diffuseblm

Bayesian linear regression model with diffuse conjugate prior for data likelihood

Description

The Bayesian linear regression model object diffuseblm specifies that the joint prior distribution of (β,σ²) is proportional to 1/σ² (the diffuse prior model).

The data likelihood is $${\displaystyle \prod _{t=1}^T\varphi \left(y_t;x_t\beta ,{\sigma }^2\right)},$$ where ϕ(y_t;x_tβ,σ²) is the Gaussian probability density evaluated at y_t with mean x_tβ and variance σ². 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 = diffuseblm(NumPredictors)

PriorMdl = diffuseblm(NumPredictors,Name,Value)

Description

PriorMdl = diffuseblm(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 diffuse model. PriorMdl is a template defining the prior distributions and dimensionality of β.

PriorMdl = diffuseblm(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

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 specify that there is no model intercept in PriorMdl, a Bayesian linear regression model containing three model coefficients, enter

PriorMdl.Intercept = false;

NumPredictors — Number of predictor variables

nonnegative integer

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 VarNames reverts to its default value.

Data Types: double

Intercept — Indicate whether to include regression model intercept

true (default) | false

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

Value	Description
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

VarNames — Predictor variable names

string vector | cell vector of character vectors

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

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

Create Diffuse Prior Model

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 .

Suppose that the regression coefficients and the disturbance variance are random variables, and you have no prior knowledge of their values or distribution. That is, you want to use the noninformative Jefferys prior: the joint prior distribution is proportional to .

These assumptions and the data likelihood imply an analytically tractable posterior distribution.

Create a diffuse prior model for the linear regression parameters, which is the default prior model type. Specify the number of predictors, p.

p = 3;
Mdl = bayeslm(p)

Mdl = 
  diffuseblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}

 
           | Mean  Std        CI95        Positive        Distribution       
-----------------------------------------------------------------------------
 Intercept |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Beta(1)   |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Beta(2)   |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Beta(3)   |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Sigma2    |  Inf  Inf  [   NaN,    NaN]    1.000   Proportional to 1/Sigma2 
 

Mdl is a diffuseblm 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. Because the prior is noninformative and the data have not been incorporated yet, the summary is trivial.

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 = 
  diffuseblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}

 
           | Mean  Std        CI95        Positive        Distribution       
-----------------------------------------------------------------------------
 Intercept |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 IPI       |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 E         |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 WR        |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Sigma2    |  Inf  Inf  [   NaN,    NaN]    1.000   Proportional to 1/Sigma2 
 

Estimate Marginal Posterior Distributions

This example is based on Create Diffuse Prior Model.

Create a diffuse 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','diffuse','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
 
           |   Mean      Std           CI95        Positive       Distribution      
------------------------------------------------------------------------------------
 Intercept | -24.2536   9.5314  [-43.001, -5.506]    0.006   t (-24.25, 9.37^2, 58) 
 IPI       |   4.3913   0.1535   [ 4.089,  4.693]    1.000   t (4.39, 0.15^2, 58)   
 E         |   0.0011   0.0004   [ 0.000,  0.002]    0.999   t (0.00, 0.00^2, 58)   
 WR        |   2.4682   0.3787   [ 1.723,  3.213]    1.000   t (2.47, 0.37^2, 58)   
 Sigma2    |  51.9790  10.0034   [35.965, 74.937]    1.000   IG(29.00, 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 [1.723, 3.213] 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 0.006.

• 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 0.15, and 58 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

  -24.2536
    4.3913
    0.0011
    2.4682

Estimate Conditional Posterior Distribution

This example is based on Create Diffuse Prior Model.

Create a diffuse 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','diffuse','VarNames',["IPI" "E" "WR"])

PriorMdl = 
  diffuseblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}

 
           | Mean  Std        CI95        Positive        Distribution       
-----------------------------------------------------------------------------
 Intercept |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 IPI       |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 E         |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 WR        |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Sigma2    |  Inf  Inf  [   NaN,    NaN]    1.000   Proportional to 1/Sigma2 
 

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 | -24.2536  1.8696  [-27.918, -20.589]    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        |   2.4682  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 = 
  diffuseblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}

 
           | Mean  Std        CI95        Positive        Distribution       
-----------------------------------------------------------------------------
 Intercept |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 IPI       |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 E         |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 WR        |  0    Inf  [   NaN,    NaN]    0.500   Proportional to one      
 Sigma2    |  Inf  Inf  [   NaN,    NaN]    1.000   Proportional to 1/Sigma2 
 

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.

Estimate Posterior Probability Using Monte Carlo Simulation

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','diffuse','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
 
           |   Mean      Std           CI95        Positive       Distribution      
------------------------------------------------------------------------------------
 Intercept | -24.2536   9.5314  [-43.001, -5.506]    0.006   t (-24.25, 9.37^2, 58) 
 IPI       |   4.3913   0.1535   [ 4.089,  4.693]    1.000   t (4.39, 0.15^2, 58)   
 E         |   0.0011   0.0004   [ 0.000,  0.002]    0.999   t (0.00, 0.00^2, 58)   
 WR        |   2.4682   0.3787   [ 1.723,  3.213]    1.000   t (2.47, 0.37^2, 58)   
 Sigma2    |  51.9790  10.0034   [35.965, 74.937]    1.000   IG(29.00, 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 = 0.5341

The posterior probability that the coefficient of real wages is less than 2.5 is about 0.53.

The marginal posterior distribution of the coefficient of WR is a , but centered at 2.47 and scaled by 0.37. 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 = 0.5333

The posterior probabilities are nearly identical.

Forecast Responses Using Posterior Predictive Distribution

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','diffuse','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 = 25.5489

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.

More About

Bayesian Linear Regression Model

A Bayesian linear regression model treats the parameters β and σ² in the multiple linear regression (MLR) model y_t = x_tβ + ε_t as random variables.

For times t = 1,...,T:

• y_t is the observed response.

• x_t is a 1-by-(p + 1) row vector of observed values of p predictors. To accommodate a model intercept, x_1t = 1 for all t.

• β is a (p + 1)-by-1 column vector of regression coefficients corresponding to the variables composing the columns of x_t.

• ε_t is the random disturbance having a mean of zero and Cov(ε) = σ²I_T×T, while ε is a T-by-1 vector containing all disturbances. These assumptions imply that the data likelihood is

  $$\ell \left(\beta ,{\sigma }^2|y,x\right)={\displaystyle \prod _{t=1}^T\varphi \left(y_t;x_t\beta ,{\sigma }^2\right).}$$

  ϕ(y_t;x_tβ,σ²) is the Gaussian probability density with mean x_tβ and variance σ² evaluated at y_t;.

Before considering the data, a joint prior distribution assumption is imposed on (β,σ²). In a Bayesian analysis, the beliefs about the distribution of the parameters are updated using information about the parameters gleaned from the likelihood of the data. The result is the joint posterior distribution of (β,σ²) or the conditional posterior distributions of the parameters.

Alternatives

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