# theta.prob

##### Posterior probabilities for theta

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

- Keywords
- math

##### 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()`

.

##### 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--112M. Abramowitz and I. A. Stegun 1965.

*Handbook of Mathematical Functions*, New York: Dover

##### See Also

##### Examples

```
# NOT RUN {
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.
# }
```

*Documentation reproduced from package untb, version 1.7-4, License: GPL*