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

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.