Optimize ELBO computation for normal prior

[xiki-tempula]

Summary

This PR significantly improves the performance of BayesMBAR when using prior='normal' by:

• Adding configurable elbo_samples with intelligent auto-determination
• Implementing early stopping for hyperparameter optimization
• Pre-computing constant terms to avoid redundant computation
• Adding input validation with clear error messages

Motivation

When running BayesMBAR with prior='normal' on large datasets (e.g., 48 states, 240K configurations), the original implementation was extremely slow or ran out of memory due to:

1. Hardcoded 1024 ELBO samples per optimization step
2. Fixed 10,000 optimization steps with no early stopping
3. Repeated computation of log(num_conf) on every likelihood evaluation

Changes

New Parameters

Parameter	Default	Description
elbo_samples	"auto"	Number of samples for ELBO estimation. Auto-determines based on problem size, or accepts a positive integer.
early_stopping_patience	100	Stops optimization after N steps without improvement
early_stopping_tol	1e-4	Minimum improvement threshold to reset patience counter

Auto elbo_samples Determination

The auto-determination uses heuristics based on:

• Problem dimensionality: scales with √(m-1) for m states
• Dataset size: reduces samples for large datasets (>50K configs) since each evaluation is more informative
• Bounded to [32, 256] for practical limits

Example auto-determined values:

States	Configs	Auto Value
10	5,000	36
48	24,000	92
48	240,000	54
100	500,000	79

Performance Impact

Optimization	Speedup
Reduced ELBO samples	~10-20x per step
Early stopping	~2-10x (stops when converged)
Pre-computed constants	~1.1x

Testing

Added new tests:

• test_auto_elbo_samples_scaling: Verifies scaling behavior and bounds
• test_elbo_samples_validation: Verifies input validation and error messages
• test_BayesMBAR_uniform_prior_accuracy: Quick accuracy check

All tests pass in ~12 seconds.

Usage

from bayesmbar import BayesMBAR

# Auto-determined (recommended)
mbar = BayesMBAR(
    energy, num_conf,
    prior='normal', mean='linear',
    state_cv=state_cv,
    elbo_samples='auto',  # Default - auto-determined
)

# Or explicit value
mbar = BayesMBAR(
    energy, num_conf,
    prior='normal', mean='linear', 
    state_cv=state_cv,
    elbo_samples=64,
    early_stopping_patience=50,
)