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A Black-Box Debiasing Framework for Conditional Sampling
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Proposes a black-box debiasing scheme for conditional sampling from an unknown prior, combining samplers applied to recursively resampled empirical distributions with polynomial weights to achieve O(n^-k) bias and O(n^-1) variance.
Problem
Approximating a posterior P_{X|Y=y*} when the prior is unknown and only n i.i.d. prior samples are available; plug-in posterior samplers (Bayes rule or generative models applied to the empirical distribution) are biased because of the nonlinearity of the map from prior to posterior.
Approach
Treat the sampler f as a black box and apply it to recursively resampled empirical distributions, then combine the outputs with binomial (possibly negative) weights from an operator expansion (I - B_n)^k, so that the expected output matches f(pi) to order n^-k while variance stays O(n^-1).
Key results
- Binary prior experiments confirm bias decay rates of O(n^-k) for k=1..4 for the debiased estimator D_{n,k} g(T/n), matching reference slopes.
- Gaussian mixture prior experiments (k=1,2) show bias and variance decay rates consistent with the theoretical O(n^-k) bias and O(n^-1) variance predictions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply order-k debiasing with recursive resampling instead of the naive plug-in approximation | Conditional posterior approximation · Conditional posterior (continuous prior) | bias rate: \mathcal{O}(n^{-1}) -> \mathcal{O}_{L_1, L_2, k}(n^{-k}) | Helped |
| Apply order-k debiasing with recursive resampling instead of the naive plug-in approximation | Conditional posterior approximation · Conditional posterior (continuous prior) | variance rate: \mathcal{O}(n^{-1}) -> \mathcal{O}_{L_1, L_2, k}(n^{-1}) | No clear effect |
| Apply order-k debiasing | Binary posterior estimator · Binary prior | bias rate: \mathcal{O}(n^{-k}) | Helped |
| Apply order-k debiasing | Gaussian mixture posterior estimator · Gaussian mixture prior | bias rate: \mathcal{O}(n^{-k}) | Helped |
| Apply order-k debiasing | Gaussian mixture posterior estimator · Gaussian mixture prior | variance rate: n^{-1} | No clear effect |
Surprising findings
- Exact unbiasedness is impossible: for a binary prior the expected plug-in posterior is a polynomial in q while the true posterior is rational, so only approximate debiasing is achievable.
Limitations
- Rigorous guarantees are proven only for the Bayes-optimal sampler under continuous priors and for smooth samplers under finite discrete priors; generalization to general generative models such as diffusion is conjectured.
- Curse of dimensionality: bounds scale with (L2/L1)^{2kd}, requiring kd much smaller than log n.
- Signed (negative) weights mean the combined output is not a single sample; rejection sampling is needed, with bound M possibly large.
Experimental setup
Metrics: bias (absolute error of expected posterior) variance log-log convergence rate
Baselines: naive plug-in posterior
Builds on: jackknife bias reduction Bernstein polynomial approximation importance-weighted autoencoder jackknife
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