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Date: Thu, 27 Oct 2011 07:08:32 +0000 (UTC)
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Subject: [jira] [Commented] (MATH-692) Cumulative probability and inverse
 cumulative probability inconsistencies
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    [ https://issues.apache.org/jira/browse/MATH-692?page=3Dcom.atlassian.j=
ira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=3D131368=
53#comment-13136853 ]=20

S=C3=A9bastien Brisard commented on MATH-692:
----------------------------------------

Hi Christian,
I've started looking into this issue. As I said, you will have to be patien=
t with me ;).
I can see there already is a default implementation of {{AbstractContinuous=
Distribution.inverseCumulativeProbability}}. So what exactly would you like=
 me to do? Is this implementation fragile? Would you like me to improve rob=
ustness? Provide full testing?

I think there might be issues when the PDF falls down to zero in a range (i=
n which case the cum exhibits a plateau). The returned value might differ f=
rom the mathematical definition you proposed. Is this what you want me to w=
ork on? Have you already identified other issues?

Best regards,
S=C3=A9bastien
               =20
> 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 probab=
ilities. More precisely, '<' and '<=3D' 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 do=
es neither exist for Distribution nor for ContinuosDistribution. Thus we ne=
ed 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 functio=
ns in the basic distribution interfaces as follows:
> Distribution:
> - cumulativeProbability(double x): returns P(X <=3D x)
> - cumulativeProbability(double x0, double x1): returns P(x0 < X <=3D x1) =
[see also 1)]
> - inverseCumulativeProbability(double p):
>   returns the quantile function inf{x in R | P(X<=3Dx) >=3D p} [see also =
2), 3), and 4)]
> 1) An aternative definition could be P(x0 <=3D X <=3D x1). But this requi=
res to put the function probability(double x) or another cumulative probabi=
lity function into the interface Distribution in order be able to calculate=
 P(x0 <=3D X <=3D x1) in AbstractDistribution.
> 2) This definition is stricter than the definition in ContinuousDistribut=
ion, because the definition there does not specify what to do if there are =
multiple x satisfying P(X<=3Dx) =3D p.
> 3) A modification could be defined for p=3D0: Returning sup{x in R | P(X<=
=3Dx) =3D 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 consid=
ering 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 <=3D x)
> - cumulativeProbability(int x0, int x1): returns P(x0 < X <=3D x1) [see a=
lso 1) above]

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