# Likelihood Free Inference

We want estimations on theta that is robust of uncertainty of nuisance parameters.

https://arxiv.org/abs/1506.02169 Approximating Likelihood Ratios with Calibrated Discriminative Classifiers

n many fields of science, generalized likelihood ratio tests are established tools for statistical inference. At the same time, it has become increasingly common that a simulator (or generative model) is used to describe complex processes that tie parameters θ of an underlying theory and measurement apparatus to high-dimensional observations x∈ℝp. However, simulator often do not provide a way to evaluate the likelihood function for a given observation x, which motivates a new class of likelihood-free inference algorithms. In this paper, we show that likelihood ratios are invariant under a specific class of dimensionality reduction maps ℝp↦ℝ. As a direct consequence, we show that discriminative classifiers can be used to approximate the generalized likelihood ratio statistic when only a generative model for the data is available. This leads to a new machine learning-based approach to likelihood-free inference that is complementary to Approximate Bayesian Computation, and which does not require a prior on the model parameters. Experimental results on artificial problems with known exact likelihoods illustrate the potential of the proposed method.

https://github.com/diana-hep/carl

https://figshare.com/articles/NIPS_2016_Keynote_Machine_Learning_Likelihood_Free_Inference_in_Particle_Physics/4291565/1 NIPS 2016 Keynote: Machine Learning & Likelihood Free Inference in Particle Physics

https://arxiv.org/pdf/1605.07826.pdf Asymptotically exact inference in likelihood-free models

Many generative models can be expressed as a differentiable function of random inputs drawn from some simple probability density. This framework includes both deep generative architectures such as Variational Autoencoders and a large class of ‘likelihood-free’ simulator models. We present a method for performing efficient MCMC inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where Approximate Bayesian Computation might otherwise be employed. We use the intuition that inference corresponds to integrating a density across the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to coherently move between inputs exactly consistent with observations. We validate the method by performing inference tasks in a diverse set of models.