Applying Bayes' Rule to AI Decisions, Explained

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Applying Bayes' Rule to AI Decisions

Introduction to Artificial Intelligence · Chapter 4: Search, Heuristics, and Game StrategySection 4.2 · lesson 24 of 6913 min · free with a free account

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In this lesson

Update beliefs with evidence and avoid the base-rate fallacy.

  • Derive Bayes' rule from conditional probability and name its parts
  • Calculate posterior probabilities for medical tests, spam filters, and fraud detectors
  • Explain the base-rate fallacy and apply step-by-step Bayesian updating
  • Compare maximum a posteriori and maximum likelihood decisions

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Transcript

Everything said in the video, section by section (1,673 words). Read along, or search it.

Opening Puzzle

A 90 Percent Accurate Test

A test for a rare disease is 90 percent accurate. Your result comes back positive. How worried should you be? Most people say: about 90 percent sure I'm ill. For this test, the true answer is only about 9 percent. Studies of doctors found the same mistake. And AI faces it all the time. A spam filter sees the word free. A fraud detector sees an odd purchase. A game program sees a raise. Each must decide how much to believe, after uncertain evidence. The tool for that is Bayes' rule.

Worked Example

A Mailbox of 1,000 Emails

Start with a mailbox of 1,000 emails. 300 are spam, and 700 are not. Of the spam, 240 contain the word free, and 60 don't. Of the good email, 35 contain free, and 665 don't. That's 275 with free, in all. A probability runs from 0, impossible, to 1, certain. P of spam is 300 out of 1,000: 0.30. Likewise, P of free is 0.275. And both at once, P of spam and free, is 240 out of 1,000: 0.24.

Definition

Conditional Probability and Its Order

Now a new question. Among only the emails with free, what share are spam? Look only at the 275 free emails: 240 of them are spam. That's a conditional probability. P of A given B is P of A and B, divided by P of B. It counts only the cases where B is true. Check it: 0.24 over 0.275 is 0.873. But order matters. P of free given spam is 240 over 300: 0.80. Mixing up the two is a classic mistake.

Derivation

Bayes' Rule in Three Steps

Bayes' rule falls out of that definition in three steps. Call the hypothesis H, and the evidence E. Step one: write the definition both ways. P of H given E is the joint, P of H and E, over P of E. And P of E given H is that same joint, over P of H. Step two: rearrange the second. The joint equals P of E given H, times P of H. Step three: substitute it into the first. That's Bayes' rule: P of H given E equals P of E given H, times P of H, over P of E.

Evidence Term

Building P of E From Two Cases

Why is this useful? Often we know one direction, and want the other. A lab tests its method on sick people, so it knows P of positive given sick. A patient wants P of sick given positive. The bottom, P of E, is rarely given. Build it from the two ways the evidence can happen: with H true, or with H false. For the mailbox: 0.80 times 0.30, plus 0.05 times 0.70, is 0.275.

Key Terms

Prior, Likelihood, Evidence, and Posterior

Each part of the rule has a name. The prior, P of H, is the belief before the evidence: here, 0.30. The likelihood, P of E given H, is how well H predicts the evidence: 0.80. The evidence term, P of E, is how likely that is overall, 0.275. It scales the answer so the probabilities add to 1. And the posterior, P of H given E, is the updated belief: 0.873. In words: posterior equals likelihood times prior, over evidence.

Odds Shortcut

Likelihood Ratio Times Prior Odds

Free is 16 times likelier in spam than in good mail: 0.80 over 0.05. That's the likelihood ratio: how strongly the evidence favors H. The shortcut: posterior odds are prior odds, times that ratio. 0.30 against 0.70 is about 0.43. Unrounded, 0.4286 times 16 is 6.86. As a probability, 6.86 over 7.86 is 0.873 again. Updating becomes a single multiplication.

Worked Example

A Rare Disease Gives 9.2 Percent

Now the test from the opening. A disease affects 1 percent of people. If you have it, the test is positive 90 percent of the time. That's its sensitivity. If you don't, it's still positive 9 percent of the time. That's the false-positive rate. Step by step. The prior is 0.01. The likelihoods are 0.90 and 0.09. The evidence: 0.90 times 0.01, plus 0.09 times 0.99. That's 0.009 plus 0.0891: 0.0981. The posterior: 0.009 over 0.0981 is 0.0917. About 9.2 percent.

Natural Frequencies

Picturing 10,000 People and the Base-Rate Fallacy

Percentages feel slippery, so picture 10,000 real people. Only 100 have the disease, and 90 of them test positive. But 9,900 are healthy, and 9 percent of them, 891 people, test positive too. So there are 981 positives, and only 90 of them are truly ill. That's 9.2 percent. Ignoring how common something is, the prior, is the base-rate fallacy. The many healthy people make more false alarms than the few sick make true ones.

Base Rates

Same Test, Different Base Rates

The same test means very different things at different base rates. If 1 person in 1,000 has it, a positive result means about a 1 percent chance. At 1 in 100, it's 9.2 percent. At 1 in 10, it's 52.6 percent. And at 1 in 2, it's 90.9 percent. The test never changed. Only the base rate did.

Quick Check

A Screening Test at 2 Percent Prevalence

Quick check. A new screening test catches 95 percent of the people who have a condition. It also flags 10 percent of the people who don't. And the condition affects 2 percent of the people tested. Someone tests positive. About how likely is it that they have the condition? Pause, and decide. A: about 16 percent. Picture 1,000 people. 20 have the condition, and 19 of them test positive. Of the 980 who don't, 10 percent, 98 people, test positive too. So only 19 of 117 positives are real: about 16 percent. 95 percent is tempting, but that's the chance of a positive given the condition: the reverse question. The low base rate drags the answer down.

AI Application

Fraud Alerts and False Alarms

Every detector for something rare faces the same issue. Say 1 transaction in 1,000 is fraud. A detector flags 95 percent of fraud, and raises a false alarm on 2 percent of honest purchases. The chance of fraud given an alert is 0.00095 over 0.02093: about 0.045. Only about 1 alert in 22 is real fraud. An accurate detector can still bury its users in false alarms.

AI Application

Spam Filters Weigh the Prior

Spam works the same way. Where only 5 percent of mail is spam, free gives 45.7 percent: close to a coin toss. In our 30 percent mailbox, 87.3. At 70 percent, 97.4. At 90, 99.3. So a good filter weighs the evidence against how common spam is.

Bayesian Updating

Each Posterior Becomes the Next Prior

Evidence rarely arrives all at once. Bayesian updating treats each posterior as the prior for the next piece of evidence. Back to the medical test: after one positive, 9.2 percent. A second test, from a lab with independent mistakes, is also positive. Now the prior is 0.0917. The evidence is 0.90 times that, plus 0.09 times 0.9083. That's 0.0826 plus 0.0817: 0.1643. The posterior is 0.0826 over 0.1643: about 50 percent.

Odds Shortcut

Three Positives Multiply the Odds

So the belief climbs from 1 percent, to 9.2, to about 50. A third positive lifts it to about 91 percent. The odds shortcut makes this easy. Each positive multiplies the odds by the same ratio: 0.90 over 0.09, or 10. Odds of 1 to 99, times 10 three times, come to about 10 to 1: 91 percent. A negative first result would instead cut it to about 0.1 percent.

Cautions

Independence and Priors of Zero or One

Some cautions. Strong evidence moves beliefs more. And independent evidence gives the same answer in any order. But independence is an assumption: the same flawed test may repeat its own mistake. And never start from a prior of exactly 0 or 1. Multiplied by anything, 0 stays 0, so that belief could never change.

Decision Rules

MAP Versus Maximum Likelihood

Often the goal is a decision. The maximum a posteriori rule, MAP, picks the hypothesis with the highest posterior. P of E is the same for every hypothesis, so MAP just compares likelihood times prior. Maximum likelihood, ML, picks whatever makes the evidence most likely, and ignores the prior. The two agree only when the priors are equal. Three machines make parts, and a defective part turns up. Machine A makes 70 percent of the parts, B makes 20, and C makes 10. Their defect rates are 1, 2, and 5 percent. The products are 0.0070, 0.0040, and 0.0050. ML picks C, the sloppiest machine. But MAP picks A. It makes so many parts that its few defects add up. Dividing by the total, 0.016, gives A about 44 percent. That's the most probable machine, not a sure one: C still has 31 percent. MAP is everywhere. To a speech recognizer, recognize speech sounds like wreck a nice beach. The first phrase is far more common, so MAP picks it. A card player who bluffs on 20 percent of hands raises with 90 percent of bluffs, and half of strong hands. So a raise is a bluff only 31 percent of the time. One limit: MAP ignores what a mistake costs. After one positive test, MAP says probably healthy. Still, test again: a missed disease costs far more.

Quick Check

Diagnosing a Stalled Robot Arm

Quick check. A robot arm stalls. Three faults could explain it. A worn gear has a prior of 0.6, and causes a stall 10 percent of the time. A loose wire has a prior of 0.3, and causes a stall 30 percent of the time. A bad chip has a prior of 0.1, and causes a stall 50 percent of the time. Which fault does MAP pick? Pause, and decide. B, the loose wire. Multiply each prior by its stall rate. The gear gives 0.06, the wire 0.09, and the chip 0.05. The wire's product is largest. Divided by the total, 0.20, its posterior is 45 percent. The chip is the tempting ML answer, and the gear is the most common fault. But MAP weighs both, and neither wins alone.

Key Takeaways

Priors, Posteriors, and MAP Summed Up

To sum up. Bayes' rule comes from conditional probability, and turns P of E given H into P of H given E. The prior, the likelihood and the evidence give the posterior. Ignoring the base rate is a fallacy: rare conditions make most alarms false. Update step by step, each posterior becoming the next prior. And MAP picks the most probable hypothesis, whatever a mistake would cost.

Reflection

Alerts From an Airport Face Scanner

Something to think about. An airport face scanner is 99 percent accurate. Among millions of travelers, only a handful are on a watch list. What would you expect of its alerts, and how should staff treat them?

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FAQ

Questions students ask us

About Applying Bayes' Rule to AI Decisions

What is Bayes' rule in simple terms?
Bayes' rule turns P(E given H) into P(H given E): the posterior equals the likelihood times the prior, divided by the evidence. In a mailbox where 30 percent of email is spam, 80 percent of spam contains free, and free appears in 27.5 percent of all email, the chance a free email is spam is 0.80 times 0.30 over 0.275, about 0.873.
What is the base-rate fallacy?
The base-rate fallacy is ignoring how common something is when judging evidence. For a disease affecting 1 percent of people, a test with 90 percent sensitivity and a 9 percent false-positive rate gives positive results to 90 of 100 sick people and 891 of 9,900 healthy ones. Only 90 of 981 positives are ill, about 9.2 percent, not 90 percent.
How does Bayesian updating work with more than one piece of evidence?
Bayesian updating uses each posterior as the prior for the next piece of evidence. Starting from a 1 percent prior, one positive medical test raises the probability to about 9.2 percent, a second independent positive to about 50 percent, and a third to about 91 percent. In odds form, each positive multiplies the odds by the likelihood ratio, here 0.90 over 0.09, or 10.
What is the difference between MAP and maximum likelihood estimation?
MAP picks the hypothesis with the highest posterior, likelihood times prior, while maximum likelihood picks the one that makes the evidence most likely and ignores the prior. If machines A, B, and C make 70, 20, and 10 percent of parts with defect rates of 1, 2, and 5 percent, maximum likelihood blames C for a defect, but MAP picks A at about 44 percent.

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