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# Bayesian Linear Regression Models

Posterior estimation, simulation, and predictor variable selection using a variety of prior models for the regression coefficients and disturbance variance

Bayesian linear regression models treat regression coefficients and the disturbance variance as random variables, rather than fixed but unknown quantities. This assumption leads to a more flexible model and intuitive inferences. For more details, see Bayesian Linear Regression.

To start a Bayesian linear regression analysis, create a standard model object that best describes your prior assumptions on the joint distribution of the regression coefficients and disturbance variance. Then, using the model and data, you can estimate characteristics of the posterior distributions, simulate from the posterior distributions, or forecast responses using the predictive posterior distribution.

Alternatively, you can perform predictor variable selection by working with the model object for Bayesian variable selection.

## Objects

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 `conjugateblm` Bayesian linear regression model with conjugate prior for data likelihood `semiconjugateblm` Bayesian linear regression model with semiconjugate prior for data likelihood `diffuseblm` Bayesian linear regression model with diffuse conjugate prior for data likelihood `empiricalblm` Bayesian linear regression model with samples from prior or posterior distributions `customblm` Bayesian linear regression model with custom joint prior distribution
 `mixconjugateblm` Bayesian linear regression model with conjugate priors for stochastic search variable selection (SSVS) `mixsemiconjugateblm` Bayesian linear regression model with semiconjugate priors for stochastic search variable selection (SSVS) `lassoblm` Bayesian linear regression model with lasso regularization

## Functions

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 `bayeslm` Create Bayesian linear regression model object
 `estimate` Fit parameters of Bayesian linear regression model to data `summarize` Distribution summary statistics of standard Bayesian linear regression model `plot` Visualize prior and posterior densities of Bayesian linear regression model parameters
 `estimate` Perform predictor variable selection for Bayesian linear regression models `summarize` Distribution summary statistics of Bayesian linear regression model for predictor variable selection `plot` Visualize prior and posterior densities of Bayesian linear regression model parameters
 `simulate` Simulate regression coefficients and disturbance variance of Bayesian linear regression model `sampleroptions` Create Markov chain Monte Carlo (MCMC) sampler options
 `forecast` Forecast responses of Bayesian linear regression model

## Topics

Bayesian Linear Regression

Learn about Bayesian analyses and how a Bayesian view of linear regression differs from a classical view.

Implement Bayesian Linear Regression

Combine standard Bayesian linear regression prior models and data to estimate posterior distribution features or to perform Bayesian predictor selection. Both workflows yield posterior models that are well suited for further analysis, such as forecasting.

Posterior Estimation and Simulation Diagnostics

Tune Markov Chain Monte Carlo sample for adequate mixing and perform a prior distribution sensitivity analysis.

Specify Gradient for HMC Sampler

Set up a Bayesian linear regression model for efficient posterior sampling using the Hamiltonian Monte Carlo sampler.

Tune Slice Sampler For Posterior Estimation

Improve a Markov Chain Monte Carlo sample for posterior estimation and inference of a Bayesian linear regression model.

Compare Robust Regression Techniques

Address influential outliers using regression models with ARIMA errors, bags of regression trees, and Bayesian linear regression.

Bayesian Lasso Regression

Perform variable selection using Bayesian lasso regression.

Bayesian Stochastic Search Variable Selection

Implement stochastic search variable selection (SSVS), a Bayesian variable selection technique.

Replacing Discouraged Syntaxes of estimate

In a future release, the `estimate` function of the Bayesian linear regression models `conjugateblm`, `semiconjugateblm`, `diffuseblm`, `empiricalblm`, and `customblm` will return only an estimated model and an estimation summary table.

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