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Sébastien Brisard commented on MATH692:

Hi Christian,
{quote}
Hi Sébastien,
the problem with the plateau is indeed one issue which needs to be solved.
{quote}
I'm working on it...
{quote}
Additionally, AbstractDistribution will need an implementation of inverseCumulativeProbability.
In fact both implementations should be the same except for the solver to be used. Thus inverseCumulativeProbability
should be implemented just once in AbstractDistribution, and invoking the solver should be
put to a separate procedure so that it can be overridden in AbstractContinuousDistribution.
{quote}
OK, for now, I'm concentrating on making the current impl in {{AbstractContinuousDistribution}}
more robust. The other impl should be easier.
{quote}
A third point is the choice of the solvers. For AbstractDistribution we need a solver which
works even for discontinuous cdfs (BisectionSolver can do the job, but maybe the implementations
of the faster IllinoisSolver, PegasusSolver, BrentSolver, or another solver can cope with
discontinuities, too). For AbstractContinuousDistribution it would be beneficial to use a
DifferentiableUnivariateRealSolver. However, the NewtonSolver cannot be used due to uncertainty
of convergence and an alternative doesn't seem to exist by now. So we have to choose one of
the other solvers for now.
{quote}
The current implementation uses a Brent solver. I think the solver itself is only one side
of the issue. The other point is the algorithm used to bracket the solution, in order to ensure
that the result is consistent with the definition of the cumprob. As for the {{DifferentiableUnivariateRealSolver}},
I'm not too sure. I guess it depends on what is meant by "continuous distribution". For me,
it means that the random variable takes values in a continuous set, and possibly its distribution
is defined by a density. However, in my view, nothing prevents occurences of Dirac functions,
in which case the cum sum is only piecewise C1. It's all a matter of definition, of course,
and I'll ask the question on the forum to check whether or not people want to allow for such
a situation.
{quote}
As all these points are interdependent, I guess it's best to solve them as a whole. If you
like, you can do this.
Best Regards,
Christian
{quote}
Yes, I'm very interested.
Best regards,
Sébastien
> Cumulative probability and inverse cumulative probability inconsistencies
> 
>
> Key: MATH692
> URL: https://issues.apache.org/jira/browse/MATH692
> 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 MATH540. 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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