# Exact Inference with Latent Variables in an Arbitrary Domain

## TL;DR

This paper establishes conditions under which exact inference in latent variable models is possible using semidefinite programming, without prior knowledge of the latent variables or their domain, supported by theoretical analysis and concentration inequalities.

## Contribution

It introduces a novel SDP-based method for exact inference in latent models without prior domain knowledge, supported by theoretical guarantees and spectral analysis.

## Key findings

- SDP approach achieves exact inference without latent domain knowledge
- KKT conditions and spectral analysis predict SDP correctness accurately
- Provides new concentration inequalities related to latent variables

## Abstract

We analyze the necessary and sufficient conditions for exact inference of a latent model. In latent models, each entity is associated with a latent variable following some probability distribution. The challenging question we try to solve is: can we perform exact inference without observing the latent variables, even without knowing what the domain of the latent variables is? We show that exact inference can be achieved using a semidefinite programming (SDP) approach without knowing either the latent variables or their domain. Our analysis predicts the experimental correctness of SDP with high accuracy, showing the suitability of our focus on the Karush-Kuhn-Tucker (KKT) conditions and the spectrum of a properly defined matrix. As a byproduct of our analysis, we also provide concentration inequalities with dependence on latent variables, both for bounded moment generating functions as well as for the spectra of matrices. To the best of our knowledge, these results are novel and could be useful for many other problems.