B function

Normalizing constant for the hyperdirichlet distribution

Numerical techniques for calculating the normalizing constant for the hyperdirichlet distribution

B(H, disallowed=NULL, give=FALSE, ...)
probability(H, disallowed=NULL, ...)
mgf(H, powers, ...) 
dhyper2(ip,H,...)
dhyper2_e(e,H,include.Jacobian=TRUE)
mean_hyper2(H, normalize=TRUE, ...)
Jacobian(e)
e_to_p(e)
p_to_e(p)

Arguments

• H: Object of class hyper2

• powers: Vector of length dim(x) whose elements are the powers of the expectation; see details section

• disallowed: Function specifying a subset of the simplex over which to integrate; default NULL means to integrate over the whole simplex. The integration proceeds over p with disallowed(p) evaluating to FALSE

• e,p: A vector; see details

• ip: A vector of probabilities corresponding to indep(p)

  where p is vector with unit sum

• include.Jacobian: Boolean, with default TRUE meaning to include the Jacobian transformation in the evaluation, and FALSE meaning to ignore it; use FALSE for likelihood work and TRUE for probability densities

• give: Boolean, with default FALSE meaning to return the value of the integral and TRUE meaning to return the full output of adaptIntegrate()

• normalize: Boolean, indicates whether return value of mean_hyper2() is normalized to have unit sum

• ...: Further arguments passed to adaptIntegrate()

Details

• Function B() returns the normalizing constant of a hyperdirichlet likelihood function. Internally, $p$ is converted to e (by e_to_p()) and the integral proceeds over a hypercube. This function can be very slow, especially if disallowed is used.
• Function dhyper2(ip,H) is a probability density function on the independent components of a unit-sum vector, that is, ip=indep(p). This function calls B() each time so might be a performance bottleneck.
• Function probability() gives the probability of an observation from a hyperdirichlet distribution satisfying !disallowed(p).
• Function mgf() is the moment generating function, taking an argument that specifies the powers of p needed: the expectation of c("$prod\n$", "$p^powers$") is returned.
• Function mean_hyper2() returns the mean value of the hyperdirichlet distribution. This is computationally slow (consider maxp() for a measure of central tendency). The function takes a normalize argument, not passed to adaptIntegrate(): this is Boolean with FALSE meaning to return the value found by integration directly, and default TRUE meaning to normalize so the sum is exactly 1

Returns

• Function B() returns a scalar: the normalization constant
• Function dhyper2() is a probability density function over indep(p)
• Function mean() returns a $k$-tuple with unit sum
• Function mgf() returns a scalar equal to the expectation of p^power
• Functions is.proper() and validated() return a Boolean
• Function probability() returns a scalar, a (Bayesian) probability

Author(s)

Robin K. S. Hankin

Note

The adapt package is no longer available on CRAN; from 1.4-3, the package uses adaptIntegrate of the cubature package.

See Also

loglik

Examples

# Two different measures of central tendency:
# mean_hyper2(chess,tol=0.1)   # takes ~10s to run
maxp(chess)                    # faster

# Using the 'disallowed' argument typically results in slow run times;
# use high tol for speed:

# probability(chess,disallowed=function(p){p[1]>p[2]},tol=0.5)
# probability(chess,disallowed=function(p){p[1]<p[2]},tol=0.5)

# Above should sum to 1 [they are exclusive and exhaustive events]