Sequential Bayesian updating¶

In SHM and long-term monitoring, data arrives in batches over time — a new measurement campaign after seasonal changes, after a reported event, or on a scheduled interval. Sequential Bayesian updating lets you incorporate each new dataset without re-processing all historical data:

\[p(\theta \mid D_1, D_2) \propto p(D_2 \mid \theta)\; \underbrace{p(\theta \mid D_1)}_{\text{new prior}}\]

The posterior from step \(t\) becomes the prior for step \(t+1\).

How it works in mcmckit¶

PosteriorPrior wraps a set of posterior samples and exposes a log_prior(theta) callable — the same interface as any hand-written prior function. This means all samplers work unchanged.

# After any sampling run
prior_next = result.as_prior(method='gaussian', discard=2000)

# Use exactly like a normal log_prior
problem_next = mc.Problem(prior=prior_next, likelihood=ll_new)
result_next  = mc.DRAM(...).run(problem_next, x0=prior_next.mean)

Choosing the method¶

method='gaussian' — fits \(\mathcal{N}(\mu, \Sigma)\) to the samples.

Fast and memory-efficient in any dimension.
prior.mean and prior.cov are immediately available.
Best when the posterior is approximately unimodal and elliptical — the typical case for structural stiffness / damping identification.

method='kde' — non-parametric kernel density estimate.

Handles multimodal or strongly skewed posteriors.
Evaluation cost grows with the number of samples and dimension.
Recommended for \(\leq\) 6–8 parameters.

Three-campaign example¶

# Campaign 0: flat prior, 5 observations
result0 = mc.DRAM(n_samples=12_000, initial_cov=0.5*np.eye(2)).run(prob0, x0=[10., 8.])

# Campaign 1: Gaussian prior from campaign 0, 5 new observations
prior1 = result0.as_prior(method='gaussian', discard=2000)
prob1  = mc.Problem(prior=prior1, likelihood=ll1)
result1 = mc.DRAM(n_samples=12_000, initial_cov=prior1.cov).run(prob1, x0=prior1.mean)

# Campaign 2: further update, 10 new observations
prior2 = result1.as_prior(method='gaussian', discard=2000)
prob2  = mc.Problem(prior=prior2, likelihood=ll2)
result2 = mc.DRAM(n_samples=12_000, initial_cov=prior2.cov).run(prob2, x0=prior2.mean)

Posterior std across campaigns (2-DOF frame, k1=10, k2=8 N/m):

Campaign 0 (flat prior,     5 obs):  k1 std = 3.46,  k2 std = 1.72
Campaign 1 (Gaussian prior, +5 obs): k1 std = 2.35,  k2 std = 1.16
Campaign 2 (Gaussian prior,+10 obs): k1 std = 2.06,  k2 std = 1.03

Uncertainty shrinks monotonically as information accumulates.

Use with TMCMC¶

TMCMC requires samples from the prior, not just density evaluations. PosteriorPrior.sample(n) provides them:

prior = result0.as_prior(method='gaussian', discard=2000)
result1 = mc.TMCMC(n_particles=500).run(
    problem1,
    prior_samples=prior.sample(500),
)

Bayes consistency¶

Sequential updating is mathematically equivalent to a single batch run with all data combined, provided the density approximation is accurate. The Gaussian method introduces a small approximation error when the posterior deviates significantly from Gaussian; KDE reduces this at higher computational cost.

API reference¶

See the full runnable script at examples/sequential.py.