Taking the logarithm of the normalized probability density function, then summing over the independent observations, gives the log-likelihoodBoth and are parameters being inferred, so their contributions to the normalizing constant must be retained. The following R implementation validates the arguments and returns one number:
ll_f <- function(p, beta, x) {
scalar <- function(v) {
is.numeric(v) && !is.complex(v) && length(v) == 1L &&
is.finite(v)
}
if (!scalar(p) || p <= 0 || p >= 1)
stop("p must lie strictly between 0 and 1")
if (!scalar(beta) || beta <= 0)
stop("beta must be positive")
if (!is.numeric(x) || is.complex(x) || !is.null(dim(x)) ||
any(!is.finite(x)) || any(x <= 0))
stop("x must be a vector of finite positive numbers")
if (length(x) == 0L) return(0)
t <- log1p(-p) - beta * x
length(x) * (log(beta) - log(-log(p))) +
sum(t - log(-expm1(t)))
}Here and
-expm1(t) computes accurately near zero. The empty-data likelihood function is one, so its log-likelihood is zero. Invalid parameter values produce errors in this evaluator; the next algorithm rejects candidates outside the parameter support before calling it. Articles by others on the same topic
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