untb (version 1.7-7-1)

theta.prob: Posterior probabilities for theta

Description

Determines the posterior probability and likelihood for theta, given a count object

Usage

theta.prob(theta, x=NULL, give.log=TRUE)
theta.likelihood(theta, x=NULL, S=NULL, J=NULL, give.log=TRUE)

Arguments

theta

biodiversity parameter

x

object of class count or census

give.log

Boolean, with FALSE meaning to return the value, and default TRUE meaning to return the (natural) logarithm of the value

S, J

In function theta.likelihood(), the number of individuals (J) and number of species (S) in the ecosystem, if x is not supplied. These arguments are provided so that x need not be specified if S and J are known.

Author

Robin K. S. Hankin

Details

The formula was originally given by Ewens (1972) and is shown on page 122 of Hubbell (2001): $$\frac{J!\theta^S}{ 1^{\phi_1}2^{\phi_2}\ldots J^{\phi_J} \phi_1!\phi_2!\ldots \phi_J! \prod_{k=1}^J\left(\theta+k-1\right)}.$$

The likelihood is thus given by $$\frac{\theta^S}{\prod_{k=1}^J\left(\theta+k-1\right)}.$$

Etienne observes that the denominator is equivalent to a Pochhammer symbol \((\theta)_J\), so is thus readily evaluated as \(\Gamma(\theta+J)/\Gamma(\theta)\) (Abramowitz and Stegun 1965, equation 6.1.22).

References

• S. P. Hubbell 2001. “The Unified Neutral Theory of Biodiversity”, Princeton University Press.

• W. J. Ewens 1972. “The sampling theory of selectively neutral alleles”, Theoretical Population Biology, 3:87--112

• M. Abramowitz and I. A. Stegun 1965. Handbook of Mathematical Functions, New York: Dover

See Also

phi, optimal.prob

Examples

theta.prob(1,rand.neutral(15,theta=2))

gg <- as.count(c(rep("a",10),rep("b",3),letters[5:9]))
theta.likelihood(theta=2,gg)

optimize(f=theta.likelihood,interval=c(0,100),maximum=TRUE,x=gg)


## An example showing that theta.prob() is indeed a PMF:

a <- count(c(dogs=3,pigs=3,hogs=2,crabs=1,bugs=1,bats=1))
x <- partitions::parts(no.of.ind(a))
f <- function(x){theta.prob(theta=1.123,extant(count(x)),give.log=FALSE)}
sum(apply(x,2,f))  ## should be one exactly.