# Fisher Efficient Inference of Intractable Models

arXiv: https://arxiv.org/abs/1805.07454 · HTML: https://mlsanity.com/paper/1805.07454/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 9 of 9)

- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Feature Shift Detection: Localizing Which Features Have Shifted via Conditional Distribution Tests](https://mlsanity.com/paper/2107.06929/) (2021)
- [A Computationally Efficient Method for Learning Exponential Family Distributions](https://mlsanity.com/paper/2110.15397/) (2021)
- [On Computationally Efficient Learning of Exponential Family Distributions](https://mlsanity.com/paper/2309.06413/) (2023)
- [Sound Source Separation Using Latent Variational Block-Wise Disentanglement](https://mlsanity.com/paper/2402.06683/) (2024)
- [Sequential Kernelized Stein Discrepancy](https://mlsanity.com/paper/2409.17505/) (2024)
- [Guidance for Prior Change via Density Ratio Estimation](https://mlsanity.com/paper/2608.21729/) (2026)
- [Correcting Mode Proportion Bias in Generalized Bayesian Inference via a Weighted Kernel Stein Discrepancy](https://mlsanity.com/paper/2503.02108/) (2025)

## Abstract (arXiv)

Maximum Likelihood Estimators (MLE) has many good properties. For example, the asymptotic variance of MLE solution attains equality of the asymptotic Cram'er-Rao lower bound (efficiency bound), which is the minimum possible variance for an unbiased estimator. However, obtaining such MLE solution requires calculating the likelihood function which may not be tractable due to the normalization term of the density model. In this paper, we derive a Discriminative Likelihood Estimator (DLE) from the Kullback-Leibler divergence minimization criterion implemented via density ratio estimation and a Stein operator. We study the problem of model inference using DLE. We prove its consistency and show that the asymptotic variance of its solution can attain the equality of the efficiency bound under mild regularity conditions. We also propose a dual formulation of DLE which can be easily optimized. Numerical studies validate our asymptotic theorems and we give an example where DLE successfully estimates an intractable model constructed using a pre-trained deep neural network.