[ENH] Add explicit energy computations for multiple distributions

[arnavk23]

Reference Issues/PRs

Towards #267

What does this implement/fix? Explain your changes.

• Implement closed-form energy for Exponential (2/λ self-energy, piecewise cross)
• Add deterministic quadrature energy for Gamma, Logistic, Weibull, Pareto, Beta
• Implement MeanScale energy using delegation and scaling
• Move energy from approximate to exact capabilities for all above distributions
• Fix escape sequence warnings in MeanScale docstrings

Does your contribution introduce a new dependency? If yes, which one?

What should a reviewer concentrate their feedback on?

All implementations use either closed-form formulas or deterministic numerical integration (scipy.integrate.quad) instead of Monte Carlo approximation.

Did you add any tests for the change?

Any other comments?

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[fkiraly]

This is super useful! May I ask where you got the energy computations from? Derived yourself or from a book/paper?

Did you test against the Monte Carlo approximate computations?

[arnavk23]

This is super useful! May I ask where you got the energy computations from? Derived yourself or from a book/paper?

Did you test against the Monte Carlo approximate computations?

Thanks! The energy computations come from a standard probability identity.

For i.i.d. X, Y with CDF F on support S, E|X−Y| = 2 ∫_S F(t)·(1−F(t)) dt.
For symmetric full-real support you can write it as 4 ∫_0^∞ F(t)·(1−F(t)) dt; for non-negative support (Gamma, Weibull, Beta, Pareto) the correct factor is 2 over the support. Exponential has a closed form E|X−Y| = 1/λ; for Logistic I compute energy_x by numerically integrating |t−x|·pdf(t), which guarantees non-negativity.

Yes—validated against Monte Carlo approximations. With N=200k samples:
Exponential(rate=2): exact 0.5000, MC 0.5005
Gamma(α=2, β=1): exact 1.5000, MC 1.4979
Beta(α=2, β=3): exact 0.2286, MC 0.2293
Logistic(μ=0, scale=1): exact 2.0000, MC 1.9975
Weibull(k=2, scale=1): exact 0.5191, MC 0.5208
Pareto(α=2.5, scale=1): exact 0.8333, MC 0.8285
All are within ~0.1–0.6% relative error.

[fkiraly]

Really great! Excellent that you know about the identity, that is very helpful :-)

Some comments:

• could you remove the call of the energy function from the distribution examples?
• it should be possible to compute logistic without quadrature, you need to integrate tanh-squared and tanh which have analytical expressions
• for Pareto, it should also be possible to derive the integral, you integrate 1/x to some power

Let me know if you do not see this and I can help work it out. (or maybe I made a mistake)

[arnavk23]

@fkiraly Removed .energy() calls from Examples.

Logistic: Analytical self-energy — Replaced quadrature with closed-form formula:
Formula: E|X-Y| = 2s for Logistic(μ, s)
Derivation: Using the identity F(t)(1-F(t)) = 1/(4 cosh²((t-μ)/(2s))), the integral simplifies to s, giving E|X-Y| = 2s.

Pareto: Analytical self-energy — Replaced quadrature with closed-form formula:
Formula: E|X-Y| = 2ma/[(a-1)(2a-1)] for Pareto(scale=m, alpha=a) with a > 1
Derivation: Direct integration of (m/t)ᵃ - (m/t)²ᵃ from m to ∞
Edge case: Returns inf for α ≤ 1

Validation:
Monte Carlo (N=200k) confirms all formulas within < 0.6% relative error:
Exponential: 0.09%, Gamma: 0.14%, Beta: 0.33%, Logistic: 0.12%, Weibull: 0.32%, Pareto: 0.58%

[fkiraly]

Great, thanks!