# Approximate Variational Inference Based on a Finite Sample of Gaussian Latent Variables

## TL;DR

This paper introduces a new variational inference scheme that simplifies approximation for complex integrals involving Gaussian latent variables, especially in cases where traditional bounds are hard to derive.

## Contribution

It proposes a novel approach to variational inference that overcomes difficulties in deriving lower bounds for certain integrals, broadening applicability.

## Key findings

- Effective on synthetic and real datasets
- Applicable to geophysical models where standard methods fail
- Simplifies variational inference in complex scenarios

## Abstract

Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lower bound is then optimised with respect to its free parameters, the so called variational parameters. However, this is not always possible as for certain integrals it is very challenging (or tedious) to come up with a suitable lower bound. Here we propose a simple scheme that overcomes some of the awkward cases where the usual variational treatment becomes difficult. The scheme relies on a rewriting of the lower bound on the model log-likelihood. We demonstrate the proposed scheme on a number of synthetic and real examples, as well as on a real geophysical model for which the standard variational approaches are inapplicable.