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Direct Amortized Likelihood Ratio Estimation
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Proposes the Direct Neural Ratio Estimator (DNRE), which takes parameter pairs as direct input to estimate the likelihood ratio in one forward pass, and benchmarks it against NRE and BNRE with random-walk MH and HMC on SBI tasks and a quadcopter design problem.
Problem
Likelihood-free (simulation-based) Bayesian inference requires estimating likelihood ratios for MCMC; prior amortized ratio estimators either need two forward passes or rely on likelihood-to-evidence ratios, and HMC has not been compared with random-walk MH for ratio-estimator-based SBI.
Approach
Train a classifier that takes the observation and both parameter values (x, θ, θ') and directly outputs the likelihood ratio p(x|θ)/p(x|θ') in a single forward pass, using swapped-input ordering for the zero label; also derive a Monte Carlo posterior estimate via logSumExp and a gradient estimator that approximates the log-likelihood gradient for HMC.
Key results
- DNRE achieved the best C2ST score on 5 of 10 SBI benchmark tasks with random-walk MH, versus 4 for NRE and 1 for BNRE.
- On the 1D Gaussian toy model, DNRE's log-ratio MSE was 0.104 / 0.122 / 0.124 across sigma 0.1/0.3/0.5, compared with NRE at 0.136 / 0.207 / 0.759 and BNRE at 3.584 / 4.289 / 3.693.
- With HMC, DNRE achieved the lowest C2ST on 5 of 10 benchmarks, and the paper reports the two samplers each winning on several tasks, with HMC viable as an alternative to random-walk MH.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use direct pair-conditioned likelihood ratio estimation (DNRE) instead of NRE or BNRE | Amortized SBI with random-walk MH · SBI Benchmark, 10 tasks | C2ST | Mixed |
| Use direct pair-conditioned likelihood ratio estimation (DNRE) instead of NRE or BNRE | Amortized SBI with HMC · SBI Benchmark, 10 tasks | C2ST | Mixed |
| Use likelihood-free HMC instead of random-walk MH | SBI ratio estimators · SBI Benchmark | C2ST | Mixed |
| Use DNRE instead of NRE | One-dimensional Gaussian ratio approximation · Gaussian model, σ = 0.1 | MSE: 0.136 -> 0.104 | Helped |
| Use DNRE instead of NRE | One-dimensional Gaussian ratio approximation · Gaussian model, σ = 0.3 | MSE: 0.207 -> 0.122 | Helped |
| Use DNRE instead of NRE | One-dimensional Gaussian ratio approximation · Gaussian model, σ = 0.5 | MSE: 0.759 -> 0.124 | Helped |
| Increase Monte Carlo sample size M for the DNRE posterior approximation | DNRE posterior approximation · Two Moons | Expected coverage | Helped |
| Use DNRE rather than the existing design | Quacopter design optimization with likelihood-free HMC · Existing quadcopter design | Structural interferences: 60 to 4 | Helped |
Surprising findings
- BNRE performs very poorly on the toy Gaussian model when log-likelihood differences exceed about two, with MSE above 3.5 across all noise levels.
- On the quadcopter ranking task, NRE and BNRE overlap heavily in top-100 designs with DNRE (89 and 87), yet DNRE shares only 8 of its top-10 with each baseline.
Limitations
- DNRE's Monte Carlo posterior estimate requires integrating over theta' samples, which is computationally intractable for higher-dimensional parameters.
- C2ST is a classifier-based metric that cannot capture all indicators of estimator quality; differences between methods are often within standard deviations.
- The gradient estimator sets the theta' contribution to zero, justified only for well-fit estimators.
Experimental setup
Datasets: SBI Benchmark (Two Moons, Gaussian Linear, Lotka-Volterra, SIR, SLCP, GLU, SLCP Distractors, Bernoulli GLM, Gaussian Mix Quadcopter design simulation data (4,490 train+validation observations, 1,059 test)
Benchmarks: SBI Benchmark Two Moons Gaussian hierarchical toy model quadcopter design
Metrics: C2ST MSE of log likelihood ratio expected coverage mean log posterior of ground truth parameters top-k overlap
Baselines: NRE BNRE random-walk Metropolis-Hastings Hamiltonian Monte Carlo
Builds on: Neural Ratio Estimation (Hermans et al.) Balanced Neural Ratio Estimation LAMPE
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