Triangular, Uniform, Bernoulli and Categorical SurvivalFunction round small upper tails to 0 (Bernoulli(1e-20).SurvivalFunction(0) = 0)

[matt-edmondson]

What's wrong

IDistribution.SurvivalFunction defaults to 1.0 - Cdf(value) (Essentials/IDistribution.cs L72). Its remarks say that a distribution with a directly computable tail should declare this member. Exponential, Geometric, LogNormal, Normal and Poisson do. Triangular, Uniform, Bernoulli and Categorical do not, although each has a closed-form tail, so they lose precision the same way #50 describes for Binomial.

Reproduction (net10.0, current main)

Call	Actual	Exact
Triangular(0, 0, 1).SurvivalFunction(0.999999999)	0	1e-18
Triangular(0, 0, 1).SurvivalFunction(0.9999999)	9.992e-15	1e-14
Uniform(-1, 1).SurvivalFunction(0.9999999999999999)	0	5.55e-17
Bernoulli(1e-20).SurvivalFunction(0)	0	1e-20
Bernoulli(0.1).SurvivalFunction(0)	0.09999999999999998	0.1
Categorical([1, 1e-20]).SurvivalFunction(0)	0	1e-20

Callers that use the survival function for rare-event probabilities, p-values or log-likelihoods get 0, and then -Infinity after a log, where the true value is small but representable.

Suggested fix

Declare SurvivalFunction on each provider:

• Triangular: for v ≥ mode, return (max − v)² / ((max − min)(max − mode)). Below the mode, 1 − Cdf is well conditioned.
• Uniform: return (max − v) / (max − min), clamped to [0, 1].
• Bernoulli: return v < 0 ? 1 : v < 1 ? p : 0.
• Categorical: precompute upper cumulative sums, adding from the last category down, and index into them.

Add rows like the table above to the distribution tests.

This is related to #50 (Binomial), but it covers different providers and needs a different fix.

[matt-edmondson]

Triage

• Category: Bug
• Priority: Low. It's a numerical-precision loss in the extreme upper tail. It matters for rare-event probabilities and log-likelihoods, but ordinary callers see values correct to about 1e-16. The fix is closed-form and local to each provider.
• Area / suggested assignment: the Triangular, Uniform, Bernoulli and Categorical distribution providers, and the distribution tests
• Related: Binomial SurvivalFunction rounds upper tails to 0: Binomial(100, 0.01).SurvivalFunction(20) = 0 (exact 9.58e-22) #50 (the same 1 − Cdf precision loss for Binomial). These could be handled in one "declare SurvivalFunction on every provider with a closed-form tail" pass.
• Duplicates: none. Binomial SurvivalFunction rounds upper tails to 0: Binomial(100, 0.01).SurvivalFunction(20) = 0 (exact 9.58e-22) #50 covers a different provider.
• In progress: no open PR covers this.

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