untb (version 1.6-9)

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 n

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

Run this code
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 a PDF:

a <- count(c(dogs=3,pigs=3,hogs=2,crabs=1,bugs=1,bats=1))
x <- 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.

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