Chance and Belief

This is Unit Six of Artificial Intelligence: chance and belief. The unit's method is worth stating first, because it makes everything here concrete. Every probability question in this unit is worked by counting — imagine a fixed number of cases distributed exactly as you believe, and count the ones that matter. That is not a teaching simplification. The formulas that appear later are shown to reproduce the count digit for digit, because a formula is the count with the counting compressed out of it. The entries run both roads on the same problem to demonstrate it. The unit also fixes the most consequential probability error there is — reading a test's accuracy as the chance you have the condition. Those are two different questions facing opposite directions, and no improvement to the first answers the second. And it repairs something from Unit Five. Rule systems went silent past their knowledge and could not hedge. A probabilistic reasoner always has an answer. By the end you will be able to do five things. One: say when chances multiply and when they add. Two: compute an updated belief by counting a room. Three: state base-rate neglect, and why accuracy cannot fix it. Four: say what Bayes' rule converts, and what it costs. Five: say what this buys over rule systems, and what it pays.

Two entries, and the second is a free error-check worth applying to any set of probabilities anybody shows you. If they do not total exactly one, something has been invented or something has gone missing.

The definition.

AI 0281 A Chance Is a Share of Cases

A fraction between zero and one — the share of relevant cases in which something happens. And a count means nothing until its denominator is stated.

The check.

AI 0282 The Books Balance

The chances of every way one situation can resolve sum to exactly one. Books totalling more mean a case was invented; totalling less means one was lost.

Four entries. Two operations, each licensed by a specific condition — and the conditions are what get skipped. Then one sweep that audits an entire tree and locates a fault upstream of wherever it shows.

The first operation.

AI 0283 Along a Branch, Multiply

When one thing must happen and then another, multiply along the path. The multiplication is a two-stage count compressed into one step.

Its condition.

AI 0284 The Independence Flag

Multiplying assumes the second event keeps its chance whatever the first did. Every casual multiplication carries that assumption — and it is either checked or it is silently assumed.

The second operation.

AI 0285 Across Branches, Add

When something can happen several mutually exclusive ways, add across them. Adding is licensed by exclusivity — the ways must genuinely be unable to happen together.

The audit.

AI 0286 The Leaf Audit

One sweep: multiply down every path, add across the endings, and confirm the total is exactly one. A tree whose endings miss one is broken somewhere above where the error shows.

Two entries. What it means to compute a chance given that something is known — which turns out to be ordinary counting in a smaller room — and the number that a repeated decision actually earns.

Given a fact.

AI 0287 Shrinking the World

A chance given something is an ordinary chance computed in a shrunken world: count only the cases where the fact holds. Nothing about anybody changed — the operation is bookkeeping rather than causation.

The long-run number.

AI 0288 The Weighted Average

Multiply each outcome by its chance and add. The result is what a repeated decision earns on average — no single instance produces it, which is what makes it a rate rather than a prediction.

Three entries moving the same arithmetic from drums and dice onto what an agent believes. And the method for computing with beliefs is the counting method — imagine a thousand cases and count.

What a probability is here.

AI 0295 Probability Is Degree of Belief

A number between zero and one measuring how strongly an agent believes a claim. A middle value never means partly true — it means the agent does not know which way it goes.

The method.

AI 0296 The 1,000-Worlds Method

Imagine a thousand cases distributed exactly as you believe, and count. The translation runs both directions — beliefs into counts before computing, counts back into beliefs after.

Compound beliefs.

AI 0297 Compound Beliefs by Counting

Running the counting method through a tree of imagined cases. Keep the panel's size fixed throughout — every count has to be a share of the same thousand.

Two entries on what learning something does to belief. The capstone works the most famous example in the subject entirely by counting, and never by quoting a rate back at the question.

What evidence does.

AI 0298 Evidence Shrinks the World

The chance of a claim given evidence is its share among only the cases where that evidence holds. Nothing in the world changed overnight — the information did, and the count moved with it.

The capstone. A flagged result.

AI 0299 The Screening Count

What a test result does to belief comes from counting the room the result selected — never from quoting the test's accuracy back at the question. Every number in it comes from counting that room.

One entry, and it names the mistake this whole unit exists to prevent. The test's accuracy and your chance of having the condition are two different questions, and they face in opposite directions.

The error.

AI 0300 Base-Rate Neglect

Reading a flagged result from a ninety-percent-accurate test as ninety percent certainty. Accuracy answers a different question from the one being asked, and no improvement to the first will ever answer the second.

Four entries. The formula, a demonstration that it is the count in compressed form, what it licenses, and the error it exists to prevent — which is the same error as the last topic, stated structurally.

The formula.

AI 0301 Bayes' Rule

Updated belief from how probable the evidence is if the claim holds, the prior, and how common the evidence is overall. The counting came first in this unit on purpose — whoever owns the count can rebuild the formula.

Both roads.

AI 0302 Same Answer, Two Roads

The formula reproduces the count exactly, because it is the count with the counting compressed out. And the difference matters: a misplaced formula term fails silently, while a miscounted room fails visibly.

What it licenses.

AI 0303 The Licensed Crossing

It converts reasoning from causes into reasoning toward them — nature supplies how often a condition produces a sign, and this turns that around. The toll for the crossing is the prior.

And the error it prevents.

AI 0304 The Transposed Conditional

Swapping the chance of the evidence given the claim for the chance of the claim given the evidence. On hearing any conditional probability, ask which room is the denominator.

Three entries closing the unit, and the first two repair specific failures from Unit Five. Rules can now hedge, and the system always has an answer — and then the bill for all of it.

Rules that hedge.

AI 0305 Rules That Hedge

A rule can now say usually — all-or-nothing statements become claims about how strongly something shifts belief. The case that embarrassed the rule system stops being a scandal and becomes a priced signal.

Always an answer.

AI 0306 Always a Posterior

There is always a posterior, sometimes a humble one, where the rule system had only silence past its edge. And hedged is not vague — a value of one half is a precise statement of balance, ready to update.

And the bill.

AI 0307 Numbers from Somewhere

Graded, calibrated, always-available belief, paid for in numbers that have to come from somewhere. The arithmetic guards the form of the belief; the content is only as good as those numbers.

Five things you can now do. You can say when chances multiply and when they add, and name the condition each requires — independence for one, exclusivity for the other. You can compute an updated belief by counting a room rather than quoting a rate, and you know the formula reproduces that count exactly. You can state base-rate neglect: the test's accuracy and your chance of having the condition are different questions facing opposite directions. You can say what Bayes' rule converts — reasoning from causes into reasoning toward them — and that the toll for that crossing is the prior. And you can say what this buys over rule systems — graded belief that is always available — and what it pays, which is numbers that must come from somewhere.