%
% Set the prior shape and scale of \sigma^2 to their assumed values.

Mdl.A = 5;
Mdl.B = 2

% bayeslm updates the prior distribution summary based on the changes in the shape and scale.
% Copyright 2012 The MathWorks, Inc.