Mixture-proposal importance sampling for Flow estimator

[cweniger]

Summary

Generalises Flow._importance_sample to draw proposals from a mixture of the conditional and marginal flows instead of a single flow, weighted by the balance heuristic (multiple importance sampling). This is a defensive mixture in Hesterberg's sense: the marginal flow is the broad component that covers the tails the (tempered) conditional under-covers, while the conditional flow gives efficiency near the mode. The mixture density is used as the weight denominator, which bounds the worst-case importance weight and raises the effective sample size (importance_sample:n_eff_min).

What changed

• New proposal_mixture_beta parameter (default 0.5): fraction of the num_proposals drawn from the conditional flow; the rest are drawn from the marginal flow. Total proposal count is unchanged, so per-call cost is the same.
• Proposals are pooled from both flows; both log_prob_cond and log_prob_marg are evaluated on the full pooled set (required for the balance heuristic, regardless of which flow drew each sample).
• Weight denominator is the balance-heuristic mixture density Σ_k (n_k / N) · g_k(θ), computed from the actual sampled counts (exact under rounding).
• Targets are unchanged: tempered posterior c^{γ/(1+γ)} for proposal mode, importance-corrected posterior c/m for posterior mode.
• Small unrelated tidy-up: added a log_prefix to the FlowDensity online-norm logger so its metrics are namespaced.

Behaviour / compatibility

• proposal_mixture_beta=1.0 exactly reproduces the previous single-proposal behaviour in both modes (verified by reduction: log_g_mix → log_prob_cond, so the proposal weight collapses to −1/(1+γ)·log c and the posterior weight to −log m). Use it as a revert switch.
• The default 0.5 changes sampling results vs. main (this is the intended mixture). Watch importance_sample:n_eff_min — it should increase.

Refs

• Hesterberg (1995), Weighted Average Importance Sampling and Defensive Mixture Distributions, Technometrics 37(2):185-194.
• Veach & Guibas (1995), Optimally Combining Sampling Techniques for Monte Carlo Rendering, SIGGRAPH '95 (balance heuristic).

Test plan

• python -m py_compile falcon/estimators/flow.py
• Smoke-test a Flow run and confirm n_eff_min is ≥ the beta=1.0 baseline.

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 0% with 21 lines in your changes missing coverage. Please review.
✅ Project coverage is 10.33%. Comparing base (a175353) to head (c7cf168).
⚠️ Report is 2 commits behind head on main.

Files with missing lines	Patch %	Lines
falcon/estimators/flow.py	0.00%	21 Missing ⚠️

Additional details and impacted files

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- Coverage   10.38%   10.33%   -0.05%     
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  Files          30       30              
  Lines        3910     3927      +17     
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  Hits          406      406              
- Misses       3504     3521      +17     

Flag	Coverage Δ
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