binomial_variance: Variance of Bayesian estimator for binomial trial with flat...

In momeara/MPStats: Tools for analyzing Morphological Profiling data

Variance of Bayesian estimator for binomial trial with flat prior

Description

The Bayesian estimator for the success probability p from a binomial trial with n successes and m failures and beta prior with rate parameters a and b is posterior ~ prior * likelihood posterior ~ Beta(a,b) * Binomial(n, m+n) posterior ~ p^(a-1) * (1-p)^(b-1) * p^n * (1-p)^m posterior ~ p^(n+a-1) * (1-p)^(m+b-1) posterior ~ Beta(n+a,m+b)

Usage

binomial_variance(n_positive, n_trials, prior_positive = 1, prior_negative = 1)

Arguments

n_positive	number of successes in binomial trial
n_trials	number of binomial trials
prior_positive	number of pseudo successes to add as prior
prior_negative	number of pseudo failures to add as prior

Details

A flat prior is a Beta(a=1, b=1), so the estimate of the posterior distribution for p is Beta(n+1, m+1). One way to think of this is that since the beta distribution is the congugate prior to the binomial distribution, the flat prior effectively adds one pseudo positive count and one pseudo negative count to the observation.

The variance of a Beta distribution is

var[Beta(a,b)] = a*b/((a+b)^2(a+b+1))

so the variance of the posteriror is

var[posterior] = var[Beta(n+1,m+1)] = (n+1)(m+1)/((n+m+2)^2 * (n+m+3))

Ref: https://stats.stackexchange.com/questions/185221/binomial-uniform-prior-bayesian-statistics

Value

variance of parameter estimate

momeara/MPStats documentation built on July 19, 2022, 3:34 p.m.