Psychology 290 Special Topics Study Course: Advanced Meta-analysis

1. Psychology 290Special Topics Study Course: Advanced Meta-analysis April 7, 2014

2. The plan for today • Review Bayes’ theorem • Review the likelihood function • Review maximum likelihood estimation • Prior and posterior distributions • The Bayesian approach to statistical inference

3. Bayes’ theorem

4. Example of Bayes’ Theorem • Suppose a woman has had a single unplanned, unprotected sexual encounter. • She takes a pregnancy test, and it is positive. • What does she want to know? • What is the probability that I am pregnant?

5. Example of Bayes’ Theorem (cont.) • Let ‘B’ denote ‘pregnant’ and ‘A’ denote ‘positive pregnancy test.’ • Suppose P(A|B) is .90, p(A|~B) is .50, and P(B) is .15. • The marginal P(A) can be expressed as P(A|B)P(B)+ P(A|~B)P(~B) = (.90)(.15) + (.50)(.85) = .56. • P(B|A) = .90*.15 / .56 = .24107.

6. Example of Bayes’ theorem (cont.) • So, on the basis of this one positive result, there is about a 1 in 4 chance that she is pregnant. • Note that the probabilities used in this example are not accurate, and the problem is oversimplified.

7. Example of Bayes’ Theorem (cont.) • Not a very satisfying answer. • Solution: retest. • Now, our prior probability of pregnancy is P(B) = .24107. • We repeat and get another positive. • P(A) = (.90)(.24107) + (.50)(.75893) = .59643 • P(B|A) = .90*.24107/ .59643 = .36377.

8. Example of Bayes’ Theorem (cont.) • If she repeats this and continues to get positive results, her probabilities of pregnancy are: test 3 = .507, test 4 = .649, test 5 = .769, test 6 = .857, test 7 = .915, test 8 = .951, test 9 = .972, and test 10 = .984. • Each time she adds a new test (= new data), her posterior probability of being pregnant changes.

9. Bayesian inference • The basic idea of Bayesian inference is to apply Bayes’ theorem to the relationship between data and our prior beliefs about parameters. • In the example, the parameter of interest was P(pregnant). • We updated our prior belief on the basis of each subsequent test result (data).

10. Bayesian inference (cont.) • P(A|B) is the density of the data (proportional to the likelihood). • P(B) is our prior belief about the parameters. • P(B|A) is our updated belief about the parameters, given the observed data. • The updated belief is called the posterior distribution.

11. The likelihood function • Joint densities (data, given parameters). • View the joint density as a function of parameters given the data  likelihood function. • Traditional use of the likelihood function: maximum likelihood estimation.

12. Properties of maximum likelihood estimates (review) • Maximum likelihood estimators are often biased. • Minimum variance estimators. • Likelihood ratio testing.

13. Prior and posterior distributions • We have already defined the prior as a belief about the distribution of the parameter(s). • Non-informative (vague) priors are used when we don’t have strong beliefs about the parameters. • The posterior distribution is a statement of our belief about the parameters, updated to account for the evidence of the data.

14. Prior and posterior distributions (cont.) • A conjugate prior is one chosen to produce a posterior that has the same form as the likelihood function. • Examples: normal-normal, beta-binomial.

15. Bayesian estimation • Bayes estimates are based on the posterior distribution. • Often, the mean of a parameter’s posterior distribution is used as an estimate of the parameter. • The math for that can become very difficult. Sometimes, the mode of the posterior is used instead (Bayes modal estimation).

16. Bayesian estimation (cont.) • The maximum likelihood estimator may be thought of as a Bayes modal estimator with an uninformative prior. • Modern computing power can remove the need for the nasty math traditionally needed for Bayesian estimation and inference. • This makes the Bayesian approach more accessible than it once was.

17. Bayesian inference • Bayesian inference involves probabilistic statements about parameters, based on the posterior distribution. • Probabilistic statements are allowed because Bayesians view the parameters as random variables. • For example, a Bayesian credibility interval allows us to make the kind of statement we wish we could make when we use confidence intervals.

18. Bayesian inference (cont.) • In the Bayesian approach, one can discuss the probability that a parameter is in a particular range by calculating the area under the posterior curve for that range. • For example, I might be able to make the statement that the probability mu exceeds 110 is .75. • That sort of statement is never possible in frequentist statistics.

19. Bayesian inference (cont.) • Bayesian inference does not involve null hypotheses. (Formally, the null hypothesis is known to be false if we take the Bayesian perspective. Why?) • Rather, we make probabilistic statements about parameters. • We can also compare models probabilistically.

20. An example using the Peabody data • Suppose we are interested in estimating the mean Peabody score for a population of 10-year-old children. • We have strong prior reasons to believe that the mean is 85. • We operationalize that prior belief by stating that m ~ N(85, 4).

21. Peabody example (cont.) • Next, we assume that Peabody itself is normally distributed:

22. Peabody example (cont.) • Recall that we want a posterior distribution for m. • Bayes’ theorem says • Note that we can ignore the denominator here, as it is just a scaling constant.

23. Peabody example (cont.) • Our posterior, then, is proportional to • Some unpleasant algebra that involves completing the square shows that this is the same as normal with mean = (85s2+4nM) / (4n + s2).

24. Peabody example (cont.) • The variance of the posterior is 4s2/ (4n + s2). • In our example, M = 81.675, an estimate of the variance is 119.2506, and n = 40. • The posterior mean, then, is (85119.2506 + 4 40 81.675) / (4 40 + 119.2506) = 83.095.

25. Peabody example (cont.) • The variance is 4 119.2506 / (4 40 + 119.2506) = 1.708. • A 95% credibility interval is given by 83.095 ± 1.96 √1.708 = (80.53, 85.66). • As Bayesians, we may say that the probability that m lies between those values is .95.

26. Peabody example (cont.) • Now let’s suppose that we want to repeat the analysis, but with an uninformative prior for the mean. • Instead of m ~ N(85, 4), we’ll use N(85, 10000000). • The posterior distribution of the mean, then, is centered at (85119.2506 + 10000000 40 81.675) / (10000000 40 + 119.2506) = 81.675.

27. Peabody example (cont.) • The variance is 10000000 119.2506 / (10000000 40 + 119.2506) = 2.98126. • A Bayesian credibility interval for the mean, then, would 81.675 ± 1.96√2.98126 = (78.29, 85.06). • Although this is identical to the confidence interval we would calculate using frequentist maximum likelihood, we are justified in giving it a Bayesian interpretation.