Simulator

Introduction

The simulator is the general term for the function mapping the model parameters, \(\boldsymbol{\theta}\), to the observed data, \(\boldsymbol{x}\). It implicitly defines a likelihood function \(p(\boldsymbol{x} | \boldsymbol{\theta})\), in the sense that it returns samples \(\boldsymbol{x}\) which follow a distribution whose density is \(p(\boldsymbol{x} | \boldsymbol{\theta})\). Therefore, once a prior on \(\boldsymbol{\theta}\) is specified, we have all the necessary ingredients to perform Bayesian inference.

Examples

Gaussian linear

Consider a simple model where the output data are the parameters plus some gaussian noise:

\[ x = \theta + \sigma \varepsilon,\]

where \(\varepsilon \sim \mathcal{N}(0, 1)\). This corresponds to the likelihood function

\[ \log p(x | \theta) \propto -\frac{1}{2} \left(\frac{x - \theta}{\sigma} \right)^2\]

The corresponding forward model (prior + simulator) can be implemented in the following steps:

1. \(\theta \sim p(\theta)\)
2. \(\varepsilon \sim \mathcal{N}(0, 1)\)
3. \(x \leftarrow \theta + \sigma \varepsilon\)
4. return \((\theta, x)\)