Updating beliefs

10.1 The update equation

Core explicit update notation, built on the bit system. For H against its complement, with evidence E and background K:

BF(E;H given K) = P(E given H,K) / P(E given not H,K).

Δb = log₂ BF.

b after = b before + Δb.

The evidence contributes a likelihood-ratio increment. A final credence and an increment can have the same numerical value while meaning different things. “I am at +3” is not “I moved up by +3.”

10.2 A compact update record

The discourse operator rik / ריק, from rikn “move,” records a revision. Its formal technical form is:

Rik H fun −3 tsu +1; mit E; shtup +4.

ריק H פֿון −3 צו +1; מיט E; שטופּ +4.

“Revise H from −3 to +1 bits on E, an increment of +4 bits.”

Spoken: Rik H fun minus dra tsu plus eyn; mit E; shtup plus fir.

These are signed ledger quantities, so section 5.10 supplies minus and plus. Bound un- and oy- remain part of an EF complex.

Shtup “push” is repurposed for the log-likelihood increment. This is a performative entry in the speaker's belief ledger and ends at the sentence boundary. A claim about whether someone else made such an update is an ordinary assertion and requires EF. A proposal that the reader should update instead takes tu or an explicitly hypothetical worked calculation.

The record does not assert that the stated increment is justified. In careful writing, immediately supply E, both likelihoods, and the assumptions used to obtain them. A posterior assertion may follow with its own evidence marking.

10.3 A worked numerical example

Let H be “this component is faulty.” Start with b = −3, so P(H) = 1/9. Suppose a test's positive result E has likelihood 0.8 under H and 0.05 under not H. Then:

BF = 0.8 / 0.05 = 16.

Δb = log₂16 = +4.

b after = −3 + 4 = +1.

P(H given E) = 2/3.

The update record above exactly describes this calculation. Notice that a positive test does not imply near-certainty. The initial odds and the false-positive rate matter. This is an invented teaching model, not a claim about any real diagnostic test.

10.4 Repeated and correlated evidence

For sequential observations E₁ through Eₙ, the correct increment at step i conditions on the earlier observations:

Δbᵢ = log₂[P(Eᵢ given H,K,E₁…Eᵢ₋₁) / P(Eᵢ given not H,K,E₁…Eᵢ₋₁)].

The increments always add when defined this way. Reusing likelihood ratios computed without the earlier evidence requires the relevant conditional-independence assumptions under both H and not H.

Two articles copying the same press release are not two independent confirmations of its content. For a report that is a completely known deterministic copy, seeing the copy after seeing the original supplies no additional content evidence. A distinct report may still add evidence about dissemination or source reliability, but that is a different proposition and model.

Proposed note: eyne-kval “one source” tags a shared origin in a technical evidence record. It is a dependency warning, not an automatic numerical discount. The likelihood model determines the increment.

10.5 Missing evidence

Not observing a signal updates against H when H made that signal more likely than its alternative, under the specified observation process. If both hypotheses predicted no signal almost equally, the update is small. Failure to look is not necessarily equivalent to looking and finding nothing. This operational reading is inspired by “Absence of Evidence Is Evidence of Absence”.

In the test model above, a negative result has BF = 0.2/0.95 = 4/19. Its increment is approximately −2.248 bits. Starting at −3, the posterior is approximately −5.248 bits, or probability 1/39. The negative branch does not simply negate the positive branch's +4-bit increment.

10.6 Anticipated evidence and expected log-odds

Before seeing a test result, coherent Bayesian probabilities satisfy:

Σₑ P(E=e given K) P(H given E=e,K) = P(H given K).

This is the property emphasized in “Conservation of Expected Evidence”. It concerns the expected posterior probability. It does not say that expected changes in log-odds are zero. The transformation from probability to log-odds is nonlinear.

In the invented test model, P(positive) = 2/15 and P(negative) = 13/15. The weighted posterior is (2/15)(2/3) + (13/15)(1/39) = 1/9, exactly the prior. An update plan that raises P(H) for every possible result has a problem unless some aspect of the context or hypothesis changed.

10.7 Soft evidence and uncertain models

A report that E happened is not automatically the same as observing E with certainty. One can model the report R directly, using P(R given H) and P(R given not H), including the source's reliability. Where a soft-evidence updating rule is appropriate, its assumptions must be stated; it is not generally valid to multiply a hard-evidence bit increment by a source's reliability percentage.

Uncertain likelihoods require a richer model or explicitly approximate bounds. If the hypothesis itself changes, label the new hypothesis H′. Do not describe a move from P(H) to P(H′) as a simple update on one fixed proposition.

10.8 Retractions and revisions

Core operator: tsurik / צוריק, used to retract an earlier assertion or ledger entry by identifier. Retraction withdraws a commitment; it does not automatically endorse its negation. A corrected assertion supplies its new value and evidence.

Distinguish three events: the world changed, new evidence changed your belief about the same world-state, or you changed what you meant. Time indexing, update records, and heys definitions make these differences expressible. They should not all be compressed into “I updated.”