ZeroProb.jl

ZeroProb.jl

Table of Contents

• Overview
• The Zero-Probability Paradox
• Solution: Alternative Relevance Measures
  • 1. Density Ratio
  • 2. Hausdorff Measure
  • 3. Epsilon-Neighborhood Probability
• Installation
• Usage Examples
  • Density Ratio Analysis
  • Hausdorff Measure for Fractals
  • Epsilon-Neighborhood Convergence
• API Reference
  • Core Types
  • Extended Types
  • Core Measures
  • Extended Measures
  • Paradoxes
  • Applications
  • Visualization
• Integration with ECHIDNA
• Development Status
• References & Bibliography
  • Measure Theory & Probability Foundations
  • Fractal Geometry
  • Extreme Value Theory & Tail Risk
  • Quantum Measurement
  • Classical Paradoxes
• License
• Contributing
• Citation
• Contact

Overview

ZeroProb.jl is a Julia library for handling zero-probability events in continuous probability spaces. It provides rigorous mathematical tools for reasoning about events with measure zero, including density-based relevance, Hausdorff measures, paradox demonstrations, tail risk modelling, and black swan event analysis.

In continuous probability distributions, events with measure zero can still occur. This library provides alternative relevance measures, classical paradox demonstrations, and application-oriented tools for extreme events.

The Zero-Probability Paradox

In continuous probability theory, individual points have measure zero (P(X = x) = 0), yet one such point must occur when sampling. This creates a paradox:

• Probability: P(X = 3.14159…​) = 0

• Reality: X can equal exactly 3.14159…​

• Problem: How do we reason about relevance when probability is always zero?

Solution: Alternative Relevance Measures

ZeroProb.jl provides three complementary approaches:

1. Density Ratio

Compare probability densities instead of probabilities:

relevance = density_ratio(f, x, baseline)
# Higher density -> more "typical" for the distribution

2. Hausdorff Measure

Use dimensional analysis for sets of measure zero:

measure = hausdorff_measure(event_set, dimension)
# Generalizes volume to fractional dimensions

3. Epsilon-Neighborhood Probability

Consider probability mass in small neighborhoods:

prob = epsilon_neighborhood_prob(X, x, eps)
# P(|X - x| < eps) as eps -> 0

Installation

From Julia REPL:

using Pkg
Pkg.add("ZeroProb")

Usage Examples

Density Ratio Analysis

using ZeroProb, Distributions

# Define distribution
dist = Normal(0, 1)

# Compare relevance of two points
x1 = 0.0    # At mean
x2 = 3.0    # 3 standard deviations out

ratio = density_ratio(dist, x1, x2)
# ratio ~ 228.0 (x1 is 228x more "typical")

Hausdorff Measure for Fractals

using ZeroProb

# Cantor set has Hausdorff dimension log(2)/log(3) ~ 0.631
cantor_set = construct_cantor_set(iterations=10)
dimension = hausdorff_dimension(cantor_set)
measure = hausdorff_measure(cantor_set, dimension)

# Even though Lebesgue measure is 0, Hausdorff measure is non-zero

Epsilon-Neighborhood Convergence

using ZeroProb, Distributions

dist = Exponential(1.0)
x = 2.0

# Compute probability in shrinking neighborhoods
eps_values = [0.1, 0.01, 0.001, 0.0001]
probs = [epsilon_neighborhood_prob(dist, x, e) for e in eps_values]

# Observe convergence rate
convergence_rate = estimate_convergence_rate(eps_values, probs)

API Reference

Core Types

ZeroProbEvent

Abstract base type for zero-probability events.

ContinuousZeroProbEvent

Zero-probability event in a continuous space.

DiscreteZeroProbEvent

Zero-probability event in a discrete approximation.

AlmostSureEvent

Event that occurs with probability 1 (complement has measure zero).

SureEvent

Event that occurs in every outcome (not just almost every).

Extended Types

TailRiskEvent

Extreme tail event with severity and return period.

QuantumMeasurementEvent

Quantum measurement outcome with Born rule probability.

InsuranceCatastropheEvent

Catastrophe event for insurance/reinsurance modelling.

Core Measures

Function	Description
probability(event)	Probability of an event (zero for measure-zero events)
relevance(event)	Alternative relevance score
density_ratio(dist, x, baseline)	Ratio of probability densities at two points
hausdorff_measure(set, dim)	Hausdorff measure of a set at given dimension
epsilon_neighborhood(dist, x, eps)	Probability mass in epsilon-ball around x
relevance_score(event)	Composite relevance score

Extended Measures

Function	Description
hausdorff_dimension(set)	Box-counting dimension estimate
estimate_convergence_rate(eps, probs)	Rate of P(ball) → 0 as eps → 0
epsilon_neighborhood_prob(dist, x, eps)	P(|X - x| < eps) for distribution
conditional_density(dist, x, condition)	Conditional density given event
radon_nikodym_derivative(mu, nu, x)	Radon-Nikodym derivative dmu/dnu at x
total_variation_distance(P, Q)	Total variation between measures
kl_divergence(P, Q)	Kullback-Leibler divergence
fisher_information(dist, x)	Fisher information at a point
entropy_contribution(dist, x)	Pointwise entropy contribution
almost_surely(event)	Test if event holds almost surely
measure_zero_test(set)	Test if a set has Lebesgue measure zero

Paradoxes

Function	Description
continuum_paradox()	P(X = x) = 0 yet outcomes exist
borel_kolmogorov_paradox()	Conditioning on measure-zero sets
rational_points_paradox()	Countable dense set with measure zero
uncountable_union_paradox()	Uncountable union of measure-zero sets
almost_sure_vs_sure()	Distinction between a.s. and sure events
construct_cantor_set(; iterations)	Build Cantor set to given depth
banach_tarski_paradox()	Equidecomposition via non-measurable sets
vitali_set_paradox()	Non-measurable Vitali set construction
gabriels_horn_paradox()	Infinite surface area, finite volume
bertrand_paradox()	Ambiguity of "random chord" in a circle
buffon_needle_problem()	Geometric probability via needle drops

Applications

Type / Function	Description
BlackSwanEvent	Type for high-impact, low-probability events
MarketCrashEvent	Type for financial market crash events
BettingEdgeCase	Type for edge cases in probabilistic betting
impact_severity(event)	Severity measure for extreme events
expected_impact(event)	Expected impact accounting for severity
expected_value(event)	Expected value of event outcome
handles_black_swan(system)	Test if a system handles black swan events
handles_zero_prob_events(system)	Test if a system handles zero-prob events
handles_zero_prob_event(system)	Singular alias for above

Visualization

Function	Description
plot_zero_probability(dist)	Visualize the zero-probability paradox
plot_continuum_paradox()	Visualize the continuum paradox
plot_density_vs_probability(dist)	Compare density and probability
plot_epsilon_neighborhood(dist, x)	Visualize shrinking neighborhoods
plot_black_swan_impact(events)	Visualize black swan impact distribution

Integration with ECHIDNA

ZeroProb.jl is designed for use with ECHIDNA’s probabilistic reasoning:

• Premise Selection: Rank premises by density ratio instead of raw probability

• Proof Search: Use epsilon-neighborhood probability to guide search

• Anomaly Detection: Flag events far from distribution support using Hausdorff measures

Development Status

Current Version: 0.1.0

• ✓ Core types and density ratio

• ✓ Hausdorff measure and dimension

• ✓ Epsilon-neighborhood probability

• ✓ Extended types (tail risk, quantum, insurance)

• ✓ Extended measures (KL divergence, Fisher information, Radon-Nikodym)

• ✓ Classical paradoxes (Cantor, Banach-Tarski, Vitali, Bertrand, Buffon)

• ✓ Black swan and extreme event applications

• ✓ Visualization functions

• ✓ Comprehensive test suite (280 tests)

• ❏ Performance optimization

• ❏ Integration examples with ECHIDNA

See ROADMAP.adoc for future development plans.

References & Bibliography

Measure Theory & Probability Foundations

• Kolmogorov, A.N. Foundations of the Theory of Probability. Chelsea Publishing, 1933/1956. — Axiomatic probability theory.

• Billingsley, P. Probability and Measure. 3rd ed., Wiley, 1995. — Measure-theoretic probability.

• Halmos, P.R. Measure Theory. Graduate Texts in Mathematics 18, Springer, 1950/1974. — Classic measure theory reference.

• Dudley, R.M. Real Analysis and Probability. Cambridge University Press, 2002. — Rigorous integration of real analysis and probability.

• Feller, W. An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed., Wiley, 1971. — Continuous distributions, densities, and convergence.

Fractal Geometry

• Falconer, K.J. Fractal Geometry: Mathematical Foundations and Applications. 3rd ed., Wiley, 2014. — Hausdorff dimension, box-counting, iterated function systems.

Extreme Value Theory & Tail Risk

• Taleb, N.N. The Black Swan: The Impact of the Highly Improbable. 2nd ed., Random House, 2010. — Black swan events and fat-tailed distributions.

• de Haan, L. & Ferreira, A. Extreme Value Theory: An Introduction. Springer, 2006. — GEV/GPD distributions, tail risk.

• Embrechts, P., Kluppelberg, C., & Mikosch, T. Modelling Extremal Events for Insurance and Finance. Springer, 1997. — Extreme events in insurance and financial applications.

Quantum Measurement

• Nielsen, M.A. & Chuang, I.L. Quantum Computation and Quantum Information. 10th anniversary ed., Cambridge University Press, 2010. — Born rule and quantum measurement theory.

Classical Paradoxes

• Bertrand, J. Calcul des probabilites. Gauthier-Villars, 1889. — Bertrand paradox origin.

• Banach, S. & Tarski, A. "Sur la decomposition des ensembles de points en parties respectivement congruentes." Fundamenta Mathematicae 6, 1924, pp. 244-277. — Banach-Tarski theorem.

License

Palimpsest-MPL License v1.0 (MPL-2.0) — see LICENSE.

Contributing

See CONTRIBUTING.md for contribution guidelines.

Citation

If you use ZeroProb.jl in research, please cite:

@software{zeroprob2026,
  author = {Jewell, Jonathan D.A.},
  title = {ZeroProb.jl: Zero-Probability Event Analysis},
  year = {2026},
  version = {0.1.0},
  url = {https://github.com/hyperpolymath/ZeroProb.jl}
}

Contact

• Author: Jonathan D.A. Jewell <jonathan.jewell@open.ac.uk>

• GitHub: https://github.com/hyperpolymath/ZeroProb.jl

• Issues: https://github.com/hyperpolymath/ZeroProb.jl/issues