Golden Ratio estimate of success probability based on one and only sample

TL;DR

This paper introduces an iterative Bayesian approach to estimate success probability from a single sample, revealing a connection to the Golden Ratio when success occurs in only one trial.

Contribution

It proposes a novel iterative Bayesian method that converges to the MLE for various models and uncovers a statistical link to the Golden Ratio in a specific success scenario.

Findings

Iterative Bayesian estimate converges to MLE for binomial, Poisson, exponential, normal models.

When success occurs in a single trial, the estimate approximates 1/φ, linking the Golden Ratio to success probability.

Existence and uniqueness of the estimator are established for the binomial model with a triangle prior.

Abstract

This paper proposes iterative Bayesian method to estimate success probability based on unique sample. The procedure is replacing the distribution characteristic of prior with Bayes estimate on the every iteration until they coincide. Iterative Bayes estimate is generally independent of hyperparameters. For binomial, Poisson, exponential and normal model, iterative limit is shown to be MLE in case the expectation of conjugate prior is replaced respectively. Particularly, suppose success appears in one and only trial, while the mode of triangle prior is replaced iterative Bayesian method gives $1/\phi \approx 0.618$ ($\phi$ is Golden Ratio) as the estimate of success probability $p$, this result reveals the truth of Golden Ratio from the point of statistics. Furthermore, under triangle prior the unique sample $X$ from binomial model $B(n,p)$ is considered. Existence and uniqueness of iterative Bayes estimator $\hat{p}_{IB}$ for parameter $p$ is given.