theta.prob {untb}

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.

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).

Note

If estimating theta, use theta.likelihood() rather than theta.probability() because the former function generally executes much faster: the latter calculates a factor that is independent of theta.

The likelihood function L(\theta) is any function of \theta proportional, for fixed observation z, to the probability density f(z,\theta). There is thus a slight notational inaccuracy in speaking of “the” likelihood function which is defined only up to a multiplicative constant. Note also that the “support” function is usually defined as a likelihood function with maximum value 1 (at the maximum likelihood estimator for \theta). This is not easy to determine analytically for J>5.

Note that S is a sufficient statistic for \theta.

Function theta.prob() does not give a PDF for \theta (so, for example, integrating over the real line does not give unity). The PDF is over partitions of J; an example is given below.

Function theta.prob() requires a count object (as opposed to theta.likelihood(), for which J and S are sufficient) because it needs to call phi().

Author(s)

Robin K. S. Hankin

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.