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== Uncertainty of prediction ==
[[File:BinomialConfBelts.jpg|thumb|<small>Uncertainty analysis with confidence belts using the binomial distribution
Predictions of occurrence based on fitted probability distributions are subject to [[uncertainty]], which arises from the following conditions:
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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 display="block">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 display="block">\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 display="block">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
<math display="block">\sigma_{P_\theta(x|X)}^2 = P_\theta(x|X) \left[ P_\theta(x|\left\{X,x\right\}) - P_\theta(x|X)
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
[[File:GEVdistrHistogr+Density.png|thumb|220px|Histogram and probability density of a data set fitting the [[GEV distribution]] ]]
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* [[Mixture distribution]]
* [[Product distribution]]
{{clear}}
== References ==
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