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Bayesian Model Update

Statistical updating method — instantiates Predictive Residual Processing

Turns each observed surprise into a revised belief — folding new evidence into a prior to yield a posterior over the model, along with honest uncertainty.

Detecting a prediction error is one thing; changing the model principledly in response is another. Bayesian Model Update is the formal rule for that revision: it combines a prior belief with the likelihood of the observed data to produce a posterior over the model, carrying calibrated uncertainty forward. Its defining move is that it maintains a distribution, not a point — so the size of each update scales with how surprising the data is relative to how confident the model already was. A sharp prior barely budges for a mild surprise; a diffuse one moves a lot. This is the mechanism that owns how the generative model state changes when a residual arrives — as distinct from the machinery that detects, encodes, or later reviews that residual.

Example

A clinician holds a prior that a patient has condition X — say prevalence ≈2% in this population. A test with known sensitivity and specificity comes back positive. Bayes' rule combines the 2% prior with the test's likelihood to yield a posterior probability of disease that — because the base rate is so low — can still be far below what the positive result intuitively suggests. A second, independent finding updates the posterior again, each observation nudging the belief and its uncertainty rather than replacing it. The model state carried forward is not "positive/negative" but a calibrated probability with a width.

How it works

  • Specify the prior — the current belief over the model or its parameters, with its uncertainty.
  • Specify the likelihood — how probable the observed data is under each hypothesis; this is where the evidence enters.
  • Combine and normalize — multiply prior by likelihood to get the posterior; the surprise in the data determines how far belief moves.
  • Carry the posterior forward as the next round's prior, so learning is incremental and uncertainty-aware.

Tuning parameters

  • Prior strength — informative vs. weak; a strong prior resists noise but can also resist the truth.
  • Likelihood model — the assumed data-generating story; misspecify it and the posterior is confidently wrong.
  • Update granularity — full Bayesian inference vs. conjugate or approximate updates, trading fidelity for tractability.
  • Outlier robustness — how much a single surprising datum is allowed to move the posterior.
  • Forgetting / decay — how fast old evidence is discounted, so the belief can track a moving world.

When it helps, and when it misleads

Its strength is principled, uncertainty-aware revision: it naturally down-weights noisy evidence, expresses how sure it is, and makes the size of an update a function of genuine surprise rather than reflex.

Its failure modes trace to its inputs. Garbage priors or a misspecified likelihood yield confident nonsense — the tidy posterior lends false authority to assumptions no one checked.[1] Full inference can be intractable, forcing approximations whose errors are easy to forget. The classic misuse is choosing the prior that delivers the conclusion you wanted, or over-trusting a crisp posterior whose likelihood is wrong. The discipline: sensitivity-test the prior, validate the likelihood against held-out data, and keep the uncertainty band instead of collapsing to a point estimate.

How it implements the components

  • generative_model_state — it maintains the belief/parameter state from which predictions are generated, and revises it as evidence lands.
  • update_rule — Bayes' rule is the update rule: the explicit procedure by which a residual becomes a changed model.
  • confidence_and_uncertainty_state — the posterior carries calibrated uncertainty, the very quantity precision-weighting and thresholding downstream depend on.

It does not set the action thresholds on that confidence (that's Confidence Threshold Table, which reads the uncertainty this produces), detect that an update is overdue (that's Model Drift Monitoring), or store and replay past errors for learning (that's Prediction-Error Replay Buffer).

Notes

Bayesian update is the general update engine. Several siblings are its specializations for particular structure: the Innovation Residual Filter is (Kalman-style) Bayesian updating for linear-Gaussian state, and Temporal-Difference Update is the incremental error-driven update used in sequential prediction. Reach for those when their structure fits; reach for full Bayes when you need the honest, general posterior.

References

[1] Prior sensitivity — a posterior can be dominated by the prior when data are thin — and the related base-rate neglect (ignoring a low prior, as in the diagnosis example) are the standard cautions. Reporting the posterior under a range of priors is the recognized check. ↩