Cumulative probability and inverse cumulative probability inconsistencies
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Key: MATH-692
URL: https://issues.apache.org/jira/browse/MATH-692
Project: Commons Math
Issue Type: Bug
Affects Versions: 2.2, 2.1, 2.0, 1.2, 1.1, 1.0, 1.3, 2.2.1, 3.0
Reporter: Christian Winter
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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