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https://issues.apache.org/jira/browse/MATH-692?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13133440#comment-13133440
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Christian Winter commented on MATH-692:
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Thanks for the feedback to all. Sébastien, thanks for offering your help. If
you like and find time for it, you could implement
AbstractDistribution.inverseCumulativeProbability(double p).
I will provide some patches next week, but adjusting
AbstractContinuousDistribution.inverseCumulativeProbability(double p) will take
some more time.
After thinking a little more about the structure of the interfaces, I'd like to
put the function probability(double x) to Distribution anyway (independently of
the thought in point 1) above).
Are there any preferences on P(x0 <= X <= x1) or P(x0 < X <= x1) for
cumulativeProbability(double x0, double x1)?
> Cumulative probability and inverse cumulative probability inconsistencies
> -------------------------------------------------------------------------
>
> Key: MATH-692
> URL: https://issues.apache.org/jira/browse/MATH-692
> Project: Commons Math
> Issue Type: Bug
> Affects Versions: 1.0, 1.1, 1.2, 1.3, 2.0, 2.1, 2.2, 2.2.1, 3.0
> Reporter: Christian Winter
> Priority: Minor
> Fix For: 3.0
>
>
> There are some inconsistencies in the documentation and implementation of
> functions regarding cumulative probabilities and inverse cumulative
> probabilities. More precisely, '<' and '<=' are not used in a consistent way.
> Besides I would move the function inverseCumulativeProbability(double) to the
> interface Distribution. A true inverse of the distribution function does
> neither exist for Distribution nor for ContinuosDistribution. Thus we need to
> define the inverse in terms of quantiles anyway, and this can already be done
> for Distribution.
> On the whole I would declare the (inverse) cumulative probability functions
> in the basic distribution interfaces as follows:
> Distribution:
> - cumulativeProbability(double x): returns P(X <= x)
> - cumulativeProbability(double x0, double x1): returns P(x0 < X <= x1) [see
> also 1)]
> - inverseCumulativeProbability(double p):
> returns the quantile function inf{x in R | P(X<=x) >= p} [see also 2), 3),
> and 4)]
> 1) An aternative definition could be P(x0 <= X <= x1). But this requires to
> put the function probability(double x) or another cumulative probability
> function into the interface Distribution in order be able to calculate P(x0
> <= X <= x1) in AbstractDistribution.
> 2) This definition is stricter than the definition in ContinuousDistribution,
> because the definition there does not specify what to do if there are
> multiple x satisfying P(X<=x) = p.
> 3) A modification could be defined for p=0: Returning sup{x in R | P(X<=x) =
> 0} would yield the infimum of the distribution's support instead of a
> mandatory -infinity.
> 4) This affects issue MATH-540. I'd prefere the definition from above for the
> following reasons:
> - This definition simplifies inverse transform sampling (as mentioned in the
> other issue).
> - It is the standard textbook definition for the quantile function.
> - For integer distributions it has the advantage that the result doesn't
> change when switching to "x in Z", i.e. the result is independent of
> considering the intergers as sole set or as part of the reals.
> ContinuousDistribution:
> nothing to be added regarding (inverse) cumulative probability functions
> IntegerDistribution:
> - cumulativeProbability(int x): returns P(X <= x)
> - cumulativeProbability(int x0, int x1): returns P(x0 < X <= x1) [see also 1)
> above]
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