LaplacesDemon (version 16.1.1)

is.proper: Logical Check of Propriety

Description

This function provides a logical check of the propriety of a univariate prior probability distribution or the joint posterior distribution.

Usage

is.proper(f, a, b, tol=1e-5)

Arguments

f

This is either a probability density function or an object of class demonoid, laplace, pmc, or vb.

a

This is the lower limit of integration, and may be negative infinity.

b

This is the upper limit of integration, and may be positive infinity.

tol

This is the tolerance, and indicates the allowable difference from one.

Value

The is.proper function returns a logical value indicating whether or not the univariate prior or joint posterior probability distribution integrates to one within its specified limits. TRUE is returned for a proper univariate probability distribution.

Details

A proper probability distribution is a probability distribution that integrates to one, and an improper probability distribution does not integrate to one. If a probability distribution integrates to any positive and finite value other than one, then it is an improper distribution, but is merely unnormalized. An unnormalized distribution may be multiplied by a constant so that it integrates to one.

In Bayesian inference, the posterior probability distribution should be proper. An improper prior distribution can cause an improper posterior distribution. When the posterior distribution is improper, inferences are invalid, it is non-integrable, and Bayes factors cannot be used (though there are exceptions).

To avoid these problems, it is suggested that the prior probability distribution should be proper, though it is possible to use an improper prior distribution and have it result in a proper posterior distribution.

To check the propriety of a univariate prior probability distribution, create a function f. For example, to check the propriety of a vague normal distribution, such as

$$\theta \sim \mathcal{N}(0,1000)$$

the function is function(x){dnormv(x,0,1000)}. Next, set the lower and upper limits of integration, a and b. Internally, this function calls integrate from base R, which uses adaptive quadrature. By using \(f(x)\) as shorthand for the specified function, is.proper will check to see if the area of the following integral is one:

$$\int^b_a f(x)dx$$

Multivariate prior probability distributions currently cannot be checked for approximate propriety. This is currently unavailable in this package.

To check the propriety of the joint posterior distribution, the only argument to be supplied is an object of class demonoid, iterquad, laplace, pmc, or vb. The is.proper function checks the logarithm of the marginal likelihood (see LML) for a finite value, and returns TRUE when the LML is finite. This indicates that the marginal likelihood is finite for all observed \(\textbf{y}\) in the model data set. This implies:

$$\int p(\theta|\textbf{y})p(\theta)d\theta < \infty$$

If the object is of class demonoid and the algorithm was adaptive, or if the object is of class iterquad, laplace, or vb and the algorithm did not converge, then is.proper will return FALSE because LML was not estimated. In this case, it is possible for the joint posterior to be proper, but is.proper will be unable to determine propriety without the estimate of LML. If desired, the LML may be estimated by the user, and if it is finite, then the joint posterior distribution is proper.

See Also

dnormv, integrate, IterativeQuadrature, LaplaceApproximation, LaplacesDemon, LML, PMC, and VariationalBayes.

Examples

# NOT RUN {
library(LaplacesDemon)
### Prior Probability Distribution
is.proper(function(x) {dnormv(x,0,1000)}, -Inf, Inf) #x ~ N(0,1000)
is.proper(function(x) {dhalfcauchy(x,25)}, 0, Inf) #x ~ HC(25)
is.proper(function(x) {dunif(x,0,1)}, 0, 1) #x ~ U(0,1)
is.proper(function(x) {dunif(x,-Inf,Inf)}, -Inf, Inf) #x ~ U(-Inf,Inf)
### Joint Posterior Distribution
##This assumes that Fit is an object of class demonoid, iterquad,
##  laplace, or pmc
#is.proper(Fit)
# }