feat: Implement Bernstein-Flow distribution

[StatMixedML]

Add Bernstein-Flow as a new normalizing flow distribution using Bernstein polynomial quantile functions. This provides shape-constrained probabilistic modeling with natural monotonicity preservation.

Key Features

• BernsteinQuantileTransform using Bernstein polynomials
• Monotonic quantile functions with degree parameter (3-20)
• Numerical inverse transform with binary search
• Interpretable coefficients representing quantile values
• Integration with existing LightGBMLSS flow framework
• Comprehensive test suite

Closes #42

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Code Review: Bernstein-Flow Distribution Implementation

I've completed a thorough review of the Bernstein-Flow distribution implementation. Here's my detailed feedback:

Review Checklist:

• [x] Read repository guidelines (CLAUDE.md)
• [x] Review BernsteinFlow.py implementation
• [x] Analyze flow_utils.py modifications
• [x] Review test coverage and quality
• [x] Check integration with existing framework
• [x] Evaluate code quality and best practices
• [x] Assess performance considerations
• [x] Check for potential bugs or security issues
• [x] Provide final recommendations

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✅ Strengths

Mathematical Soundness

• Solid theoretical foundation: Proper implementation of Bernstein polynomials as quantile functions
• Monotonicity preservation: Clever use of cumsum and softplus to ensure beta[k] <= beta[k+1] (BernsteinFlow.py:66)
• Numerical stability: Appropriate clamping of inputs to avoid boundary issues (BernsteinFlow.py:81, 94)

Code Architecture

• Clean integration: Follows existing LightGBMLSS patterns and extends NormalizingFlowClass properly
• Comprehensive validation: Thorough input parameter validation with descriptive error messages
• Fallback mechanisms: Graceful handling of optional scipy dependency (BernsteinFlow.py:48-60)

Test Coverage

• Extensive test suite: 181 lines covering initialization, validation, mathematical properties, and integration
• Edge case testing: Validates parameter bounds, monotonicity, basis function properties
• Integration testing: Confirms compatibility with LightGBMLSS model class

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🔧 Areas for Improvement

Performance Considerations

Issue: Inefficient loop-based Bernstein polynomial evaluation

# BernsteinFlow.py:86-89 - Sequential loop
for k in range(self.degree + 1):
    basis = self._bernstein_basis(u, k)
    result += betas[k] * basis

Recommendation: Vectorize computation using PyTorch operations for better GPU utilization and performance.

Numerical Stability

Issue: Potential numerical issues with high-degree polynomials

• Manual binomial coefficient computation (BernsteinFlow.py:57-60) may lose precision for large degrees
• Binary search tolerance of 1e-6 may be insufficient for high-precision applications

Recommendations:

• Use torch.lgamma for more stable log-binomial coefficient computation
• Make tolerance configurable or adaptive based on data precision

Code Quality Issues

Issue: Redundant binomial coefficient computation

# BernsteinFlow.py:125-135 - Duplicated logic from _compute_binomial_coefficients
try:
    from scipy.special import comb
    binomial_coeff = float(comb(degree, k))
except ImportError:
    # Manual calculation repeated...

Recommendation: Extract to a shared utility method to reduce code duplication.

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🐛 Potential Bugs

Parameter Initialization

Issue: Fixed initialization may cause convergence problems

# BernsteinFlow.py:38 - May not suit all data distributions
init_values = torch.linspace(support_bounds[0], support_bounds[1], degree + 1)

Recommendation: Consider data-adaptive initialization using empirical quantiles.

Derivative Computation

Issue: Complex derivative logic with potential edge cases

• The derivative computation in _bernstein_derivative (BernsteinFlow.py:92-117) is intricate and may have edge cases for degree=0 or boundary values Recommendation: Add more comprehensive unit tests for derivative computation across different degrees and input ranges.

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🔒 Security Assessment

✅ No security concerns identified

• No external network calls or file operations
• Input validation prevents malformed parameters
• Proper error handling without information leakage

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🧪 Test Coverage Assessment

✅ Excellent coverage with minor gaps:

• Core functionality well-tested (initialization, transforms, validation)
• Missing: High-degree polynomial edge cases, extreme input values, performance benchmarks
• Suggestion: Add property-based tests for mathematical invariants

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📋 Final Recommendations

High Priority

1. Vectorize polynomial evaluation for better performance
2. Extract duplicate binomial coefficient logic to utility method
3. Add adaptive parameter initialization option

Medium Priority

4. Improve numerical stability for high-degree polynomials
5. Add more derivative edge case tests
6. Consider making tolerances configurable

Low Priority

7. Add performance benchmarks comparing to other flow methods
8. Document computational complexity in docstrings

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🎯 Overall Assessment

This is a well-implemented, mathematically sound addition to the LightGBMLSS framework. The code demonstrates good understanding of normalizing flows and Bernstein polynomials. The implementation follows project conventions and includes comprehensive testing.

Recommendation: ✅ Approve with minor improvements

The identified issues are primarily performance optimizations and code quality improvements rather than fundamental problems. The implementation is ready for merge with the suggested enhancements addressed in follow-up commits.

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