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With the binomial distribution one can obtain a [[prediction interval]]. Such an interval also estimates the risk of failure, i.e. the chance that the predicted event still remains outside the confidence interval. The confidence or risk analysis may include the [[return period]] ''T=1/Pe'' as is done in [[hydrology]].
=== [[Variance]] of [[Bayesian inference|Bayesian]] fitted probability functions ===
[[File:CumList.png|thumb|left|List of probability distributions ranked by goodness of fit.<ref>[https://www.waterlog.info/cumfreq.htm Software for probability distribution fitting]</ref>]]▼
A Bayesian approach can be used for fitting a model <math>P(x|\theta)</math> having a prior distribution <math>P(\theta)</math> for the parameter <math>\theta</math>. When one has samples <math>X</math> that are independently drawn from the underlying distribution then one can derive the so-called posterior distribution <math>P(\theta|X)</math>. This posterior can be used to update the probability mass function for a new sample <math>x</math> given the observations <math>X</math>, one obtains
<math>P_\theta (x | X) := \int d\theta\ P(x|\theta)\ P(\theta|X) </math>.
The variance of the newly obtained probability mass function can also be determined. The variance for a Bayesian probability mass function can be defined as
<math>\sigma_{P_\theta(x|X)}^2 := \int d\theta\ \left[ P(x|\theta) - P_\theta(x|X) \right]^2\ P(\theta|X)</math>.
This expression for the variance can be substantially simplified (assuming independently drawn samples). Defining the "self probability mass function" as
<math>P_\theta(x|\left\{X,x\right\}) = \int d\theta\ P(x|\theta)\ P(\theta|\left\{X, x\right\})</math>,
one obtains for the variance <ref>{{Cite journal |last=Pijlman |last2=Linnartz |date=2023 |title=Variance of Likelihood of data |url=https://sitb2023.ulb.be/proceedings/ |journal=SITB 2023 proceedings |pages=page 34}}</ref>
<math>\sigma_{P_\theta(x|X)}^2 = P_\theta(x|X) \left[ P_\theta(x|\left\{X,x\right\}) - P_\theta(x|X) \right]</math>.
▲The expression for variance involves an additional fit that includes the sample <math>x</math> of interest.[[File:CumList.png|thumb|left|List of probability distributions ranked by goodness of fit.<ref>[https://www.waterlog.info/cumfreq.htm Software for probability distribution fitting]</ref>]]
[[File:GEVdistrHistogr+Density.png|thumb|220px|Histogram and probability density of a data set fitting the [[GEV distribution]] ]]
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