CS1800 Probability 3: Bayes Rule. Professor Kevin Gold

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1 CS1800 Probability 3: Bayes Rule Professor Kevin Gold

2 Brief Review of Probability - Counting Outcomes and Axioms Recall from before that we can calculate probabilities by counting success outcomes and dividing by the total number of outcomes. Chance that a deck of 5 cards numbered 1-5 ends up in order after shuffling: 1/5! = 1/120 Chance that it s in order or reverse order: 2/120 = 1/60 Much of the discussion today will just take some probabilities as a given, like you think there s a 1% chance your friend is lying. This is fine too, even though we re not counting anything, as long as we obey the axioms of probability: Number in range [0,1], generally representing our degree of belief Union of all outcomes has probability 1 Pr(A v B) = Pr(A) + Pr(B) - Pr(A ^ B)

3 Brief Review of Probability - Independence Events are independent if learning the outcome of one, does not affect our belief about the other Two rolls of a die, events happening far from each other The definition of independence also tells us something useful we can do: Pr(A ^ B) = Pr(A)Pr(B) for all outcomes of A and B iff A and B are independent Probability that two rolls of a 6-sided die are both 6: (1/6)*(1/6) = 1/36, same result as counting 1 success (6,6) out of 36 roll pairs Pr(6-sided die is even) = 1/2 and Pr(6-sided die is odd) = 1/2, but Pr(particular 6-sided die roll is both even and odd) = 0, so those events are not independent (if they re about the same roll)

4 Brief Review of Probability - Conditional Probability Pr(A B) is the probability that event A happens given that we know B is true Pr(rains today dark clouds in the sky) > Pr(rains today) The definition is Pr(A B) = Pr(A ^ B)/Pr(B) We re shrinking the sample space to include just outcomes where B is true Pr(exactly 2 out of 3 coin flips heads at least one head flipped) = (C(2,3)/2 3 )/(1-(1/2) 3 ) = (3/8)/(7/8) = 3/7 Which we could also get by observing actual at-least-one-h outcomes: TTH, THT, HTT, THH, HTH, THH, HHH From the definition, it follows that Pr(A ^ B) = Pr(A B)Pr(B), which is a generalization of the multiplication rule used for independent events Pr (rains today ^ dark clouds) = Pr(dark clouds)pr(rains today dark clouds) We can calculate the probability of both events happening even though they aren t independent, as long as we know the conditional probability

5 The Uses of Bayes Rule Bayes Rule or Bayes Theorem is a powerful mathematical tool that allows you to decide which of several explanations for some data is best It explains how to combine: your prior degree of belief in each hypothesis (probabilities) some evidence a model of how likely it is that each hypothesis would produce the available evidence (conditional probabilities) And arrive at a posterior likelihood of each hypothesis, in light of the data

6 Specific Applications of Bayes Rule Medical Diagnosis Speech Recognition Evidence: Symptoms Hypotheses: Underlying diseases Evidence: Sound signal Hypotheses: Words

7 Specific Applications of Bayes Rule Robot Navigation Political Polling Evidence: Local images and depth camera results Hypotheses: Locations Evidence: Multiple polls Hypotheses: True underlying likely-voter counts

8 Bayes Rule in Lay Terms The likelihood of the hypothesis in light of the evidence is proportional to two factors: Our prior degree of belief in the hypothesis, before we obtained the evidence The likelihood that we would see this evidence if the hypothesis were true For each competing hypothesis, we can multiply these two factors and compare the results to determine which hypothesis is the most likely explanation.

9 Bayes Rule as a Formula The likelihood of the hypothesis in light of the evidence is proportional to two factors: Our prior degree of belief in the hypothesis, before we obtained the evidence The likelihood that we would see this evidence if the hypothesis were true For each competing hypothesis, we can multiply these two factors and compare the results to determine which hypothesis is the most likely explanation. Pr(hypothesis evidence) proportional to Pr(hypothesis)*Pr(evidence hypothesis) prior likelihood of evidence

10 Bayes Rule as an Equality The proportional to symbol means that the left side is equal to the right, multiplied by some constant. This constant is the same for all hypotheses, which lets us compare the results of the prior*likelihood calculation across different hypotheses. Pr(hypothesis evidence) = αpr(hypothesis)*pr(evidence hypothesis)

11 Bayes Rule as an Equality We can be more precise about the equation and remove the propotional to we know what the scaling factor is (it s 1/Pr(evidence)) But for many practical purposes, we don t know this factor at first, and we may not need to calculate it exactly Pr(hypothesis evidence) = Pr(hypothesis)*Pr(evidence hypothesis) Pr(evidence)

12 Bayes Rule as an Abstraction Lastly, we don t particularly need to talk about hypotheses and evidence Bayes Rule is true generally However, replacing everything with A s and B s makes it much less clear what we re usually trying to do Pr(A B) = Pr(A)*Pr(B A) Pr(B)

13 A Quick Proof of the Abstract Version of Bayes Rule Bayes rule follows naturally from the definition of conditional probability. Pr(A ^ B) = Pr(A B) Pr(B) = Pr(B A)Pr(A) by the definition of conditional probability Drop the Pr(A ^ B) and divide both sides by Pr(B) to get Pr(A B) = Pr(B A)Pr(A) / Pr(B) (which is Bayes rule)

14 Rewind a Bit I think the best balance of being memorable and accurate is the second way: the posterior Pr(hypothesis evidence) is proportional to Pr(hypothesis)*Pr(evidence hypothesis) the prior times the likelihood of the evidence (the first probability flipped )

15 Using Bayes Rule: A Basic Example I just rolled a die that was drawn at random from a bag containing an equal number of 6-sided dice (numbered 1-6) and 20-sided dice (numbered 1-20). I tell you the die roll is a 6. Which hypothesis is more likely that it s 6-sided, or 20-sided? Pr( 6 6-sided die) = 1/6 Pr( 6 20-sided die) = 1/20 Pr(6-sided die 6 ) Pr( 6 6-sided)Pr(6-sided) = 1/6 * 1/2 = 1/12 Pr(20-sided 6 ) Pr( 6 20-sided)Pr(20-sided) = 1/20*1/2 = 1/40 The die is more likely to be 6-sided since 1/12 > 1/40.

16 Changing the Prior Suppose that the bag contained 4 20-sided dice and just 2 6-sided dice before I drew a die and rolled a 6. Which die is more likely now? Pr( 6 6-sided die) = 1/6 Pr( 6 20-sided die) = 1/20 Pr(6-sided die 6 ) Pr( 6 6-sided)Pr(6-sided) = 1/6 * 1/3 = 1/18 Pr(20-sided 6 ) Pr( 6 20-sided)Pr(20-sided) = 1/20*2/3 = 1/30 Despite the larger number of 20-sided dice, the die is still more likely to be 6-sided due to how much more unlikely a 6 is on a 20-sided die.

17 Changing the Prior, II Suppose that the bag contained 9 20-sided dice and just 1 6-sided die before I drew a die and rolled a 6. Which die is more likely now? Pr( 6 6-sided die) = 1/6 Pr( 6 20-sided die) = 1/20 Pr(6-sided die 6 ) Pr( 6 6-sided)Pr(6-sided) = 1/6 * 1/10 = 1/60 = Pr(20-sided 6 ) Pr( 6 20-sided)Pr(20-sided) = 1/20*9/10 = 9/200 = With enough 20-sided dice to begin with, it finally becomes more likely that the die was actually 20-sided, despite the evidence.

18 Finding Exact Probabilities The numbers we derived tell us which hypothesis is most likely, but they are not the true probabilities. We need to divide each quantity by Pr(evidence) to get the true probabilities Pr(evidence) = Pr(evidence ^ hypothesis1) + Pr(evidence ^ hypothesis2) + + Pr(evidence ^ hypothesisn) = Pr(evidence hypothesis1)pr(hypothesis1) + Pr(evidence hypothesis2)pr(hypothesis2) + Pr(evidence hypothesisn)pr(hypothesisn) (this assumes we ve exhausted the possible explanations for the evidence) these are the numbers we calculated before! In other words, we need to divide each of our results by the sum of the results to get the true probabilities

19 Finding Exact Probabilities - Example I just rolled a die that was drawn at random from a bag containing an equal number of 6-sided dice (numbered 1-6) and 20-sided dice (numbered 1-20). I tell you the die roll is a 6. What is the probability of each hypothesis in light of the evidence (6-sided vs 20-sided)? Pr(6-sided die 6 ) Pr( 6 6-sided)Pr(6-sided) = 1/6 * 1/2 = 1/12 Pr(20-sided 6 ) Pr( 6 20-sided)Pr(20-sided) = 1/20*1/2 = 1/40 Likelihood of the evidence (overall chance of a 6) = Pr( 6 6-sided)Pr(6-sided) + Pr( 6 20-sided)Pr(20-sided) = 1/12 + 1/40 Pr(6-sided die 6 ) = 1/12/(1/12 + 1/40) = 0.76 Pr(20-sided die 6 ) = 1/40/(1/12 + 1/40) = 0.24 Notice how dividing by the sum forces the actual probabilities to sum to 1.

20 A Science-Themed Example Bayes rule can be used in scientific applications to balance prior knowledge with new observations. Scientists estimate that an asteroid is 60% likely to be composition A, and 40% likely to be composition B. Then they get some readings that would have had a 20% probability of being generated under composition A, and a 50% probability of being generated under composition B. Which hypothesis is more likely now? Pr(A readings) Pr(readings A) Pr(A) = 0.2 * 0.6 = 0.12 Pr(B readings) Pr(readings B) Pr(B) = 0.5 * 0.4 = 0.2 So hypothesis B is more likely - the new readings shifted our uncertain opinion Specifically, we should think Pr(B readings) = 0.2/( ) = 20/32 = (still pretty uncertain)

21 This Really Happened the Other Day I was playing a board game with friends (Gloomhaven) when a friend drew a curse card. I m not sure whether I shuffled since adding this to the [top of the] deck, my friend said. There were 20 other cards in the deck (all not curse cards), and I trust my friend to be honest about his uncertainty. Pr(no shuffle) Pr(curse no shuffle)pr(no shuffle) = 1*0.5 = 0.5 Pr(shuffle) Pr(curse shuffle)pr(shuffle) = 1/21*0.5 = I m pretty sure you didn t shuffle, then, I said. If I d actually done the math, I could have 0.5/( ) = 95% confidence.

22 A Practice Problem A friend flips a coin to determine who will pay for lunch, and it comes up heads you pay. The first time this happens, it seems normal. But the coin comes up heads 8 times in a row. Before, you would have said there was only a 1% chance that your friend would use a two-headed coin. But what should the probability be in light of all these heads results? Bayes Rule reminder: Pr(hypothesis evidence) Pr(evidence hypothesis)pr(hypothesis)

23 A Practice Problem A friend flips a coin to determine who will pay for lunch, and it comes up heads you pay. The first time this happens, it seems normal. But the coin comes up heads 8 times in a row. Before, you would have said there was only a 1% chance that your friend would use a two-headed coin. But what s the more likely hypothesis now? What are the probabilities? Pr(two-headed results) Pr(results two-headed)pr(two-headed) = 1*0.01 = 0.01 Pr(not two-headed results) Pr(results not two-headed)pr(not two-headed) = 1/256*0.99 = It s now more likely that the coin is two-headed. Pr(two-headed results) = 0.01/( ) = 0.72

24 Bayes Rule Tells Us It s Hard to Detect Rare Events Confidently Bayes rule tells us to pay attention to not just the immediate evidence, but base rates of events (our prior belief before receiving evidence) Our methods for detecting rare events rare diseases, terrorists, etc. all have false positive rates where false alarms happen. Combined with Bayes rule, our prior belief that the event is rare should make it extremely difficult for an even slightly unreliable technology to convince us otherwise Bayes rule tells us that most detections of rare events will be false positives, unless the method being used is extremely accurate. General Population F a l s e P o s i t i v e bad stuff

25 Security Example Suppose we have an airport face-detection based security measure for catching terrorists that has a 99% chance of going boop if a terrorist passes through, but has a 2% chance of going boop on a regular person. Suppose that 1 in 10,000 people is a terrorist. Our detector is going boop - what is the chance that it s a false alarm? Pr (terrorist boop) P(boop terrorist) P(terrorist) = 0.99 * = Pr (not terrorist boop) P(boop not terrorist) P(not terrorist) = 0.02 * = basically / = or 99.5% chance the boop is a false alarm.

26 A Real Example of the Difficulty of Reliable Detection From Nate Silver s The Signal and the Noise (p. 245): Studies show that if a woman does not have cancer, a mammogram will incorrectly claim that she does only about 10 percent of the time. If she does have cancer, on the other hand, they will detect it about 75 percent of the time. When you see those statistics, a positive mammogram seems like very bad news indeed. But if you apply Bayes s theorem to these numbers, you ll come to a different conclusion: the chance that a woman in her forties has breast cancer given that she s had a positive mammogram is still only about 10 percent.for this reason, many doctors recommend that women do not begin getting regular mammograms until they are in their fifties and the prior probability of having breast cancer is higher.

27 Updating Beliefs With More Evidence We can sometimes get around the problem of strong priors by incorporating multiple pieces of evidence. As long as the pieces of evidence are conditionally independent (independent besides their shared dependence on the hypothesis), we can treat the probabilities derived after one Bayesian calculation as our new prior going forward. This approach allows scientists to incorporate multiple experimental results, robots to combine observations across different sensors, and law enforcement to use multiple lines of evidence

28 Updating Beliefs Example Besides the face recognition software, suppose we also employ voice recognition software that has a 1% false positive rate and 5% false negative rate - and it goes bing (positive) on our suspect from before From the face recognition results we have new priors of Pr(normal) = 0.995, Pr(terrorist) = Pr(terrorist bing) Pr(bing terrorist)pr(terrorist) = 0.95*0.005 = Pr(not terrorist bing) Pr(bing not terrorist)pr(not terrorist) = 0.01*0.995 = Pr(terrorist bing) = /( ) = 0.32 or 32%, which maybe at least makes this worth investigating further

29 Nate Silver and Being Bayesian Nate Silver, founder of fivethirtyeight.com, rose to fame in 2008 for successfully predicting 49 of 50 states outcomes in that presidential election He s argued that his primary innovation was simply being Bayesian about the polls - instead of choosing one poll to believe, he would constantly update his belief with the evidence of each new poll His book The Signal and the Noise is all about how different scientific fields seem to be progressing or not, depending on how thoroughly they ve adopted Bayesian methods

30 A Quote from The Signal and the Noise (p. 452) Bayes s theorem encourages us to be disciplined about how we weigh new information.most of the time, we do not appreciate how noisy the data is, and so our bias is to place too much weight on the newest data point. But we can have the opposite bias when we become too personally or professionally invested in a problem, failing to change our minds when the facts do The more often you are willing to test your ideas, the sooner you can begin to avoid these problems and learn from your mistakes.

31 Application to the Monty Hall Problem Speaking of being too attached to priors: here s a classic probability problem that seems slightly paradoxical, named after the host of the show Let s Make a Deal A game show host offers you the choice of three doors. Behind one is a new car you d like to win. Behind the other two doors are goats. The host lets you pick a door, which you basically do at random. Then the host says, I m going to choose a door that you didn t pick, and show you what s behind that door. He opens a door and reveals a goat. (You re certain he would not reveal the car at this stage, and that he must reveal a door you did not pick.) pick Knowing what you know now, would you like to switch doors?

32 Bayes and the Monty Hall Problem pick For the sake of clarity, let s assume (without loss of generality) we picked door 2 and the host opened door 3. There are now only two valid hypotheses - car behind 1, car behind 2. Our evidence is that door 3 was opened. What was the likelihood door 3 was opened if 2 had the car, and we are right? What was the likelihood door 3 was opened if 1 had the car, and we were wrong?

33 Bayes and the Monty Hall Problem Door 1 Has a Car World can t reveal can Door 2 Has a Car World can can t can pick pick What was the likelihood door 3 was opened if 2 had the car, and we are right? 1/2 since the host could open either door 1 or door 3 with equal likelhood What was the likelihood door 3 was opened if 1 had the car, and we were wrong? 1 since the host is forced to open the non-pick, non-car door

34 Bayes and the Monty Hall Problem pick Pr(door 1 has car door 3 revealed) Pr(door 3 revealed door 1 car)pr(door 1 car) =1*1/3 = 1/3 Pr(door 2 has car door 3 revealed) Pr(door 3 revealed door 2 has car)pr(door 2 car) = 1/2*1/3 = 1/6 So in fact, door 1 is the more likely hypothesis (with probability (1/3)/(1/3 + 1/6) = 2/3), and we should switch

35 Monty Hall Without Bayes Rule Bayes rule is just a convenient shortcut when reasoning about conditional probabilities We could see the same results by mapping out all the possibilities car door your pick switch loses switch wins switch wins switch wins switch loses switch wins switch wins switch wins switch loses

36 Bayes is Old News But Also Currently Hot Bayes original work appeared in 1763, two years after his death Laplace, competitor with Newton for the development of calculus, developed the theory further (1774) But when statistics took off, its inventors emphasized the importance of a single experiment, and didn t like the idea of incorporating priors (especially Fisher, ) Non-Bayesian frequentist statistics is still the most common kind taught in undergraduate curricula, though Bayesian statistics is gaining traction Artificial intelligence got a major boost in the 80 s thanks to the efforts of Judea Pearl, who showed how to use Bayes rule to incorporate evidence over time and across sources of evidence Sebastian Thrun demonstrated how useful Bayesian reasoning could be to self-driving cars in the 2005 DARPA Grand Challenge; Bayesian reasoning drives Google s self-driving cars today

37 Summary Bayes rule tells us that the likelihood of a hypothesis in light of the evidence is proportional to the product of: our prior probability of the hypothesis the likelihood of the evidence if the hypothesis is true In other words, Pr(hypothesis evidence) Pr(evidence hypothesis)pr(hypothesis) Given a complete set of hypotheses, we can calculate their probabilities by dividing these results by their sum, thus scaling them to sum to 1 Bayes rule is a useful way to take into account everything we know and believe, including prior beliefs, base rates of uncommon events, and experiment results