| is.proper {LaplacesDemon} | R Documentation |
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 |
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. |
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.
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.
Author(s)
Statisticat, LLC. software@bayesian-inference.com
See Also
dnormv,
integrate,
IterativeQuadrature,
LaplaceApproximation,
LaplacesDemon,
LML,
PMC, and
VariationalBayes.
Examples
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)