18.05 S22 Class 11: Bayesian Updating with Discrete Priors

Bayesian Updating with Discrete Priors Class 11, 18.05 1 Learning Goals 1. Be able to apply Bayes’ theorem to compute probabilities. 2. Be able to define and to identify the roles of prior probability, likelihood (Bayes term), posterior probability, data and hypothesis in the application of Bayes’ Theorem. 3. Be able to use a Bayesian update table to compute posterior probabilities. 2 Review of Bayes’ theorem Recall that Bayes’ theorem allows us to ‘invert’ conditional probabilities. If ℋ and 𝒟 are events, then: 𝑃 (𝒟 | ℋ)𝑃 (ℋ) 𝑃 (ℋ | 𝒟) = 𝑃 (𝒟) Our view is that Bayes’ theorem forms the foundation for inferential statistics. We will begin to justify this view today. 2.1 The base rate fallacy When we first learned Bayes’ theorem we worked an example about screening tests showing that 𝑃 (𝒟|ℋ) can be very different from 𝑃 (ℋ|𝒟). In the appendix we work a similar example. If you are not comfortable with Bayes’ theorem you should read the example in the appendix now. 3 Terminology and Bayes’ theorem in tabular form We now use a coin tossing problem to introduce terminology and a tabular format for Bayes’ theorem. This will provide a simple, uncluttered example that shows our main points. Example 1. There are three types of coins which have different probabilities of landing heads when tossed. • Type 𝐴 coins are fair, with probability 0.5 of heads • Type 𝐵 coins are bent and have probability 0.6 of heads • Type 𝐶 coins are bent and have probability 0.9 of heads Suppose I have a drawer containing 5 coins: 2 of type 𝐴, 2 of type 𝐵, and 1 of type 𝐶. I reach into the drawer and pick a coin at random. Without showing you the coin I flip it once and get heads. What is the probability it is type 𝐴? Type 𝐵? Type 𝐶? 1 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 2 Solution: Let 𝐴, 𝐵, and 𝐶 be the event that the chosen coin was type 𝐴, type 𝐵, and type 𝐶. Let 𝒟 be the event that the toss is heads. The problem asks us to find 𝑃 (𝐴|𝒟), 𝑃 (𝐵|𝒟), 𝑃 (𝐶|𝒟). Before applying Bayes’ theorem, let’s introduce some terminology. • Experiment: pick a coin from the drawer at random, flip it, and record the result. • Data: the result of our experiment. In this case the event 𝒟 = ‘heads’. We think of 𝒟 as data that provides evidence for or against each hypothesis. • Hypotheses: we are testing three hypotheses: the coin is type 𝐴, 𝐵 or 𝐶. • Prior probability: the probability of each hypothesis prior to tossing the coin (collecting data). Since the drawer has 2 coins of type 𝐴, 2 of type 𝐵 and 1 of type 𝐶 we have 𝑃 (𝐴) = 0.4, 𝑃 (𝐵) = 0.4, 𝑃 (𝐶) = 0.2. • Likelihood: (This is the same likelihood we used for the MLE.) The likelihood function is 𝑃 (𝒟|ℋ), i.e., the probability of the data assuming that the hypothesis is true. Most often we will consider the data as fixed and let the hypothesis vary. For example, 𝑃 (𝒟|𝐴) = probability of heads if the coin is type 𝐴. In our case the likelihoods are 𝑃 (𝒟|𝐴) = 0.5, 𝑃 (𝒟|𝐵) = 0.6, 𝑃 (𝒟|𝐶) = 0.9. The name likelihood is so well established in the literature that we have to teach it to you. However in colloquial language likelihood and probability are synonyms. This leads to the likelihood function often being confused with the probability of a hypothesis. Because of this we’d prefer to use the name Bayes’ term. However since we are stuck with ‘likelihood’ we will try to use it very carefully and in a way that minimizes any confusion. • Posterior probability: the probability (posterior to) of each hypothesis given the data from tossing the coin. 𝑃 (𝐴|𝒟), 𝑃 (𝐵|𝒟), 𝑃 (𝐶|𝒟). These posterior probabilities are what the problem asks us to find. We now use Bayes’ theorem to compute each of the posterior probabilities. We are going to write this out in complete detail so we can pick out each of the parts (Remember that the data 𝒟 is that the toss was heads.) First we organize the probabilities into a tree: 0.4 0.2 0.4 𝐵 𝐴 𝐶 0.5 0.5 0.6 0.4 0.9 0.1 𝐻 𝑇 𝐻 𝑇 𝐻 𝑇 Probability tree for choosing and tossing a coin. 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 Bayes’ theorem says, e.g. 𝑃 (𝐴|𝒟) = 3 𝑃 (𝒟|𝐴)𝑃 (𝐴) . The denominator 𝑃 (𝒟) is computed 𝑃 (𝒟) using the law of total probability: 𝑃 (𝒟) = 𝑃 (𝒟|𝐴)𝑃 (𝐴) + 𝑃 (𝒟|𝐵)𝑃 (𝐵) + 𝑃 (𝒟|𝐶)𝑃 (𝐶) = 0.5 ⋅ 0.4 + 0.6 ⋅ 0.4 + 0.9 ⋅ 0.2 = 0.62. Now each of the three posterior probabilities can be computed: 𝑃 (𝒟|𝐴)𝑃 (𝐴) 0.5 ⋅ 0.4 0.2 = = 𝑃 (𝒟) 0.62 0.62 0.6 ⋅ 0.4 0.24 𝑃 (𝒟|𝐵)𝑃 (𝐵) = = 𝑃 (𝐵|𝒟) = 𝑃 (𝒟) 0.62 0.62 0.9 ⋅ 0.2 0.18 𝑃 (𝒟|𝐶)𝑃 (𝐶) = = 𝑃 (𝐶|𝒟) = 𝑃 (𝒟) 0.62 0.62 𝑃 (𝐴|𝒟) = Notice that the total probability 𝑃 (𝒟) is the same in each of the denominators and that it is the sum of the three numerators. We can organize all of this very neatly in a Bayesian update table: hypothesis prior likelihood Bayes numerator ℋ 𝐴 𝐵 𝐶 total 𝑃 (ℋ) 0.4 0.4 0.2 1 𝑃 (𝒟|ℋ) 0.5 0.6 0.9 NO SUM 𝑃 (𝒟|ℋ)𝑃 (ℋ) 0.2 0.24 0.18 𝑃 (𝒟) = 0.62 posterior (numerator/𝑃 (𝒟)) 𝑃 (ℋ|𝒟) 0.3226 0.3871 0.2903 1 The Bayes numerator is the product of the prior and the likelihood. We see in each of the Bayes’ formula computations above that the posterior probability is obtained by dividing the Bayes numerator by 𝑃 (𝒟) = 0.62. We also see that the law of law of total probability says that 𝑃 (𝒟) is the sum of the entries in the Bayes numerator column. Bayesian updating: The process of going from the prior probability 𝑃 (ℋ) to the posterior 𝑃 (ℋ|𝒟) is called Bayesian updating. Bayesian updating uses the data to alter our understanding of the probability of each of the possible hypotheses. 3.1 Important things to notice 1. There are two types of probabilities: Type one is the standard probability of data, e.g. the probability of heads is 𝑝 = 0.9. Type two is the probability of the hypotheses, e.g. the probability the chosen coin is type 𝐴, 𝐵 or 𝐶. This second type has prior (before the data) and posterior (after the data) values. 2. The posterior (after the data) probabilities for each hypothesis are in the last column. We see that coin 𝐵 is now the most probable, though its probability has decreased from a prior probability of 0.4 to a posterior probability of 0.39. Meanwhile, the probability of type 𝐶 has increased from 0.2 to 0.29. 3. The Bayes numerator column determines the posterior probability column. To compute the latter, we simply divided each numerator by 𝑃 (𝒟), i.e. rescaled the Bayes numerators so that they sum to 1. 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 4 4. If all we care about is finding the most likely hypothesis, the Bayes numerator works as well as the normalized posterior. 5. The likelihood column does not sum to 1. The likelihood function is not a probability function. 6. The posterior probability represents the outcome of a ‘tug-of-war’ between the likelihood and the prior. When calculating the posterior, a large prior may be deflated by a small likelihood, and a small prior may be inflated by a large likelihood. 7. The maximum likelihood estimate (MLE) for Example 1 is hypothesis 𝐶, with a likelihood 𝑃 (𝒟|𝐶) = 0.9. The MLE is useful, but you can see in this example that it is not the entire story, since type 𝐵 has the greatest posterior probability. Terminology in hand, we can express Bayes’ theorem in various ways: 𝑃 (𝒟|ℋ)𝑃 (ℋ) 𝑃 (𝒟) 𝑃 (data|hypothesis)𝑃 (hypothesis) 𝑃 (hypothesis|data) = 𝑃 (data) 𝑃 (ℋ|𝒟) = With the data fixed, the denominator 𝑃 (𝒟) just serves to normalize the total posterior probability to 1. So we can also express Bayes’ theorem as a statement about the proportionality of two functions of ℋ (i.e, of the last two columns of the table). 𝑃 (hypothesis|data) ∝ 𝑃 (data|hypothesis)𝑃 (hypothesis) This leads to the most elegant form of Bayes’ theorem in the context of Bayesian updating: posterior ∝ likelihood × prior 3.2 Prior and posterior probability mass functions Earlier in the course we saw that it is convenient to use random variables and probability mass functions. To do this we had to assign values to events (head is 1 and tails is 0). We will do the same thing in the context of Bayesian updating. Our standard notations will be: • 𝜃 is the value of the hypothesis. • 𝑝(𝜃) is the prior probability mass function of the hypothesis. • 𝑝(𝜃|𝒟) is the posterior probability mass function of the hypothesis given the data. • 𝑝(𝒟|𝜃) is the likelihood function. (This is not a pmf!) In Example 1 we can represent the three hypotheses 𝐴, 𝐵, and 𝐶 by 𝜃 = 0.5, 0.6, 0.9. For the data we’ll let 𝑥 = 1 mean heads and 𝑥 = 0 mean tails. Then the prior and posterior probabilities in the table define the prior and posterior probability mass functions. 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 Hypothesis 𝐴 𝐵 𝐶 𝜃 0.5 0.6 0.9 prior pmf 𝑝(𝜃) 𝑃 (𝐴) = 𝑝(0.5) = 0.4 𝑃 (𝐵) = 𝑝(0.6) = 0.4 𝑃 (𝐶) = 𝑝(0.9) = 0.2 5 posterior pmf 𝑝(𝜃|𝑥 = 1) 𝑃 (𝐴|𝒟) = 𝑝(0.5|𝑥 = 1) = 0.3226 𝑃 (𝐵|𝒟) = 𝑝(0.6|𝑥 = 1) = 0.3871 𝑃 (𝐶|𝒟) = 𝑝(0.9|𝑥 = 1) = 0.2903 Here are plots of the prior and posterior pmf’s from the example. 𝑝(𝜃) 𝑝(𝜃|𝑥 = 1) .4 .4 .2 .2 𝜃 .5 .6 𝜃 .9 .5 .6 .9 Prior pmf 𝑝(𝜃) and posterior pmf 𝑝(𝜃|𝑥 = 1) for Example 1 If the data was different then the likelihood column in the Bayesian update table would be different. We can plan for different data by building the entire likelihood table ahead of time. In the coin example there are two possibilities for the data: the toss is heads or the toss is tails. So the full likelihood table has two likelihood columns: hypothesis likelihood 𝑝(𝑥|𝜃) 𝜃 𝑝(𝑥 = 0|𝜃) 𝑝(𝑥 = 1|𝜃) 0.5 0.5 0.5 0.6 0.4 0.6 0.9 0.1 0.9 Important convention. Notice that in the above table we used the value of 𝜃 as the hypothesis. Of course, hypothesizing ‘𝜃 = 0.5’ is exactly the same as hypothesizing ‘the coin is type A’. It is also useful in settings where we haven’t named all the possible hypotheses. Example 2. Using the notation 𝑝(𝜃), etc., redo Example 1 assuming the flip was tails. Solution: Since the data has changed, the likelihood column in the Bayesian update table is now for 𝑥 = 0. That is, we must take the 𝑝(𝑥 = 0|𝜃) column from the likelihood table. hypothesis 𝜃 0.5 0.6 0.9 total prior 𝑝(𝜃) 0.4 0.4 0.2 1 likelihood 𝑝(𝑥 = 0 | 𝜃) 0.5 0.4 0.1 NO SUM Bayes numerator 𝑝(𝑥 = 0 | 𝜃)𝑝(𝜃) 0.2 0.16 0.02 0.38 posterior 𝑝(𝜃 | 𝑥 = 0) 0.5263 0.4211 0.0526 1 Now the probability that 𝜃 = 0.5, i.e. the coin is type A, has increased from 0.4 to 0.5263, while the probability that 𝜃 = 0.9, i.e the coin is type C, has decreased from 0.2 to only 0.0526. Here are the corresponding plots: 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 6 𝑝(𝜃|𝑥 = 0) 𝑝(𝜃) .4 .4 .2 .2 𝜃 .5 .6 𝜃 .9 .5 .6 .9 Prior pmf 𝑝(𝜃) and posterior pmf 𝑝(𝜃|𝑥 = 0) 3.3 Food for thought. Suppose that in Example 1 you didn’t know how many coins of each type were in the drawer. You picked one at random and got heads. How would you go about deciding which hypothesis (coin type) if any was most supported by the data? 4 Updating again and again In life we are continually updating our beliefs with each new experience of the world. In Bayesian inference, after updating the prior to the posterior, we can take more data and update again! For the second update, the posterior from the first data becomes the prior for the second data. Example 3. Suppose you have picked a coin as in Example 1. You flip it once and get heads. Then you flip the same coin and get heads again. What is the probability that the coin was type A? Type B? Type C? Solution: As we update several times the table gets big, so we use a smaller font to fit it in: hypothesis 𝜃 0.5 0.6 0.9 total prior 𝑝(𝜃) 0.4 0.4 0.2 1 likelihood 1 𝑝(𝑥1 = 1|𝜃) 0.5 0.6 0.9 NO SUM Bayes numerator 1 𝑝(𝑥1 = 1|𝜃)𝑝(𝜃) 0.2 0.24 0.18 likelihood 2 𝑝(𝑥2 = 1|𝜃) 0.5 0.6 0.9 NO SUM Bayes numerator 2 𝑝(𝑥2 = 1|𝜃)𝑝(𝑥1 = 1|𝜃)𝑝(𝜃) 0.1 0.144 0.162 0.406 posterior 2 𝑝(𝜃|𝑥1 = 1, 𝑥2 = 1) 0.2463 0.3547 0.3990 1 Note that the second Bayes numerator is computed by multiplying the first Bayes numerator and the second likelihood; since we are only interested in the final posterior, there is no need to normalize until the last step. As shown in the last column and plot, after two heads the type C hypothesis has finally taken the lead! 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 𝑝(𝜃) 𝑝(𝜃|𝑥1 = 1, 𝑥2 = 1) 𝑝(𝜃|𝑥 = 1) .4 .4 .4 .2 .2 .2 𝜃 .5 .6 7 𝜃 .9 .5 .6 𝜃 .9 .5 .6 .9 first posterior 𝑝(𝜃|𝑥1 = 1), second posterior 𝑝(𝜃|𝑥1 = 1, 𝑥2 = 1) prior 𝑝(𝜃), 5 Appendix: the base rate fallacy Example 4. A screening test for a disease is both sensitive and specific. By that we mean it is usually positive when testing a person with the disease and usually negative when testing someone without the disease. Let’s assume the true positive rate is 99% and the false positive rate is 2%. Suppose the prevalence of the disease in the general population is 0.5%. If a random person tests positive, what is the probability that they have the disease? Solution: As a review we first do the computation using trees. Next we will redo the computation using tables. Let’s use notation established above for hypotheses and data: let ℋ+ be the hypothesis (event) that the person has the disease and let ℋ− be the hypothesis they do not. Likewise, let 𝒯+ and 𝒯− represent the data of a positive and negative screening test respectively. We are asked to compute 𝑃 (ℋ+ |𝒯+ ). We are given 𝑃 (𝒯+ |ℋ+ ) = 0.99, 𝑃 (𝒯+ |ℋ− ) = 0.02, 𝑃 (ℋ+ ) = 0.005. From these we can compute the false negative and true negative rates: 𝑃 (𝒯− |ℋ+ ) = 0.01, 𝑃 (𝒯− |ℋ− ) = 0.98 All of these probabilities can be displayed quite nicely in a tree. 0.005 0.995 ℋ+ ℋ− 0.99 0.01 0.02 0.98 𝒯+ 𝒯− 𝒯+ 𝒯− Bayes’ theorem yields 𝑃 (ℋ+ |𝒯+ ) = 𝑃 (𝒯+ |ℋ+ )𝑃 (ℋ+ ) 0.99 ⋅ 0.005 = = 0.19920 ≈ 20% 𝑃 (𝒯+ ) 0.99 ⋅ 0.005 + 0.02 ⋅ 0.995 Now we redo this calculation using a Bayesian update table: 18.05 Class 11, Bayesian Updating with Discrete Priors, Spring 2022 hypothesis ℋ ℋ+ ℋ− total prior 𝑃 (ℋ) 0.005 0.995 1 likelihood 𝑃 (𝒯+ |ℋ) 0.99 0.02 NO SUM Bayes numerator 𝑃 (𝒯+ |ℋ)𝑃 (ℋ) 0.00495 0.01990 𝑃 (𝒯+ ) = 0.02485 8 posterior 𝑃 (ℋ|𝒯+ ) 0.19920 0.80080 1 The table shows that the posterior probability 𝑃 (ℋ+ |𝒯+ ) that a person with a positive test has the disease is about 20%. This is far less than the sensitivity of the test (99%) but much higher than the prevalence of the disease in the general population (0.5%).