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Bayesian linear regression model with custom joint prior distribution

The Bayesian linear regression
model object `customblm`

contains a log of the pdf of the
joint prior distribution of
(*β*,*σ*^{2}). The log pdf
is a custom function that you declare.

The data likelihood is $$\prod _{t=1}^{T}\varphi \left({y}_{t};{x}_{t}\beta ,{\sigma}^{2}\right)},$$ where
*ϕ*(*y _{t}*;

In general, when you create a Bayesian linear regression model object, it specifies the joint prior distribution and characteristics of the linear regression model only. That is, the model object is a template intended for further use. Specifically, to incorporate data into the model for posterior distribution analysis, pass the model object and data to the appropriate object function.

`PriorMdl = customblm(NumPredictors,'LogPDF',LogPDF)`

`PriorMdl = customblm(NumPredictors,'LogPDF',LogPDF,Name,Value)`

creates a Bayesian linear
regression model object (`PriorMdl`

= customblm(`NumPredictors`

,'`LogPDF`

',LogPDF)`PriorMdl`

) composed of
`NumPredictors`

predictors and an intercept, and sets
the `NumPredictors`

property. `LogPDF`

is a function representing the log of the joint prior distribution of
(*β*,*σ*^{2}).
`PriorMdl`

is a template that defines the prior
distributions and the dimensionality of *β*.

sets properties (except
`PriorMdl`

= customblm(`NumPredictors`

,'`LogPDF`

',LogPDF,`Name,Value`

)`NumPredictors`

) using name-value pair arguments.
Enclose each property name in quotes. For example,
`customblm(2,'LogPDF',@logprior,'Intercept',false)`

specifies the function that represents the log of the joint prior density of
(*β*,*σ*^{2}),
and specifies a regression model with 2 regression coefficients, but no
intercept.

`estimate` | Fit parameters of Bayesian linear regression model to data |

`simulate` | Simulate regression coefficients and disturbance variance of Bayesian linear regression model |

`forecast` | Forecast responses of Bayesian linear regression model |

`plot` | Visualize prior and posterior densities of Bayesian linear regression model parameters |

`summarize` | Distribution summary statistics of standard Bayesian linear regression model |

The `bayeslm`

function can create any supported prior model object for Bayesian linear regression.