theta.prob

Posterior probabilities for theta

Determines the posterior probability and likelihood for theta, given an ecosystem.

Usage

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

Details

The probability is given 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().

References

• S. P. Hubbell 2001.The Unified Neutral Theory of Biodiversity, Princeton University Press.
• M. Abramowitz and I. A. Stegun 1965.Handbook of Mathematical Functions, New York: Dover

See Also

phi, optimal.prob

Aliases

• theta.prob
• theta.likelihood

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)

a <- untb(start=rep(1,1000),gens=1000,prob=1e-3)



## First, an example showing that theta.prob() is a PDF:
library(untb)
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)))}
sum(apply(x,2,f))  ## should be one exactly.