Probability distribution fitting: Difference between revisions

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== Uncertainty of prediction ==

[[File:BinomialConfBelts.jpg|thumb|<small>Uncertainty analysis with confidence belts using the binomial distribution </small><ref>Frequency predictions and their binomial confidence limits. In: International Commission on Irrigation and Drainage, Special Technical Session: Economic Aspects of Flood Control and non-Structural Measures, Dubrovnik, Yugoslavia, 1988. [http://www.waterlog.info/pdf/binomial.pdf On line]</ref>]]

Predictions of occurrence based on fitted probability distributions are subject to [[uncertainty]], which arises from the following conditions:

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<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 |last1=Pijlman |last2=Linnartz |date=2023 |title=Variance of Likelihood of data |url=https://sitb2023.ulb.be/proceedings/ |journal=SITB 2023 Proceedings |pages=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]] ]]