summarize
Distribution summary statistics of standard Bayesian linear regression model
Description
To obtain a summary of a Bayesian linear regression model for predictor selection, see summarize
.
summarize(
displays a tabular summary of the random regression coefficients and disturbance variance of the standard Bayesian linear regression model
Mdl
)Mdl
at the command line. For each parameter, the summary includes the:
Standard deviation (square root of the variance)
95% equitailed credible intervals
Probability that the parameter is greater than 0
Description of the distributions, if known
returns a structure array that stores a:SummaryStatistics
= summarize(Mdl
)
Table containing the summary of the regression coefficients and disturbance variance
Table containing the covariances between variables
Description of the joint distribution of the parameters
Examples
Summarize Posterior Distribution
Consider the multiple linear regression model that predicts the US real gross national product (GNPR
) using a linear combination of industrial production index (IPI
), total employment (E
), and real wages (WR
).
For all time points, is a series of independent Gaussian disturbances with a mean of 0 and variance .
Assume these prior distributions:
. 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
and the variable names.
p = 3; VarNames = ["IPI" "E" "WR"]; PriorMdl = bayeslm(p,'ModelType','conjugate','VarNames',VarNames);
PriorMdl
is a conjugateblm
Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance.
Summarize the prior distribution.
summarize(PriorMdl)
| 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)
The function displays a table of summary statistics and other information about the prior distribution at the command line.
Load the Nelson-Plosser data set and create variables for the predictor and response data.
load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable.GNPR;
Estimate the posterior distributions. Suppress the estimation display.
PosteriorMdl = estimate(PriorMdl,X,y,'Display',false);
PosteriorMdl
is a conjugateblm
model object that contains the posterior distributions of and .
Obtain summary statistics from the posterior distribution.
summary = summarize(PosteriorMdl);
summary
is a structure array containing three fields: MarginalDistributions
, Covariances
, and JointDistribution
.
Display the marginal distribution summary and covariances by using dot notation.
summary.MarginalDistributions
ans=5×5 table
Mean Std CI95 Positive Distribution
_________ __________ ________________________ _________ __________________________
Intercept -24.249 8.7821 -41.514 -6.9847 0.0032977 {'t (-24.25, 8.65^2, 68)'}
IPI 4.3913 0.1414 4.1134 4.6693 1 {'t (4.39, 0.14^2, 68)' }
E 0.0011202 0.00032931 0.00047284 0.0017676 0.99952 {'t (0.00, 0.00^2, 68)' }
WR 2.4683 0.34895 1.7822 3.1543 1 {'t (2.47, 0.34^2, 68)' }
Sigma2 44.135 7.802 31.427 61.855 1 {'IG(34.00, 0.00069)' }
summary.Covariances
ans=5×5 table
Intercept IPI E WR Sigma2
__________ ___________ ___________ ___________ ______
Intercept 77.125 0.77133 -0.0023655 0.5311 0
IPI 0.77133 0.019994 -6.5001e-06 -0.02948 0
E -0.0023655 -6.5001e-06 1.0844e-07 -8.0013e-05 0
WR 0.5311 -0.02948 -8.0013e-05 0.12177 0
Sigma2 0 0 0 0 60.871
The MarginalDistributions
field is a table of summary statistics and other information about the posterior distribution. Covariances
is a table containing the covariance matrix of the parameters.
Input Arguments
Mdl
— Standard Bayesian linear regression model
conjugateblm
model object | semiconjugateblm
model object | diffuseblm
model object | empiricalblm
model object | customblm
model object
Standard Bayesian linear regression model, specified as a model object in this table.
Model Object | Description |
---|---|
conjugateblm | Dependent, normal-inverse-gamma conjugate prior or posterior model returned by bayeslm or estimate |
semiconjugateblm | Independent, normal-inverse-gamma semiconjugate prior model returned by bayeslm |
diffuseblm | Diffuse prior model returned by bayeslm |
empiricalblm | Prior or posterior model characterized by random draws from respective distributions, returned by bayeslm or estimate |
customblm | Prior distribution function that you declare returned by bayeslm |
Output Arguments
SummaryStatistics
— Parameter distribution summary
structure array
Parameter distribution summary, returned as a structure array containing the information in this table.
Structure Field | Description |
---|---|
MarginalDistributions | Table containing a summary of the parameter distributions. Rows correspond to parameters. Columns correspond to the:
Row names are the names in |
Covariances | Table containing covariances between parameters. Rows and columns correspond to the intercept (if one exists) the regression coefficients, and disturbance variance. Row and column names are the same as the row names in |
JointDistribution | A string scalar that describes the distributions of the regression coefficients ( |
For distribution descriptions:
N(Mu,V)
denotes the normal distribution with meanMu
and variance matrixV
. This distribution can be multivariate.IG(A,B)
denotes the inverse gamma distribution with shapeA
and scaleB
.t(Mu,V,DoF)
denotes the Student’s t distribution with meanMu
, varianceV
, and degrees of freedomDoF
.
More About
Bayesian Linear Regression Model
A Bayesian linear regression model treats the parameters β and σ2 in the multiple linear regression (MLR) model yt = xtβ + εt as random variables.
For times t = 1,...,T:
yt is the observed response.
xt is a 1-by-(p + 1) row vector of observed values of p predictors. To accommodate a model intercept, x1t = 1 for all t.
β is a (p + 1)-by-1 column vector of regression coefficients corresponding to the variables that compose the columns of xt.
εt is the random disturbance with a mean of zero and Cov(ε) = σ2IT×T, while ε is a T-by-1 vector containing all disturbances. These assumptions imply that the data likelihood is
ϕ(yt;xtβ,σ2) is the Gaussian probability density with mean xtβ and variance σ2 evaluated at yt;.
Before considering the data, you impose a joint prior distribution assumption on (β,σ2). In a Bayesian analysis, you update the distribution of the parameters by using information about the parameters obtained from the likelihood of the data. The result is the joint posterior distribution of (β,σ2) or the conditional posterior distributions of the parameters.
Version History
Introduced in R2017a
See Also
Objects
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