# Approximating Bayes

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This is the note for Martin, G. M., Frazier, D. T., & Robert, C. P. (2024). Approximating Bayes in the 21st Century. Statistical Science, 39(1), 20–45. https://doi.org/10.1214/22-STS875

approximate Bayes methods:

- produce computational solutions to certain “intractable” statistical problems that challenge exact methods like MCMC

four main approximate techniques:

- approximate Bayesian computation (ABC)
- Bayesian synthetic likelihood (BSL)
- variational Bayes (VB)
- integrated nested Laplace approximation (INLA)

simulation-based approaches: ABC and BSL optimization-based approaches: VB and INLA

## Bayesian computation in a nutshell

- data generating process: $p(y\mid \theta)$
- prior belief: $p(\theta)$

the quantities that underpin the whole of Bayesian analysis can be expressed as expectations

\[\bbE[g(\theta)\mid y]\]and

\[\bbE[g(\theta)\mid \cM]\]all Bayesian computational techniques:

- deterministic integratiomn methods
- exact simulation methods
- approximate methods

### Simulation-based approaches

#### approximate bayesian computation (ABC)

in cases where, despite the complexity of the problem preventing the evaluation of $p(y\mid \theta)$, $p(y\mid\theta)$ and $p(\theta)$ can still be simulated from

#### Bayesian synthetic likelihood (BSL)

### Optimization-based approaches

#### Variational Bayes (VB)

#### Integrated nested Laplace approximation

Laplace asymptotic approximation