Concepts: the reasoning behind the fit workflow

The runbook (camdl docs workflow) gives the order of operations. This explains why — the identifiability logic, what priors are actually for, and the epistemic stance that runs through all of it: a failing diagnostic is information, not a thing to tune away. Read once at onboarding; the runbook is enough for the per-fit loop.

Synthetic recovery is necessary, not sufficient

Fitting synthetic data generated from known truth tests the pipeline: did the optimizer and sampler find the parameters you put in? It does not test whether your model describes the world. A model can recover its own synthetic data perfectly and still be wrong about reality — the synthetic data came from that same wrong model.

So recovery is a gate you must pass before real data, but passing it proves only that the machinery works. Whether the model is right is a separate question, answered by real-data checks (predictive checks and external oracles, below).

Identifiability: what the data can and cannot pin

Some parameter combinations are not separable from the data alone. The WA-State seeding example: the introduction time τ and the seed size n_seed trade off along a ridge — a small late introduction and a large early one produce the same observed growth curve. The data constrains their joint effect (the early trajectory) but not the two parameters separately.

This is structural non-identifiability, and more sampling does not fix it — the likelihood is genuinely flat along the ridge. You see it as a flat profile likelihood, a posterior that pins to a bound, or chains that disagree (R̂ ✗). The diagnostics in camdl docs workflow are how you detect it; the fix is below.

Why priors are load-bearing, not a nuisance

A prior is how you supply the information the data lacks.

On an identified parameter, a weak prior barely moves the posterior — the data dominates.
On a non-identified direction (the ridge), the prior is what regularizes it: it selects the part of the ridge consistent with outside knowledge — a reference class (“how large are cryptic introductions, typically?”). This is legitimate, as long as the prior is honest (grounded in knowledge, not data hindsight) and you report its influence with a prior-sensitivity sweep.

The trap is the mirror image: because the prior shows up in the posterior, a prior chosen to make the sampler converge launders your ignorance into a false-confident answer. camdl refuses silent flat priors for exactly this reason — the prior is a scientific claim, so it must be explicit and defensible, not a knob you turn until the chains agree.

A failing gate is information

Put the pieces together with the WA fit:

Uninformative priors → the n_seed posterior pins to its upper bound and camdl fit summary reports max R̂ = 1.216 ✗.
The instinct — “fix the convergence,” run more sweeps, loosen the threshold — is backwards. The failing R̂ correctly reported that the data do not identify (τ, n_seed) jointly. There was no well-defined posterior to converge to.
The fix was to add structure (weakly-informative priors), after which R̂ passed honestly — because the posterior was now actually well-defined and the ridge had collapsed to a bowl.
The result validated against an independent oracle: the τ posterior agreed with the Bedford genomic estimate (after the report-time-vs-infection-time correction).

Lowering the R̂ threshold in step 2 would have produced a “converged” fit that was meaningless. Tuning a diagnostic away — wider tolerance, fewer chains, a prior picked for convergence — is epistemic laundering: it manufactures the appearance of a good answer. For software whose outputs inform public-health decisions, that is the failure mode to fear most. A diagnostic that fails is telling you something true about your model–data–prior triple; listen to it.

`dt` and discretization bias

The chain_binomial backend discretizes continuous-time dynamics into steps of size dt. A coarse dt systematically biases the dynamics — and therefore the estimates. The subtlety: synthetic recovery at the same coarse dt will not reveal the bias, because it is baked identically into both the data generation and the fit.

The post-fit Richardson check (re-score θ̂ at dt, dt/2, dt/4) is what catches it: if θ̂ moves as you refine dt, your MLE is discretization- dependent and you must refit at a finer step. Heuristic for a starting value: a few steps per mean dwell time; halve and re-check the verdict when unsure.

Validating against the world

Recovery and convergence say the machinery works and the posterior is well-defined. They do not say the model is right. Three real-data checks, in increasing strength:

Prior-predictive — do the priors, before seeing data, imply epidemiologically plausible epidemics? (If the 95% prior envelope sits far above the data, the priors are too loose.)
Posterior-predictive — does the fitted model reproduce the observed data within its predictive bands?
External oracle — the strongest test: agreement with an independent estimate (a genomic TMRCA, a serosurvey, a published result). When the oracle disagrees, that is a finding to surface — never a number to tune toward.