Sequential Monte Carlo (SMC)

Sequential Monte Carlo (SMC)#

SMC is the recommended default sampler for EOS inference. It anneals a particle population from the prior (\(\lambda = 0\)) to the posterior (\(\lambda = 1\)) through a sequence of tempered distributions \(\pi_\lambda(\theta) \propto p(\theta)\, \mathcal{L}(\theta)^\lambda\), with \(\lambda = 1/T\) the so-called inverse temperature.

How it works. At each annealing step, the next temperature \(\lambda\) is chosen adaptively so that the effective sample size (ESS) after importance reweighting stays at target_ess x N. Particles are resampled with systematic resampling, then refreshed by running n_mcmc_steps MCMC transitions with one of two kernels: Gaussian Random Walk (SMC-RW) or NUTS (SMC-NUTS). The loop terminates when \(\lambda = 1\).

Evidence is accumulated as \(\log Z = \sum_t \Delta \log Z_t\), where each increment is computed from the importance weights at step t.

The sampler is implemented in blackjax. For more information about the inner workings, check out the blackjax source code here: blackjax-devs/blackjax. The blackjax SMC api is available in the documentation here: https://blackjax-devs.github.io/blackjax/autoapi/blackjax/smc/index.html.

Gaussian Random Walk kernel (smc-rw)#

The proposal covariance is estimated from the current particle cloud and scaled by random_walk_sigma2:

\[\Sigma_\text{prop} = \sigma^2 \cdot \hat{\Sigma}_\text{particles}\]

The covariance shape adapts every tempering step; only the overall scale is fixed. This makes the kernel well-suited to posteriors whose shape changes significantly during annealing.

This is the default sampler and runs comfortably on a laptop without a GPU.

sampler:
  type: smc-rw
  n_particles: 5000       # number of SMC particles
  n_mcmc_steps: 10        # number of MCMC steps per annealing level
  target_ess: 0.9         # target ESS to compute next temperature
  random_walk_sigma: 0.5  # proposal = sigma^2 * empirical covariance

NUTS kernel (smc-nuts) — experimental#

Warning

SMC-NUTS has not been thoroughly validated. Cross-check results with SMC-RW.

Uses the No-U-Turn Sampler as the refreshment kernel, which exploits gradient information via JAX automatic differentiation. The inverse mass matrix is adapted each annealing step using an eigen-decomposition of the Hessian evaluated at the highest-log-posterior particle, with SoftAbs regularisation. The step size is adapted with a simple dual-averaging update targeting target_acceptance.

sampler:
  type: smc-nuts
  n_particles: 10000
  n_mcmc_steps: 5
  target_ess: 0.9
  init_step_size: 0.01       # initial leapfrog step size
  mass_matrix_base: 0.2      # diagonal mass matrix baseline
  mass_matrix_param_scales:  # optional per-parameter overrides
    K_sat: 0.5
  target_acceptance: 0.7
  adaptation_rate: 0.3

Diagnostics#

After sampling, plot_diagnostics() produces a three-panel figure showing the temperature schedule, ESS evolution, and acceptance rate over annealing steps. These are saved automatically to the output directory as smc_diagnostics.png.

API reference#