A belief state needs an observation model; a planning score is not a probability

A finite partially observed model names hidden states, possible actions, transition probabilities, and observation likelihoods. The belief distribution summarizes uncertainty under those assumptions. The POMDP literature derives its update by prediction through transitions followed by conditioning on the observed result. Those guarantees depend on the specified model; an open-ended language task rarely arrives with every state and likelihood known. A useful engineering record can instead list unresolved hypotheses, source evidence, and qualitative uncertainty without fabricating a calibrated probability for each one.

When numerical probabilities are used, document how they were estimated and evaluate calibration on relevant held-out outcomes. A softmax over arbitrary plan scores sums to one, but that algebra alone does not make its values probabilities of real-world success. Missing states, changing tools, and selective observations can invalidate calibration from another environment. A zero likelihood for an actual observation signals an unsupported case or model mismatch; do not divide by zero or silently assign certainty because the old model could not represent what occurred.

Two tool responses can reflect the same cached measurement, mirror the same source, or share a common failure. Treating them as conditionally independent can make a posterior unjustifiably extreme. Preserve observation IDs and source dependencies. If a second measurement is genuinely conditionally independent given a static state, repeated Bayes updates can be valid; if the state changes between probes, include the transition model. The number of agreeing text passages is not by itself the number of independent observations.

Belief and utility are also different. A decision combines state probabilities with a declared outcome or cost model; the most likely state need not determine the best action when consequences are asymmetric. A probability estimate can be well calibrated while the utility model ignores an important user requirement. Conversely, uncertainty can justify gathering a discriminating observation rather than immediately acting on the highest-scored plan. Keep evidence quality, state uncertainty, action costs, and acceptance constraints separate so a numerical preference cannot silently override a required precondition.

Use a small known environment to verify the filtering arithmetic, then test repeated cached evidence, unexpected observations, changed likelihoods, and omitted alternatives. Measure both calibration and decision outcomes where the interpretation permits them. For an open-ended task, inspect whether the agent preserves unresolved hypotheses and requests or gathers the evidence that would distinguish them. A detailed numerical belief record is not useful if its likelihoods are invented. Report the uncertainty model and its limits with empirical performance rather than claiming formal POMDP optimality for an approximate language-agent controller.

The sampling lab displays an independent Bernoulli sample-mean law, not a belief filter. Reset, set both p=0.4 and n=25, predict mean 0.4 and variance 0.0096, then Play or Step. Reset and set both p=0.4 and n=100 for variance 0.0024. Exact binomial bars assume independent equal-probability trials; playback highlights outcomes, not random draws. The normal comparison is approximate and its Berry–Esseen value bounds standardized CDF error, not each bar or a posterior. The model supplies no tool likelihood, hidden state, correlated evidence, or probability calibration.