Automated Discovery of Maximum Entropy Distribution Families via Genetic Programming with Monte Carlo Validation

Abstract

We present a novel computational framework for the automated discovery of analytical maximum entropy distribution families using genetic programming (GP) validated against parameterizable data generators. Unlike traditional approaches that fit parameters to known distribution forms, our method evolves the analytical structure of maximum entropy distributions themselves, discovering entirely new families that arise from specific measurement constraint combinations. The framework employs Monte Carlo sampling across generator parameter spaces to validate whether evolved analytical forms truly represent maximum entropy solutions under given constraints. This approach enables systematic exploration of the space of possible distributions and their associated measurement relationships, potentially revolutionizing statistical modeling by automating the discovery process that has historically required mathematical insight. We demonstrate the method’s capability to rediscover known distributions and reveal novel forms, establishing a foundation for creating domain-specific distribution families and advancing our understanding of the fundamental relationships between measurement constraints and maximum entropy solutions.

Keywords: genetic programming, maximum entropy, distribution discovery, Monte Carlo methods, symbolic regression, automated scientific discovery

1. Introduction

The principle of maximum entropy, first formalized by Jaynes (1957), provides a fundamental framework for constructing probability distributions that make the fewest assumptions beyond what is constrained by available measurements. This principle has proven invaluable across numerous scientific domains, from statistical mechanics to machine learning, yet the discovery of new maximum entropy distribution families has remained largely dependent on mathematical intuition and theoretical analysis.

Traditional statistical modeling relies on a relatively small catalog of well-established distribution families—normal, exponential, beta, gamma, and others—each discovered through decades of mathematical development. However, as modern applications generate increasingly complex and domain-specific measurement constraints, the limitations of this fixed catalog become apparent. The question arises: can we systematically discover new analytical forms that represent maximum entropy solutions for previously unexplored constraint combinations?

We propose a revolutionary approach that combines genetic programming with Monte Carlo validation to automate the discovery of maximum entropy distribution families. Our method transcends parameter fitting by evolving the analytical structure of distributions themselves, validated against families of parameterizable data generators that serve as ground truth oracles. This framework enables systematic exploration of the vast space of possible analytical forms while ensuring discovered distributions truly satisfy maximum entropy principles.

1.1 Contributions

Our primary contributions include:

  1. Novel Discovery Framework: The first automated system for discovering analytical maximum entropy distribution families rather than optimizing parameters of known forms

  2. Monte Carlo Validation Protocol: A rigorous validation methodology using parameterizable data generators as ground truth oracles for testing evolved distributions

  3. Systematic Exploration: A principled approach to exploring the relationship between measurement schema and resulting analytical forms

  4. Generalization Validation: A framework for testing distribution discovery across generator parameter spaces to ensure robust generalization

  5. Foundation for Statistical Innovation: A computational pathway for expanding the catalog of usable probability distributions in a systematic, automated manner

2.1 Maximum Entropy Principle

The maximum entropy principle, established by Jaynes (1957), states that given constraints from measurements, the probability distribution that maximizes entropy while satisfying those constraints is the most unbiased choice. For a continuous distribution with constraints E[f_i(x)] = α_i, the maximum entropy solution takes the exponential family form:

1

p(x) = exp(λ₀ + Σᵢ λᵢfᵢ(x)) / Z(λ)

where λᵢ are Lagrange multipliers and Z(λ) is the normalization constant. The challenge lies in determining the appropriate functional forms fᵢ(x) for specific measurement constraints and discovering the resulting analytical expressions.

2.2 Genetic Programming for Scientific Discovery

Genetic programming has demonstrated remarkable success in automated scientific discovery, from rediscovering physical laws to evolving mathematical expressions. Koza (1992) established the foundation for using GP in symbolic regression, while more recent work has applied evolutionary methods to discover differential equations and physical relationships.

However, existing GP applications to statistical modeling have focused primarily on:

Our approach represents a fundamental departure by targeting the discovery of probability distribution families directly.

2.3 Monte Carlo Methods in Validation

Monte Carlo methods have long been used for sampling from known distributions and validating statistical models. Recent advances in Monte Carlo Tree Search have been applied to symbolic regression, demonstrating the power of combining stochastic search with validation. Our innovation lies in using families of parameterizable data generators as validation oracles, enabling rigorous testing of whether evolved forms truly represent maximum entropy solutions.

3. Methodology

3.1 Framework Overview

Our framework consists of four integrated components:

  1. Genetic Programming Engine: Evolves candidate analytical expressions for maximum entropy distributions
  2. Measurement Schema: Defines the constraint types and measurement protocols
  3. Generator Family: Parameterizable data generation processes that serve as ground truth
  4. Monte Carlo Validator: Tests evolved expressions against generator outputs across parameter spaces

3.2 Genetic Programming for Distribution Discovery

3.2.1 Representation

We represent candidate maximum entropy distributions as expression trees with the following components:

Function Set F:

Terminal Set T:

Constraint Syntax: Expressions must maintain the exponential family structure:

1

p(x) = exp(Σᵢ λᵢfᵢ(x)) / Z

where the fᵢ(x) terms are evolved by the GP system.

3.2.2 Fitness Evaluation

The fitness of each candidate distribution involves multiple criteria:

  1. Entropy Maximization: Verification that the candidate actually maximizes entropy under given constraints
  2. Constraint Satisfaction: Adherence to measurement-derived constraints
  3. Normalization: Proper integration to unity
  4. Generator Agreement: Match with parameterizable generator families
  5. Parsimony: Preference for simpler expressions (Occam’s razor)

3.3 Monte Carlo Validation Protocol

3.3.1 Generator Families

We define families of parameterizable data generators G(θ) where θ represents the parameter vector. Each generator can produce synthetic datasets with known properties:

1

D_θ = {x₁, x₂, ..., xₙ} ~ G(θ)

Examples include:

3.3.2 Validation Procedure

For each evolved candidate distribution p*(x):

  1. Parameter Sampling: Sample θ ~ P(Θ) from generator parameter space
  2. Data Generation: Generate D_θ ~ G(θ)
  3. Constraint Extraction: Compute empirical constraints C_emp from D_θ
  4. Distribution Fitting: Fit candidate p*(x) to satisfy C_emp
  5. Comparison: Evaluate match between p*(x) and true generator distribution
  6. Aggregation: Repeat across parameter space and aggregate results

3.3.3 Validation Metrics

We employ multiple validation metrics:

3.4 Measurement Schema Integration

3.4.1 Constraint Types

Our framework accommodates various measurement constraint types:

Moment Constraints:

Quantile Constraints:

Functional Constraints:

Composite Constraints:

3.4.2 Adaptive Constraint Discovery

The system can adaptively discover which constraints are most informative for specific generator families:

  1. Constraint Ranking: Evaluate information content of different constraints
  2. Constraint Selection: Choose minimal sufficient constraint sets
  3. Constraint Refinement: Iteratively improve constraint specifications

4. Experimental Design

4.1 Validation Studies

4.1.1 Rediscovery Experiments

To validate our framework, we first attempt to rediscover known maximum entropy distributions:

Gaussian Distribution: From mean and variance constraints

Exponential Distribution: From mean constraint with positive support

Beta Distribution: From first two moments on [0,1]

4.1.2 Novel Discovery Experiments

After validation, we explore discovery of potentially novel forms:

Multi-Modal Constraints:

High-Dimensional Constraints:

Domain-Specific Constraints:

4.2 Performance Metrics

4.2.1 Discovery Success Rate

4.2.2 Novel Form Quality

4.2.3 Computational Efficiency

4.3 Baseline Comparisons

We compare against several baseline approaches:

  1. Traditional Maximum Entropy: Analytical derivation where possible
  2. Numerical Maximum Entropy: Optimization-based approaches
  3. Symbolic Regression: Standard GP without maximum entropy constraints
  4. Neural Density Estimation: Modern machine learning approaches

5. Expected Results and Implications

5.1 Immediate Outcomes

Rediscovery Validation: We expect successful rediscovery of standard distributions (normal, exponential, gamma) with high fidelity, demonstrating the framework’s correctness.

Novel Form Discovery: The discovery of previously unknown analytical maximum entropy forms for specific constraint combinations, particularly in multi-modal and high-dimensional cases.

Computational Feasibility: Demonstration that the approach scales reasonably with problem complexity and provides practical computational performance.

5.2 Scientific Implications

Distribution Catalog Expansion: A systematic method for expanding the available catalog of probability distributions beyond historically discovered forms.

Constraint-Distribution Mapping: Deep insights into the fundamental relationships between measurement constraints and resulting maximum entropy forms.

Domain-Specific Distributions: The ability to automatically generate statistically principled distributions tailored to specific application domains.

5.3 Practical Applications

Custom Statistical Models: Automated creation of optimal distributions for domain-specific applications in finance, biology, engineering, and social sciences.

Advanced Machine Learning: Enhanced probabilistic models with distributions specifically designed for particular data characteristics.

Scientific Discovery: A tool for discovering statistical relationships in experimental data where traditional distributions prove inadequate.

6. Challenges and Limitations

6.1 Computational Complexity

Genetic Programming Scale: Managing the exponential growth of the expression space as complexity increases.

Monte Carlo Validation Cost: Balancing validation thoroughness with computational feasibility across large parameter spaces.

Convergence Guarantees: Ensuring the system converges to globally optimal solutions rather than local maxima.

6.2 Theoretical Considerations

Maximum Entropy Verification: Rigorous proof that discovered forms actually maximize entropy under given constraints.

Uniqueness Questions: Addressing cases where multiple analytical forms might satisfy the same constraints.

Constraint Completeness: Ensuring that constraint sets uniquely determine maximum entropy solutions.

6.3 Practical Limitations

Generator Fidelity: The quality of discoveries depends on the representativeness of the generator families used for validation.

Expression Complexity: Balancing the expressiveness of the GP system with the interpretability of resulting distributions.

Domain Knowledge Integration: Incorporating prior knowledge about reasonable analytical forms without overly constraining the search.

7. Future Extensions

7.1 Advanced Techniques

Hierarchical Discovery: Multi-level approaches that discover distributions for related constraint families simultaneously.

Transfer Learning: Using distributions discovered for simple constraints to bootstrap discovery for more complex cases.

Ensemble Methods: Combining multiple discovered forms to handle complex constraint combinations.

7.2 Theoretical Development

Convergence Theory: Mathematical analysis of when and why the approach converges to correct maximum entropy solutions.

Constraint Sufficiency: Theory for determining minimal constraint sets that uniquely specify maximum entropy distributions.

Expressiveness Bounds: Understanding the limits of what analytical forms can be discovered through genetic programming.

7.3 Application Domains

Time Series: Discovery of maximum entropy distributions for temporal constraint patterns.

Spatial Processes: Extension to spatially correlated data with geometric constraints.

Network Data: Maximum entropy distributions on graph structures with topological constraints.

Multivariate Extensions: Scaling to high-dimensional constraint combinations.

8. Conclusion

We have presented a groundbreaking framework for the automated discovery of maximum entropy distribution families using genetic programming validated through Monte Carlo methods with parameterizable data generators. This approach represents a fundamental shift from parameter optimization to structure discovery, potentially revolutionizing how we develop and apply statistical models.

The framework’s ability to systematically explore the space of analytical forms while ensuring adherence to maximum entropy principles opens unprecedented possibilities for statistical modeling. By automating what has historically been a manual, insight-driven process, we enable the discovery of domain-specific distributions tailored to emerging applications and novel constraint combinations.

The integration of genetic programming with rigorous Monte Carlo validation provides both the exploratory power needed for discovery and the statistical rigor required for scientific acceptance. As experimental validation proceeds, we anticipate this methodology will establish new standards for statistical model development and scientific discovery.

The implications extend beyond statistics to any field requiring probabilistic modeling. From machine learning applications needing custom distributions to scientific domains with unique measurement constraints, this framework promises to democratize the creation of optimal statistical models while maintaining theoretical rigor.

Future work will focus on scaling the approach to increasingly complex constraint combinations, developing theoretical guarantees for convergence and optimality, and establishing the framework as a standard tool for statistical discovery across scientific disciplines.

References

Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106(4), 620-630.

Koza, J. R. (1992). Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press.

Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3), 379-423.

[Additional references would be included based on the specific literature review and experimental results]

Appendix A: Mathematical Formulation

A.1 Maximum Entropy Formulation

Given constraints E[f_i(X)] = α_i for i = 1, …, m, the maximum entropy distribution is:

1

p*(x) = arg max_p H(p) = -∫ p(x) log p(x) dx

subject to:

1
2

∫ p(x) f_i(x) dx = α_i, ∀i
∫ p(x) dx = 1

The solution has the form:

1

p*(x) = exp(λ₀ + Σᵢ λᵢfᵢ(x))

where λᵢ are determined by the constraint equations.

A.2 Genetic Programming Operators

Crossover: Exchange subtrees between parent expressions while maintaining exponential family structure.

Mutation: Replace subtrees with randomly generated expressions from the same function/terminal sets.

Selection: Tournament selection based on multi-objective fitness combining entropy maximization, constraint satisfaction, and parsimony.

A.3 Monte Carlo Integration

For validation, we compute:

1

∫ g(x) p*(x) dx ≈ (1/N) Σₙ g(x_n) / p*(x_n)

where x_n are samples from the generator distribution, enabling comparison between theoretical and empirical expectations.

Appendix B: Implementation Details

B.1 Software Architecture

B.2 Parameter Settings

B.3 Computational Requirements