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conjugateblm

Bayesian linear regression model with conjugate prior for data likelihood

Description

The Bayesian linear regression model object `conjugateblm` specifies that the joint prior distribution of the regression coefficients and the disturbance variance, that is, (β, σ2) is the dependent, normal-inverse-gamma conjugate model. The conditional prior distribution of β|σ2 is multivariate Gaussian with mean μ and variance σ2V. The prior distribution of σ2 is inverse gamma with shape A and scale B.

The data likelihood is $\prod _{t=1}^{T}\varphi \left({y}_{t};{x}_{t}\beta ,{\sigma }^{2}\right),$ where ϕ(yt;xtβ,σ2) is the Gaussian probability density evaluated at yt with mean xtβ and variance σ2. The specified priors are conjugate for the likelihood, and the resulting marginal and conditional posterior distributions are analytically tractable. For details on the posterior distribution, see Analytically Tractable Posteriors.

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.

Creation

Syntax

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

Description

example

````PriorMdl = conjugateblm(NumPredictors)` creates a Bayesian linear regression model object (`PriorMdl`) composed of `NumPredictors` predictors and an intercept. The joint prior distribution of (β, σ2) is the dependent normal-inverse-gamma conjugate model. `PriorMdl` is a template defining the prior distributions and dimensionality of β.```

example

````PriorMdl = conjugateblm(NumPredictors,Name,Value)` uses additional options specified by one or more `Name,Value` pair arguments. `Name` is a property name, except `NumPredictors`, and `Value` is the corresponding value. `Name` must appear inside single quotes (`''`). You can specify several `Name,Value` pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.```

Properties

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You can set property values when you create the model object using name-value pair argument syntax, or after model creation using dot notation. For example, to set a more diffuse prior covariance matrix for `PriorMdl`, a Bayesian linear regression model containing three model coefficients, enter

`PriorMdl.V = 100*eye(3);`

Number of predictor variables in the Bayesian multiple linear regression model, specified as a nonnegative integer.

`NumPredictors` must be the same as the number of columns in your predictor data, which you specify during model estimation or simulation.

When specifying `NumPredictors`, exclude any intercept term for the value.

After creating a model, if you change the value `NumPredictors` using dot notation, then all these parameters revert to the default values:

• Variables names (`VarNames`)

• The prior mean of β (`Mu`)

• The prior covariance matrix of β (`V`)

Data Types: `double`

Indicate whether to include regression model intercept, specified as the comma-separated pair consisting of `'Intercept'` and a value in this table.

ValueDescription
`false`Exclude an intercept from the regression model. Hence, β is a `p`-dimensional vector, where `p` is the value of the `NumPredictors` property.
`true`Include an intercept in the regression model. Hence, β is a (`p` + 1)-dimensional vector. During estimation, simulation, and forecasting, MATLAB® prepends the predictor data with an appropriately-sized vector of ones.

If you include a column of ones in the predictor data for an intercept term, then set `Intercept` to `false`.

Example: `'Intercept',false`

Data Types: `logical`

Predictor variable names for displays, specified as a string vector or cell vector of character vectors. `VarNames`  must contain `NumPredictors` elements. `VarNames(j)` is the name of variable in column `j` of the predictor data set, which you specify during estimation, simulation, and forecasting.

The default is `{'Beta(1)','Beta(2),...,Beta(p)}`, where `p` is the value of `NumPredictors`.

Example: `'VarNames',["UnemploymentRate"; "CPI"]`

Data Types: `string` | `cell` | `char`

Mean parameter of the Gaussian prior on β, specified as a numeric scalar or vector.

If `Mu` is a vector, then it can have `NumPredictors` or ```NumPredictors + 1``` elements.

• For `NumPredictors` elements, `conjugateblm` sets the prior mean of the `NumPredictors` predictors only. Predictor variables correspond to the columns in the predictor data set, which is specified during estimation, simulation, or forecasting. `conjugateblm` ignores the intercept in the model, that is, any intercept retains the default or its current prior mean.

• For `NumPredictors + 1` elements, the first element corresponds to the prior mean of the intercept, and all other elements correspond to the predictor variables.

Example: `'Mu',[1; 0.08; 2]`

Data Types: `double`

Conditional covariance matrix of Gaussian prior on β, specified as a `c`-by-`c` symmetric, positive definite matrix. `c` can be `NumPredictors` or ```NumPredictors + 1```.

• If `c` is `NumPredictors`, then `conjugateblm` sets the prior covariance matrix to

`$\left[\begin{array}{cccc}1e5& 0& \cdots & 0\\ 0& & & \\ ⋮& & V& \\ 0& & & \end{array}\right].$`

That is, `conjugateblm` attributes the default prior covariances to the intercept, and `V` to the coefficients of the predictor variables in the data. Rows and columns of `V` correspond to columns (variables) in the predictor data.

• If `c` is ```NumPredictors + 1```, then `conjugateblm` sets the entire prior covariance to `V`. The first row and column correspond to the intercept. All other rows and columns correspond to columns in the predictor data.

The default value is a flat prior. For an adaptive prior, specify `diag(Inf(Intercept + NumPredictors,1))`. Adaptive priors indicate zero precision, that is, you want the prior distribution to have as little influence on the posterior distribution as possible.

`V` is the prior covariance of β up to a factor of σ2.

Example: `'V',diag(Inf(3,1))`

Data Types: `double`

Shape parameter of inverse gamma prior on σ2, specified as a numeric scalar.

`A` must be at least ```-(Intercept + NumPredictors)/2```.

With `B` held fixed, as `A` increases, the inverse gamma distribution becomes taller and more concentrated. This characteristic weighs the prior model of σ2 more heavily than the likelihood during posterior estimation.

For the functional form of the inverse gamma distribution, see Analytically Tractable Posteriors.

Example: `'A',0.1`

Data Types: `double`

Scale parameter of inverse gamma prior on σ2, specified as a positive scalar or `Inf`.

With `A` held fixed, as `B` increases, the inverse gamma distribution becomes taller and more concentrated. This characteristic weighs the prior model of σ2 more heavily than the likelihood during posterior estimation.

Example: `'B',5`

Data Types: `double`

Object Functions

 `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

Examples

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Consider the multiple linear regression model that predicts U.S. real gross national product (`GNPR`) using a linear combination of industrial production index (`IPI`), total employment (`E`), and real wages (`WR`).

For all , is a series of independent Gaussian disturbances with a mean of 0 and variance .

Assume that the prior distributions are:

• . is a 4-by-1 vector of means and is a scaled 4-by-4 positive definite covariance matrix.

• . and 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')```
```Mdl = conjugateblm with properties: NumPredictors: 3 Intercept: 1 VarNames: {4x1 cell} Mu: [4x1 double] V: [4x4 double] A: 3 B: 1 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Beta(1) | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Beta(2) | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Beta(3) | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Sigma2 | 0.5000 0.5000 [ 0.138, 1.616] 1.000 IG(3.00, 1) ```

`Mdl` is a `conjugateblm` Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance. At the command window, `bayeslm` displays a summary of the prior distributions.

You can set writable property values of created models using dot notation. Specify the regression coefficient names to the corresponding variable names.

`Mdl.VarNames = ["IPI" "E" "WR"]`
```Mdl = conjugateblm with properties: NumPredictors: 3 Intercept: 1 VarNames: {4x1 cell} Mu: [4x1 double] V: [4x4 double] A: 3 B: 1 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) IPI | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) E | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) WR | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Sigma2 | 0.5000 0.5000 [ 0.138, 1.616] 1.000 IG(3.00, 1) ```

This example is based on Create Normal-Inverse-Gamma Conjugate Prior Model.

Create a normal-inverse-gamma conjugate prior model for the linear regression parameters. Specify the number of predictors, `p`, and the names of the regression coefficients.

```p = 3; PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',["IPI" "E" "WR"]);```

Load the Nelson-Plosser data set. Create variables for the response and predictor series.

```load Data_NelsonPlosser X = DataTable{:,PriorMdl.VarNames(2:end)}; y = DataTable{:,'GNPR'};```

Estimate the marginal posterior distributions of and .

`PosteriorMdl = estimate(PriorMdl,X,y);`
```Method: Analytic posterior distributions Number of observations: 62 Number of predictors: 4 Log marginal likelihood: -259.348 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | -24.2494 8.7821 [-41.514, -6.985] 0.003 t (-24.25, 8.65^2, 68) IPI | 4.3913 0.1414 [ 4.113, 4.669] 1.000 t (4.39, 0.14^2, 68) E | 0.0011 0.0003 [ 0.000, 0.002] 1.000 t (0.00, 0.00^2, 68) WR | 2.4683 0.3490 [ 1.782, 3.154] 1.000 t (2.47, 0.34^2, 68) Sigma2 | 44.1347 7.8020 [31.427, 61.855] 1.000 IG(34.00, 0.00069) ```

`PosteriorMdl` is a `conjugateblm` model object storing the joint marginal posterior distribution of and given the data. `estimate` displays a summary of the marginal posterior distributions to the command window. Rows of the summary correspond to regression coefficients and the disturbance variance, and columns to characteristics of the posterior distribution. The characteristics include:

• `CI95`, which contains the 95% Bayesian equitailed credible intervals for the parameters. For example, the posterior probability that the regression coefficient of `WR` is in [1.782, 3.154] is 0.95.

• `Positive`, which contains the posterior probability that the parameter is greater than 0. For example, the probability that the intercept is greater than 0 is 0.003.

• `Distribution`, which contains descriptions of the posterior distributions of the parameters. For example, the marginal posterior distribution of `IPI` is t with a mean of 4.39, a standard deviation of 0.14, and 68 degrees of freedom.

Access properties of the posterior distribution using dot notation. For example, display the marginal posterior means by accessing the `Mu` property.

`PosteriorMdl.Mu`
```ans = 4×1 -24.2494 4.3913 0.0011 2.4683 ```

This example is based on Create Normal-Inverse-Gamma Conjugate Prior Model.

Create a normal-inverse-gamma conjugate prior model for the linear regression parameters. Specify the number of predictors, `p`, and the names of the regression coefficients.

```p = 3; PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',["IPI" "E" "WR"])```
```PriorMdl = conjugateblm with properties: NumPredictors: 3 Intercept: 1 VarNames: {4x1 cell} Mu: [4x1 double] V: [4x4 double] A: 3 B: 1 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) IPI | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) E | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) WR | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Sigma2 | 0.5000 0.5000 [ 0.138, 1.616] 1.000 IG(3.00, 1) ```

Load the Nelson-Plosser data set. Create variables for the response and predictor series.

```load Data_NelsonPlosser X = DataTable{:,PriorMdl.VarNames(2:end)}; y = DataTable{:,'GNPR'};```

Estimate the conditional posterior distributions of given and the data.

```[Mdl,condPostMeanBeta,CondPostCovBeta] = estimate(PriorMdl,X,y,... 'Sigma2',2);```
```Method: Analytic posterior distributions Conditional variable: Sigma2 fixed at 2 Number of observations: 62 Number of predictors: 4 | Mean Std CI95 Positive Distribution -------------------------------------------------------------------------------- Intercept | -24.2494 1.8695 [-27.914, -20.585] 0.000 N (-24.25, 1.87^2) IPI | 4.3913 0.0301 [ 4.332, 4.450] 1.000 N (4.39, 0.03^2) E | 0.0011 0.0001 [ 0.001, 0.001] 1.000 N (0.00, 0.00^2) WR | 2.4683 0.0743 [ 2.323, 2.614] 1.000 N (2.47, 0.07^2) Sigma2 | 2 0 [ 2.000, 2.000] 1.000 Fixed value ```
```Warning: Current syntax supports 6 output arguments, and will be removed in a future release. For supported output arguments, see <a href="matlab:helpview(fullfile(docroot,'econ','econ.map'),'blmestimate')">estimate</a>. ```

`estimate` returns the 4-by-1 vector of means and the 4-by-4 covariance matrix of the conditional posterior distribution of the in `condPostMeanBeta` and `CondPostCovBeta`, respectively. Also, `estimate` displays a summary of the conditional posterior distribution of .

The warning indicates that, in a future release, the syntaxes of `estimate` will change. At this time, do not update your code. For more details, see Replacing Discouraged Syntaxes of estimate.

Display `Mdl`.

`Mdl`
```Mdl = conjugateblm with properties: NumPredictors: 3 Intercept: 1 VarNames: {4x1 cell} Mu: [4x1 double] V: [4x4 double] A: 3 B: 1 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) IPI | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) E | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) WR | 0 70.7107 [-141.273, 141.273] 0.500 t (0.00, 57.74^2, 6) Sigma2 | 0.5000 0.5000 [ 0.138, 1.616] 1.000 IG(3.00, 1) ```

Because `estimate` computes the conditional posterior distribution, it returns the original prior model, not the posterior, in the first position of the output argument list.

This example is based on Estimate Marginal Posterior Distributions.

Create a prior model for the regression coefficients and disturbance variance, then estimate the marginal posterior distributions.

```p = 3; PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',["IPI" "E" "WR"]); load Data_NelsonPlosser X = DataTable{:,PriorMdl.VarNames(2:end)}; y = DataTable{:,'GNPR'}; PosteriorMdl = estimate(PriorMdl,X,y);```
```Method: Analytic posterior distributions Number of observations: 62 Number of predictors: 4 Log marginal likelihood: -259.348 | Mean Std CI95 Positive Distribution ----------------------------------------------------------------------------------- Intercept | -24.2494 8.7821 [-41.514, -6.985] 0.003 t (-24.25, 8.65^2, 68) IPI | 4.3913 0.1414 [ 4.113, 4.669] 1.000 t (4.39, 0.14^2, 68) E | 0.0011 0.0003 [ 0.000, 0.002] 1.000 t (0.00, 0.00^2, 68) WR | 2.4683 0.3490 [ 1.782, 3.154] 1.000 t (2.47, 0.34^2, 68) Sigma2 | 44.1347 7.8020 [31.427, 61.855] 1.000 IG(34.00, 0.00069) ```

Extract the posterior mean of from the posterior model, and the posterior covariance of from the estimation summary returned by `summarize`.

```estBeta = PosteriorMdl.Mu; Summary = summarize(PosteriorMdl); estBetaCov = Summary.Covariances{1:(end - 1),1:(end - 1)};```

Suppose that if the coefficient of real wages is below 2.5, then a policy is enacted. Although the posterior distribution of `WR` is known, and so you can calculate probabilities directly, you can estimate the probability using Monte Carlo simulation instead.

Draw `1e6` samples from the marginal posterior distribution of .

```NumDraws = 1e6; rng(1); BetaSim = simulate(PosteriorMdl,'NumDraws',NumDraws);```

`BetaSim` is a 4-by- `1e6` matrix containing the draws. Rows correspond to the regression coefficient and columns to successive draws.

Isolate the draws corresponding to the coefficient of real wages, and then identify which draws are less than 2.5.

```isWR = PosteriorMdl.VarNames == "WR"; wrSim = BetaSim(isWR,:); isWRLT2p5 = wrSim < 2.5;```

Find the marginal posterior probability that the regression coefficient of `WR` is below 2.5 by computing the proportion of draws that are less than 2.5.

`probWRLT2p5 = mean(isWRLT2p5)`
```probWRLT2p5 = 0.5362 ```

The posterior probability that the coefficient of real wages is less than 2.5 is about `0.54`.

The marginal posterior distribution of the coefficient of `WR` is a , but centered at 2.47 and scaled by 0.34. Directly compute the posterior probability that the coefficient of `WR` is less than 2.5.

```center = estBeta(isWR); stdBeta = sqrt(diag(estBetaCov)); scale = stdBeta(isWR); t = (2.5 - center)/scale; dof = 68; directProb = tcdf(t,dof)```
```directProb = 0.5361 ```

The posterior probabilities are nearly identical.

This example is based on Estimate Marginal Posterior Distributions.

Create a prior model for the regression coefficients and disturbance variance, then estimate the marginal posterior distributions. Hold out the last 10 periods of data from estimation so that they can be used for forecasting real GNP. Turn the estimation display off.

```p = 3; PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',["IPI" "E" "WR"]); load Data_NelsonPlosser fhs = 10; % Forecast horizon size X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)}; y = DataTable{1:(end - fhs),'GNPR'}; XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data yFT = DataTable{(end - fhs + 1):end,'GNPR'}; % True future responses PosteriorMdl = estimate(PriorMdl,X,y,'Display',false);```

Forecast responses using the posterior predictive distribution and using the future predictor data `XF`. Plot the true values of the response and the forecasted values.

```yF = forecast(PosteriorMdl,XF); figure; plot(dates,DataTable.GNPR); hold on plot(dates((end - fhs + 1):end),yF) h = gca; hp = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],... h.YLim([1,1,2,2]),[0.8 0.8 0.8]); uistack(hp,'bottom'); legend('True GNPR','Forecasted GNPR','Forecast Horizon','Location','NW') title('Real Gross National Product: 1909 - 1970'); ylabel('rGNP'); xlabel('Year'); hold off```

`yF` is a 10-by-1 vector of future values of real GNP corresponding to the future predictor data.

Estimate the forecast root mean squared error (RMSE).

`frmse = sqrt(mean((yF - yFT).^2))`
```frmse = 25.5397 ```

Forecast RMSE is a relative measure of forecast accuracy. Specifically, you estimate several models using different assumptions. The model with the lowest forecast RMSE is the best performing model of the ones being compared.

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Algorithms

You can reset all model properties using dot notation, for example, ```PriorMdl.V = diag(Inf(3,1))```. For property resets, `conjugateblm` does minimal error checking of values. Minimizing error checking has the advantage of reducing overhead costs for Markov chain Monte Carlo simulations, which results in efficient execution of the algorithm.

Alternatives

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