betaHPD {pscl} | R Documentation

## compute and optionally plot beta HDRs

### Description

Compute and optionally plot highest density regions for the Beta distribution.

### Usage

       betaHPD(alpha,beta,p=.95,plot=FALSE,xlim=NULL,debug=FALSE)

### Arguments

`alpha` | scalar, first shape parameter of the Beta density. Must be greater than 1, see details
`beta` | scalar, second shape parameter of the Beta density. Must be greater than 1, see details
`p` | scalar, content of HPD, must lie between 0 and 1
`plot` | logical flag, if `TRUE` then plot the density and show the HDR
`xlim` | numeric vector of length 2, the limits of the density's support to show when plotting; the default is `NULL`, in which case the function will confine plotting to where the density is non-neglible
`debug` | logical flag, if `TRUE` produce messages to the console

### Details

The Beta density arises frequently in Bayesian models of binary events, rates, and proportions, which take on values in the open unit interval. For instance, the Beta density is a conjugate prior for the unknown success probability in binomial trials. With shape parameters α > 1 and β > 1, the Beta density is unimodal.

In general, suppose θ \in Θ \subseteq R^k is a random variable with density f(θ). A highest density region (HDR) of f(θ) with content p \in (0,1] is a set \mathcal{Q} \subseteq Θ with the following properties:

\int_\mathcal{Q} f(θ) dθ = p

and

f(θ) > f(θ^*) \, \forall\ θ \in \mathcal{Q}, θ^* \not\in \mathcal{Q}.

For a unimodal Beta density (the class of Beta densities handled by this function), a HDR of content 0 < p < 1 is simply an interval \mathcal{Q} \in (0,1).

This function uses numerical methods to solve for the end points of a HDR for a Beta density with user-specified shape parameters, via repeated calls to the functions `dbeta`, `pbeta` and `qbeta`. The function `optimize` is used to find points v and w such that

f(v) = f(w)

subject to the constraint

\int_v^w f(θ; α, β) dθ = p,

where f(θ; α, β) is a Beta density with shape parameters α and β.

In the special case of α = β > 1, the end points of a HDR with content p are given by the (1 \pm p)/2 quantiles of the Beta density, and are computed with the `qbeta` function.

Again note that the function will only compute a HDR for a unimodal Beta density, and exit with an error if `alpha<=1 | beta <=1`. Note that the uniform density results with α = β = 1, which does not have a unique HDR with content 0 < p < 1. With shape parameters α<1 and β>1 (or vice-versa, respectively), the Beta density is infinite at 0 (or 1, respectively), but still integrates to one, and so a HDR is still well-defined (but not implemented here, at least not yet). Similarly, with 0 < α, β < 1 the Beta density is infinite at both 0 and 1, but integrates to one, and again a HDR of content p<1 is well-defined in this case, but will be a set of two disjoint intervals (again, at present, this function does not cover this case).

### Value

If the numerical optimization is successful an vector of length 2, containing v and w, defined above. If the optimization fails for whatever reason, a vector of `NAs` is returned.

The function will also produce a plot of the density with area under the density supported by the HDR shaded, if the user calls the function with `plot=TRUE`; the plot will appear on the current graphics device.

Debugging messages are printed to the console if the `debug` logical flag is set to `TRUE`.

### Author(s)

Simon Jackman simon.jackman@sydney.edu.au. Thanks to John Bullock who discovered a bug in an earlier version.

### See Also

`pbeta`, `qbeta`, `dbeta`, `uniroot`

### Examples

    betaHPD(4,5)
    betaHPD(2,120)
    betaHPD(120,45,p=.75,xlim=c(0,1))

[Package pscl version 1.5.2 Index]