Bayesian linear regression model with lasso regularization

The Bayesian linear regression model object `lassoblm`

specifies the
joint prior distribution of the regression coefficients and the disturbance variance
(*β*, *σ*^{2}) for implementing
*Bayesian lasso regression*
[1]. For *j* =
1,…,`NumPredictors`

, the conditional prior distribution of
*β _{j}*|

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 and feature selection, pass the model object and data to the appropriate object function.

`PriorMdl = lassoblm(NumPredictors)`

`PriorMdl = lassoblm(NumPredictors,Name,Value)`

creates a Bayesian linear regression
model object (`PriorMdl`

= lassoblm(`NumPredictors`

)`PriorMdl`

) composed of
`NumPredictors`

predictors and an intercept, and sets the
`NumPredictors`

property. The joint prior distribution of
(*β*, *σ*^{2}) is
appropriate for implementing Bayesian lasso regression [1]. `PriorMdl`

is a template that defines the prior distributions and specifyies the values of the
lasso regularization parameter *λ* and the dimensionality of
*β*.

sets properties (except
`PriorMdl`

= lassoblm(`NumPredictors`

,`Name,Value`

)`NumPredictors`

) using name-value pair arguments. Enclose each
property name in quotes. For example, `lassoblm(3,'Lambda',0.5)`

specifies a shrinkage of `0.5`

for the three coefficients (not the
intercept).

`estimate` | Perform predictor variable selection for Bayesian linear regression models |

`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 Bayesian linear regression model for predictor variable selection |

`Lambda`

is a tuning parameter. Therefore, perform Bayesian lasso regression using a grid of shrinkage values, and choose the model that best balances a fit criterion and model complexity.For estimation, simulation, and forecasting, MATLAB

^{®}does not standardize predictor data. If the variables in the predictor data have different scales, then specify a shrinkage parameter for each predictor by supplying a numeric vector for`Lambda`

.

The `bayeslm`

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

[1]
Park, T., and G. Casella. "The Bayesian Lasso."
*Journal of the American Statistical Association*. Vol. 103, No. 482,
2008, pp. 681–686.