MCMC Hierarchical Model

TransPlan's third inference engine uses Markov Chain Monte Carlo to fit a hierarchical Bayesian survival model to SRTR data. This approach quantifies honest uncertainty in model parameters rather than treating point estimates as ground truth.

Motivation

Monte Carlo simulation uses fixed SRTR data values (median waits, mortality rates, delisting rates) as known constants. But these estimates carry statistical uncertainty. A per-center estimate of kidney mortality is derived from finite sample sizes, with confidence intervals that overlap. The MCMC model learns the joint posterior distribution of all parameters, propagating this uncertainty through to outcome probabilities.

Model Specification

A three-level Bayesian hierarchical model:

1. National hyperpriors: population-level priors on organ-specific baseline mortality, delisting, and graft survival
2. Center random effects: deviations from national rates per center, pooled via exchangeability
3. Patient-level covariates: blood type, urgency, age as multipliers on the center effects

The model includes 92 free parameters per organ (6 organs, so ~550 parameters total):

• Baseline mortality and delisting rates
• one random effect per center (up to 248 in full granularity; ~50 in state mode) per outcome
• Blood type offsets (8 types)
• Urgency offsets (4 levels)
• Age scaling
• Correlation between mortality and delisting (shared frailty via LKJ-Cholesky)

Fitting Process

Fitting occurs offline using PyMC's NUTS (No-U-Turn Sampler), a variant of Hamiltonian Monte Carlo. For each organ:

1. SRTR historical data (15 biannual releases from 2018 to 2025) is loaded
2. The model is specified in PyMC with informed priors
3. NUTS samples 4 chains of 2000 iterations each (discarding 1000 warmup)
4. Convergence is assessed: R-hat < 1.01, ESS > 400 per parameter
5. The posterior trace is saved as an ArviZ NetCDF file (~10-50 MB per organ)

This process runs once per organ and takes approximately 20 to 40 minutes per organ on a modern CPU.

Query-Time Inference

At query time:

1. The cached posterior trace for the requested organ is loaded
2. 50 draws are sampled uniformly from the posterior
3. For each draw, a complete Monte Carlo simulation is run using posterior parameter values
4. Results are aggregated into median probabilities and credible intervals

Total query time: approximately 400ms to 800ms for 248 centers.

Uncertainty Quantification

Unlike Monte Carlo (which has sampling error) or BBN (which is deterministic), MCMC produces Bayesian credible intervals that reflect parameter uncertainty.

For a given center and patient:

• Posterior predictive 95% credible interval = the middle 95% of per-draw p24 values
• This interval reflects: uncertainty in city effects, blood type multipliers, mortality/delisting rates, and their interactions, all learned from data

Shared Frailty

The MCMC model includes a bivariate normal with LKJ-Cholesky prior linking city-level mortality and delisting offsets. This replaces the fixed copula in Monte Carlo, allowing the model to learn the correlation between these competing risks from data.

The learned correlation typically ranges from 0.2 to 0.5 across organs, indicating moderate positive dependence.

Validation

MCMC and Monte Carlo rankings were cross-validated on representative patient profiles. Spearman rank correlation: 0.48 to 0.68 by organ. Posterior predictive checks compare MCMC posterior predictions against observed SRTR statistics and show good agreement on national medians, city factors, blood type multipliers, and monotonicity of urgency effects.

Access

Use the /simulate endpoint with ?inference_mode=mcmc to run MCMC inference. The frontend includes a dropdown to select the inference engine. Results are returned in the same schema as Monte Carlo, with credible intervals replacing bootstrap confidence intervals.

Limitations

The hierarchical model assumes center effects are exchangeable (pooled via a normal prior). In reality, centers differ substantially in unobserved ways (case complexity, referral networks, center specialization). The model partially handles this via the LKJ frailty, but unmeasured confounding remains.

MCMC is sensitive to prior specification. The priors are now empirical per organ and metric (#317), measured from the release panel rather than the flat Beta(2,2) guess they replaced — that guess was simultaneously over-trusting some panels and under-trusting others.

The prior-sensitivity caveat this section used to carry has since been answered directly: a simulation-based calibration study draws parameters from the prior, simulates data, refits, and checks that the resulting rank statistics are uniform. All four monitored parameters pass (KS p = 0.41-0.57), which is evidence the priors, likelihood and sampler are jointly calibrated — not merely that the sampler converges. See docs/sbc-report.md.

Sensitivity analysis is still worthwhile for policy-relevant conclusions, but the specific worry that the priors are silently overconfident has been tested rather than left open.

Fitting requires substantial computational resources (hours per organ), and the model is refit when a new SRTR release is ingested rather than on a fixed calendar.

It is worth being explicit about why the fit uses the most recent release rather than pooling all 15. A panel-likelihood study (#358) tested exactly that: pooling across releases with cross-release shrinkage loses to the current single-release design, because centers genuinely drift over time, so older releases describe a center that no longer exists. The single-release choice is therefore an evidence-backed decision, not a simplification awaiting improvement. See docs/panel-fit-report.md.