The three accounts in every update

Bayes' rule becomes intuitive once probability is converted into odds. The prior establishes a starting balance. The likelihood ratio records one evidence transaction. The posterior is the balance after that transaction.

The prior is a population claim

In the bearing example, the prior probability of a fault is 0.01. That number does not express an attitude such as optimism or pessimism. It refers to a defined class of bearings under defined installation, inspection and time conditions. Change the supplier, age, operating temperature or defect definition and the relevant prior can change. A prior without its reference population is an orphaned number.

Convert probability p into odds by comparing the probability that the hypothesis is true with the probability that it is false:

Odds are useful because evidence can multiply them directly. A probability scale is bounded between zero and one, so equal numerical additions do not have equal meaning near its centre and edges. Odds measure relative support. Log-odds, introduced shortly, turn multiplication into addition.

The prior can come from historical rates, a hierarchical model, an earlier posterior, expert elicitation or a policy-defined reference class. Each source creates different obligations. Historical rates need temporal and population fit. Elicited priors need an elicitation protocol and sensitivity range. An earlier posterior needs lineage showing that the old evidence remains applicable and has not re-entered the new update through another route.

The likelihood ratio is directional evidence

Let E denote a positive alert. The alert's positive likelihood ratio compares how often that alert would appear under two rival states:

For the detector, P(alert | fault) is 0.90 and P(alert | sound) is 0.09. The likelihood ratio is therefore 10. A red alert is ten times as likely when the bearing is faulty as when it is sound. That is substantial evidence. It still begins from odds of 1 to 99, so multiplying by ten produces odds of 10 to 99, which convert to about 9.2 per cent.

A likelihood ratio of one is neutral: the evidence is equally likely in both states. A value above one supports H relative to the stated alternative. A value below one supports the alternative. This direction matters. The same observation can favour a hypothesis against one rival and barely distinguish it from another. Evidence never carries weight in isolation from a comparison.

Even a correctly defined ratio is an estimated quantity with modelling choices and uncertainty. Iyer and Lund's NIST analysis cautions against treating an expert's likelihood ratio as a theory-free transfer of evidential weight to a separate decision maker; its fitness depends on an explicit lattice of assumptions and uncertainty characterisation.[18] The operational lesson is direct: record the model, denominator, uncertainty and intended decision context beside the multiplier.

Neyman and Pearson's 1933 work established the central role of likelihood ratios in testing simple hypotheses under controlled error constraints.[2] Bayesian posterior odds and frequentist likelihood-ratio tests use related mathematical objects for different inferential contracts. The posterior combines a prior with relative evidence. A Neyman-Pearson test can choose a rejection rule with a specified long-run error property without assigning a posterior probability to a hypothesis. The distinction prevents one framework's guarantees from being imported into the other.

The posterior is a conditional claim

Once the odds are updated, convert them back into probability:

The posterior is conditional on everything encoded in the model: the hypothesis definition, prior reference class, detector behaviour, observed result and assumptions about how the evidence was sampled. It does not certify that those conditions are true. A perfectly executed calculation over a misspecified model can be precisely wrong.

That boundary is easy to miss when a system emits only a final score. A score of 0.62 could arise from a weak prior plus strong independent evidence, a strong prior plus neutral evidence, or a duplicated signal counted several times. These routes have different fragility. The closing balance is interpretable only with the transaction history that produced it.

Natural frequencies and odds solve different cognitive problems

The 1,000-bearing table exposes the denominator that probability notation can hide. It shows 9 true alerts beside 89 false alerts in one population. This format prevents a reader from silently replacing the question “How many alerted bearings are faulty?” with “How many faulty bearings alert?” Research on diagnostic reasoning has repeatedly found that people struggle with this inversion, including trained professionals, when the relevant base rates and conditional probabilities are presented abstractly.[5]

Natural frequencies are therefore an excellent inspection view for a single update. They expose sample size, prevalence, hits, misses, false alerts and true negatives. They also reveal when rounded percentages imply impossible or unstable counts. A reviewer can ask whether ten faults were observed or merely expected, whether the false-positive estimate came from 990 sound cases or ninety, and whether the reference sample resembles the current population.

Odds and likelihood ratios serve the next problem: carrying evidence through a sequence without rebuilding a full frequency tree after every observation. A 2025 preregistered crossover trial found a nuanced format effect: natural frequencies better supported single-test positive predictive value in the tested participants, while odds and likelihood ratios better supported the sequential calculation used in that study.[12] The result does not establish one universally superior notation. It suggests a design rule.

Use natural frequencies to audit the ingredients of an update, then use odds to transport those ingredients through time. A well-designed interface should let a reviewer switch views. The frequency view answers “Where did this multiplier come from?” The odds ledger answers “What did this multiplier do after the evidence already recorded?” Keeping both views available reduces the chance that compact mathematics conceals a weak denominator or that a large frequency tree obscures a long sequential trajectory.

Evidence is a multiplier, not a verdict

A likelihood ratio records movement between rival explanations. On a log-odds scale, each item contributes a signed increment. This makes a sequence of updates readable as an evidential trajectory rather than a succession of opaque scores.

The likelihood-ratio balance

Take logarithms of the odds identity. Multiplication becomes addition:

This form reveals a useful geometry. Evidence does not drag belief directly to a fixed probability. It moves belief by a distance. The same evidence increment has the same weight wherever the process starts, while its visible effect on probability varies because the probability scale compresses near zero and one.

The visual gain is operational. A reasoning trace can record an initial log-odds value, then one signed increment per admissible evidence item. Reviewers can see which item produced a large movement, which item contradicted the current hypothesis, and whether a threshold crossing depended on one fragile transaction.

Abraham Wald's work on sequential tests formalised decisions that continue sampling until accumulated likelihood evidence reaches a stopping boundary.[3] A Bayesian workflow may use different loss functions and posterior thresholds, but the same sequential discipline is valuable: observe, update, check whether another observation is worth its cost, then stop, defer or gather more evidence. The order is controlled by the decision problem, not by a ritual demand to collect every possible signal.

The same evidence can produce different posterior probabilities

The thought experiment's two factories can be plotted across a continuum of priors. With a fixed likelihood ratio of ten, the posterior is below 10 per cent when the prior is 1 per cent, equals 50 per cent when the prior odds are 1 to 10, and exceeds 90 per cent only when the prior is already above about 47 per cent. Strong evidence can still leave a rare hypothesis unlikely.

This is the mechanism behind base-rate neglect. People can focus on how representative an observation seems and underweight how often the underlying state occurs. Kahneman and Tversky documented prior-probability neglect in prediction tasks, while later sequential work has modelled how underweighting the base rate shapes entire belief trajectories.[4][11] The error is not cured by attaching more decimal places to a test score. The missing account must be restored.

A sequence is one joint update written in stages

For evidence items E1 through En, the exact sequential identity conditions each new likelihood ratio on the evidence already seen:

Taking logs gives one opening balance plus a sum of conditional evidence weights. If the likelihood model is coherent and the same complete evidence set is used, changing the presentation order does not change the final posterior. It changes only the intermediate path. Order effects in human judgement or software often signal approximation, forgotten conditioning, changing hypotheses or evidence that was selected in response to earlier results.

Simple likelihood ratios and Bayes factors

The bearing example compares two simple states with fixed test behaviour. Real hypotheses often contain unknown parameters or several mechanisms. A Bayes factor extends the same posterior-odds logic by averaging each model's likelihood over its parameter prior. Kass and Raftery give the standard account of Bayes factors as data-based multipliers between models.[7]

The distinction matters because a Bayes factor depends on the model and parameter priors inside each hypothesis, while a simple likelihood ratio may compare fixed distributions. Calling both “the evidence score” hides different sensitivities. The ledger should name the exact object: a measured diagnostic likelihood ratio, a model-based conditional likelihood ratio, a Bayes factor, or an estimated classifier score calibrated to posterior probability.

The arithmetic is compact because the model carries the complexity. That is a strength when the model is explicit and checked. It is a risk when assumptions disappear behind a single multiplier.

flowchart LR
  A["The detector that is right<br/>about faults and wrong about<br/>this fault"] --> B["The three accounts in every update"]
  B --> C["Evidence is a multiplier, not<br/>a verdict"]
  C --> D["The clean multiplication trap"]
  D -. "re-test" .-> A
  classDef first fill:#0a1424,stroke:#007D68,stroke-width:2px,color:#e6f3ff
  classDef step fill:#0a1424,stroke:#3f7da0,stroke-width:1px,color:#e6f3ff
  class A first
  class B,C,D step

The clean multiplication trap

Multiplying marginal likelihood ratios is exact only under the right dependence structure. Two messages can look like two observations while carrying one underlying piece of information.

Independence is a model claim, not a formatting fact

Suppose a vibration monitor raises an alert and a second dashboard raises “bearing instability”. If both messages were derived from the same accelerometer window, the second message may add little evidence beyond the first. Treating the labels as separate tests and multiplying their marginal likelihood ratios counts the shared telemetry twice.

The safe sequential identity already showed the repair: use the likelihood of the second item conditional on the first. If the second alert is almost guaranteed once the first has fired under both hypotheses, its conditional likelihood ratio will be close to one. Its visible drama does not create additional discrimination.

This is not a merely theoretical concern. van Walraven and colleagues showed that combining correlated serial diagnostic tests under an independence assumption can overestimate disease probability, with the error increasing as correlation rises.[10] Work on likelihood ratios for multiple pieces of evidence likewise treats dependence as a joint modelling problem rather than a licence to multiply convenient marginal values.[17] The same mechanism appears in machine intelligence systems when several agents retrieve the same document, several classifiers consume the same embedding, or a model's explanation is later reintroduced as though it were fresh evidence.

A multi-agent system is especially vulnerable because organisational separation can masquerade as epistemic separation. Agent A and Agent B may have different names, prompts and roles while sharing the same model, context packet and retrieval result. Agreement then measures repeated processing of one evidence stream, not independent corroboration. The evidence ledger must carry source lineage far enough to reveal common ancestry.

Worked scenario: an account-takeover investigation

The example also shows why evidence and action should remain separate. A 62 per cent posterior does not, by itself, specify whether to hold a payment, request authentication, create a review task or continue monitoring. Those actions depend on loss, reversibility, customer harm, review capacity, legal constraints and the value of another observation.

Negative control: when a ledger adds no predictive advantage

A calibrated discriminative model can estimate P(H|x) directly from the full feature vector. It need not decompose the answer into hand-authored likelihood ratios. On a stable task with representative data, complete features and a reliable calibration set, such a model may match or outperform a naive generative decomposition. Ng and Jordan's comparison of logistic regression and naive Bayes illustrates that generative and discriminative classifiers can occupy different sample-size and asymptotic regimes.[16]

This is the article's serious negative control. Train a joint discriminative model and a decomposed evidence-ledger model on the same synthetic or public task. Match data, feature access and tuning budget. Compare calibration, log loss, decision cost and shift performance. If the direct model is equally accurate, equally stable and can provide sufficient provenance through other means, the ledger has not earned a predictive superiority claim.

The ledger can still earn its place through compositional updating, sensitivity analysis, intervention planning or auditability. Those are separate benefits and should be measured separately. Evidence accounting is an operating discipline, not a guarantee that a generative classifier will win.

Bayes' rule cannot check its own premises

The identity always returns an answer when its inputs are numeric. It does not know whether the likelihood model can reproduce relevant patterns in the observed data. Posterior predictive assessment provides one family of checks: simulate replicated data from the fitted model and compare decision-relevant discrepancies with what was observed.[8] Stan's official documentation turns this principle into an implementation workflow for prior and posterior predictive checks.[15]

Prior-data conflict requires another check. Evans and Moshonov describe methods for asking whether observed data are surprising under the prior predictive distribution.[9] A conflict does not automatically prove that the prior is wrong; the sampling model may also be inadequate. It does tell the reviewer that the opening balance and evidence model are pulling from poorly overlapping worlds.

From posterior to governed decision

A posterior belongs to reasoning. An action belongs to a decision system with costs, authority, reversibility and recovery. The evidence ledger should connect the two without collapsing them.

A probability does not contain a decision

Suppose two maintenance actions face the same 62 per cent posterior fault probability. Replacing a bearing costs little and prevents a catastrophic failure in the first machine. Stopping the second machine would interrupt a critical process, while a non-invasive inspection is available within minutes. The appropriate actions differ even though the inference is identical.

Decision theory adds utilities or losses to uncertain states and candidate actions. A simple rule chooses the action with the lowest expected loss, provided the loss model is adequate and the decision right is clear. In an operational system, the rule also needs constraints: who may act, how long the evidence remains fresh, whether the action is reversible, what confirmation is required, and how the effect will be verified.

Bayes' rule updates belief; it does not grant authority. A model proposal may cross an evidential threshold while still requiring deterministic permission checks, human approval or another observation. The threshold should appear in the ledger as a decision policy, not be smuggled into the probability model.

The typed evidence-update receipt

A production system should preserve enough structure to reconstruct why the posterior moved. The smallest useful record is not a free-form explanation. It is a typed receipt whose fields have declared meanings, units and provenance.

Scroll within the table on a small screen to inspect every column.

Diagnostic-accuracy reporting standards such as STARD emphasise transparent descriptions of participants, index tests, reference standards and analysis because test performance is inseparable from study design and population.[13] Cochrane's diagnostic-test-accuracy handbook similarly treats applicability, bias and between-study variation as central to interpreting evidence.[14] An AI or agentic evidence ledger should preserve equivalent scope information rather than copying a sensitivity or likelihood ratio from a paper into a new environment without a transport argument.

A six-question decision instrument

Before accepting a posterior in research, architecture or governance, ask six questions in order. Each question can stop the release. The order matters because a precise calculation should not distract from an ill-posed hypothesis or inapplicable prior.

1. What exactly are the competing worlds? Define H, the alternative and any material third state. If “not H” combines mechanisms with radically different evidence behaviour, split the alternative or use a richer model.
2. Where did the opening odds come from? Name population, time, sampling route and uncertainty. Recalculate under a plausible prior range when the reference class is contested.
3. What process generated this observation? Preserve raw evidence, transformations and selection. A model-generated summary is evidence about its source only through a validated extraction or inference process.
4. How was evidential weight estimated? Record numerator, denominator, sample sizes, uncertainty, calibration and applicable operating region. Avoid transporting average test characteristics into a subgroup without evidence.
5. What has already been counted? Trace common sources, shared models, derived features and feedback. Use joint or conditional likelihoods when independence does not hold.
6. Which decision changes, and under whose authority? Overlay losses, thresholds, reversibility and evidence freshness. Verify the action's effect separately from the inference that proposed it.

This instrument changes architecture. The prior belongs in versioned world state or a governed parameter store, not inside an untraceable prompt. Evidence events need provenance identifiers. Conditional likelihood logic belongs in a testable deterministic or statistical component. The posterior and its sensitivity range belong in a decision receipt. Authority checks remain outside the probabilistic updater.

Sensitivity before certainty

When priors or likelihood ratios are uncertain, compute a range rather than selecting one convenient point. In the account-takeover scenario, suppose the prior could plausibly lie between 1 and 4 per cent and the joint device evidence LR between 6 and 10. Before the independent beneficiary event, the posterior spans roughly 5.7 to 29.4 per cent. That range may cross a review threshold even when the point estimate does not.

Sensitivity analysis also reveals where measurement effort has value. If the posterior is stable across the credible prior range but swings sharply with the dependency correction, invest in joint evidence validation. If uncertainty is dominated by the base rate after a population shift, collect current prevalence data. More precision on a weak, non-dominant term may not change the decision.

Open hypothesis: ledgers should improve repair, not merely explanation

A falsifiable prediction follows. When a compound AI system's posterior becomes miscalibrated after a source, population or dependency change, teams using typed evidence-update receipts should locate and repair the responsible assumption faster than teams retaining only final scores and free-form rationales. The hypothesis would be weakened if both groups diagnose faults equally quickly, or if receipt maintenance cost exceeds the avoided decision loss.

A useful experiment would inject four faults into matched pipelines: stale prior, shifted likelihood, duplicated evidence and missing alternative. Measure time to detection, localisation accuracy, posterior recovery and operator confidence calibration. The artefact below provides a small environment for the duplicated-evidence case.