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Preferential Normalizing Flows

NeurIPS 2024 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Normalizing flow training and design · 1 citations (OpenAlex)

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Learns a normalizing flow over an expert's belief density from preferential k-wise comparisons/rankings using a RUM with exponential noise and a decision-theoretic empirical functional prior, trained by function-space MAP.

Problem

Eliciting a flexible high-dimensional belief density from an expert using only noisy preferential judgments (comparisons or rankings), where the density can neither be sampled nor evaluated, and flows tend to collapse or diverge in probability mass with small data.

Approach

Model expert choices as a random utility model with exponential noise so the k-wise winner distribution is a tempered, tilted belief density; learn a normalizing flow by function-space MAP with an empirical functional prior that up-weights the winning points (prior proportional to product of exp(f) at winners), which regularizes collapse and divergence.

Key results

• On synthetic targets, the flow improves log-likelihood and Wasserstein distance over a factorized normal baseline; e.g., Twogaussians20D log-likelihood -0.08 (flow) vs -6.35 (normal) and Wasserstein 4.55 vs 11.07.
• On Abalone7D, flow reaches log-likelihood -2.16 and Wasserstein 0.34 vs -5.53 and 0.53 for the normal baseline, though MMTV is slightly worse (0.29 vs 0.26).
• Wasserstein distance for Onemoon2D decreases from 0.70 at k=2 to 0.11 at k=10 with n=100, showing pairwise comparisons already give a usable but less accurate estimate.

Measured effects reported in the paper

Surprising findings

• Maximum likelihood without the functional prior yields diverging mass; e.g., the mean number of rooms in the learned LLM belief reaches about 125.
• Accuracy can slightly deteriorate for very large n, attributed to prior misspecification from treating finite k as infinite in the prior.

Limitations

• The method is efficient only for exponential noise with s=1, which gives the analytic prior; other choices need explicit normalization.
• Candidates are assumed sampled i.i.d. from a known lambda; active query selection is not analyzed.
• LLM expert does not follow the RUM model exactly, and results are on simulated or LLM experts rather than real humans.

Experimental setup

Datasets: Abalone California housing Onemoon2D (synthetic) Gaussian6D (synthetic) Twogaussians10D (synthetic) Twogaussians20D (synthetic) Funnel10D (synthetic)

Benchmarks: Onemoon2D Gaussian6D Twogaussians10D Twogaussians20D Funnel10D Abalone7D California housing LLM elicitation

Metrics: log-likelihood Wasserstein distance mean marginal total variation (MMTV)

Baselines: factorized normal distribution learned from preferential data

Builds on: normalizing flows Plackett-Luce model function-space MAP Neural Spline Flow RealNVP

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