Advanced Inference

Contents

This section demonstrates Fugue's inference methods beyond adaptive MCMC:

Sequential Monte Carlo - Particle-based inference with an evidence estimate
Approximate Bayesian Computation - Likelihood-free inference from forward simulation
Variational Inference - Fast, optimization-based posterior approximation

Fugue's headline feature list advertises "Multiple Inference Methods: MCMC, SMC, Variational Inference, ABC." The Bayesian Coin Flip tutorial and most of the Statistical Modeling tutorials use adaptive_mcmc_chain. This section covers the other three, each on the same running example so you can compare their posteriors directly.

A gradient-based fifth option: HMC

Fugue also ships Hamiltonian Monte Carlo (hmc_chain), which mixes far better than single-site MH on correlated continuous posteriors by moving all continuous sites jointly using (finite-difference) gradient information. It isn't covered by its own tutorial page yet, but is fully documented — with a runnable example — in the hmc module rustdoc and listed alongside the other methods in the README.

The running example

All three tutorials in this section perform inference on the same conjugate Normal-Normal model, so their results are directly comparable:

$μ \sim N (0, 1), y ∣ μ \sim N (μ, 0. 5^{2}), y_{obs} = 1.5$

Because both the prior and likelihood are Gaussian, the posterior has a closed form (precision-weighted combination of prior and likelihood):

$μ ∣ y_{obs} \sim N (1.2, 0.2), σ_{post} = 0.2 \approx 0.4472$

Having ground truth in hand means each tutorial can check its inference method against an exact number instead of asking you to eyeball a histogram — the same property that makes each method's corresponding example (examples/smc_inference.rs, examples/abc_inference.rs, examples/vi_inference.rs) a genuine regression test rather than a "runs without panicking" demo.

When to reach for which method

Method	Function	Good fit when...
MCMC (adaptive MH)	`adaptive_mcmc_chain`	General-purpose default; works on discrete and continuous sites
HMC	`hmc_chain`	Continuous, correlated posteriors where MH mixes slowly
SMC	`adaptive_smc`	You also want a log-evidence estimate, or the posterior is multimodal/hard to reach by local moves
VI	`optimize_meanfield_vi_with_config`	You need speed over exactness, or a differentiable, tunable approximation
ABC	`abc_smc_weighted`	The likelihood is intractable but you can simulate from the model

Try it yourself

cargo run --example smc_inference cargo run --example abc_inference cargo run --example vi_inference