postM: Calculate posterior distribution of M and extract statistics...

In dwp: Density-Weighted Proportion

Calculate posterior distribution of M and extract statistics (M* and CI)

Description

Calculation of the posterior distribution of total mortality (M) given the carcass count, overall detection probability (g), and prior distribtion; calculation of summary statistics from the posterior distribution of M, including M* and credibility intervals.

Usage

postM(x, g, prior = "IbinRef", mmax = NA)

postM.ab(x, Ba, Bb, prior = "IbinRef", mmax = NULL)

calcMstar(pMgX, alpha)

MCI(pMgX, crlev = 0.95)

Arguments

x	carcass count
g	overall carcass detection probability
prior	prior distribution of M
mmax	cutoff for prior of M (large max requires large computing resources but does not help in the estimation)
Ba, Bb	parameters for beta distribution characterizing estimated g
pMgX	posterior distribution of M
crlev, alpha	credibility level (1-\alpha) and its complement (\alpha)

Details

The functions postM and postM.ab return the posterior distributions of M|(X, g) and M|(X, Ba, Bb), respectively, where Ba and Bb are beta distribution parameters for the estimated detection probability. postM and postM.ab include options to to specify a prior distribution for M and a limit for truncating the prior to disregard implausibly large values of M and make the calculations tractable in certain cases where they otherwise might not be. Use postM when g is fixed and known; otherwise, use postM.ab when uncertainty in g is characterized in a beta distribution with parameters Ba and Bb. The non-informative, integrated reference prior for binomial random variables is the default (prior = "IbinRef"). Other options include "binRef", "IbetabinRef", and "betabinRef", which are the non-integrated and integrated forms of the binomial and betabinomial reference priors (Berger et al., 2012). For X > 2, the integrated and non-integrated reference priors give virtually identical posteriors. However, the non-integrated priors assign infinite weight to m = 0 and return a posterior of Pr(M = 0| X = 0, \hat{g}) = 1, implying absolute certainty that the total number of fatalities was 0 if no carcasses were observed. In addition, a uniform prior may be specified by prior = "uniform". Alternatively, a custom prior may be given as a 2-dimensional array with columns for m and Pr(M = m), respectively. The first column (m) must be sequential integers starting at m = 0. The second column gives the probabilities associated with m, which must be non-negative and sum to 1. The named priors ("IbinRef", "binRef", "IbetabinRef", and "betabinRef") are functions of m and defined on m=0,1,2,... without upper bound. However, the posteriors can only be calculated for a finite number of m's up to a maximum of mmax, which is set by default to the smallest value of m such that Pr(X \leq x | m, \hat{g}) < 0.0001, where x is the observed carcass count, or, alternatively, mmax may be specified by the user.

Value

The functions postM and postM.ab return the posterior distributions of M | (X, g) and M | (X, Ba, Bb), respectively. The functions calcMstar and MCI return M^* value and credibility interval for the given posterior distribution, pMgX (which may be the return value of postM or postM.ab) and \alpha value or credibility level.

dwp documentation built on July 9, 2023, 5:59 p.m.