1. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 71 Lecture 7 Bayesian methods: a refresher 7.1 Principles of the Bayesian approach 7.2 The beta distribution 7.3 A simple example 7.4 Decision theory 7.5 Choosing a design

2. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 72 7.1 Principles of the Bayesian approach Bayesian methods are: Subjective Combine opinion with data Becoming accepted in scientific research Consider a clinical trial comparing experimental E with control C, in which  = advantage of E over C The advantage  is now treated as a random variable, not as a fixed quantity We learn of its distribution (incorporating uncertainty), not of its value

3. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 73 Subjective distribution The probability that  is less than or equal to , H(  ) = P (    ) is a function summarising your opinion of where  lies It rises from 0 to 1 as  goes from –  to 

4. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 74 Evaluating a null hypothesis Compute p = P (   0) = H(0) If p is small, then you have little belief in E not having an advantage over C

5. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 75 Estimation (1) Put h(  ) = dH(  )/d  The value of  in which you have the greatest belief

6. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 76 Estimation (2) is the mean of your subjective distribution for  If  > 0, then is the area between 1 and H(  )

7. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 77 Credibility interval A 95% credibility interval (  L,  U ) is found from H(  L ) = 0.025 H(  U ) = 0.975 You believe that  has a 95% chance of lying in (  L,  U )

8. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 78 Prior and posterior distributions Before seeing data from your study, you have a prior opinion about , expressed as H 0 (  ) = P 0 (    ), h 0 (  ) = dH 0 (  ) /d  where h 0 is the prior density for  After observing data x, you have a posterior opinion about , expressed as H(  | x) = P (    | x), h(  | x) = dH(  | x) /d  where h(  | x) is the posterior density for 

9. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 79 A model for the data f(x;  ) = P(X = x |  =  )  that is the probability of observing the data x actually recorded, if  were equal to  If the data are continuous, let f(x;  ) denote the corresponding multivariate density function f(x;  ) is called the likelihood of  given x and is sometimes denoted by L( , x)

10. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 710 Bayes theorem This shows how to combine prior opinion with data to produce posterior opinion As the denominator is a constant in , we need only know that and that, being a density,

11. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 711 Bayesian inference Express prior density h 0 (  ) from introspection or by elicitation of expert opinion Draw inferences about  Observe data x Deduce posterior density h(  | x) using Bayes theorem

12. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 712 7.2 The beta distribution X has the beta distribution, parameters r and s (r, s > 0) (X ~ beta (r, s)), if X has density where is the beta function X has mean = r / (r + s) and mode = (r – 1) / (r + s – 2)

13. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 713 The beta distribution has a wide variety of shapes: beta(2,2) beta(5,9) beta(10,3) beta(1,1) beta(0.5,2) beta(0.5,0.5) Beta distributions, or mixtures of them, can represent virtually any opinion

14. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 714 7.3 A simple example In an open label pilot study, all patients will be treated with an experimental drug Patients’ responses will be SUCCESS or FAILURE The parameter of interest is  = P( SUCCESS )

15. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 715 Data to be collected Observations x 1,…, x 10 are to be taken on 10 patients, where The likelihood will be where S = x 1 +…+ x 10 = number of successes

16. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 716 Prior density: beta (2, 2)

17. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 717 Data observed x = (1, 1, 0, 0, 0, 1, 0, 0, 0, 0 ) s = 3 The likelihood is

18. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 718 Posterior density Thus the posterior is beta (5, 9), for which

19. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 719 PriorPosterior Likelihood

20. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 720 Notes on the prior 1. BETA  DATA  BETA The posterior is from the same distributional family as the prior. This is a conjugate prior 2.The proportion of successes is The posterior modal estimate of  is prior beta (2, 2)  one extra SUCCESS and one extra FAILURE

21. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 721 7.4 Decision theory Key ingredients: 1.A model f n (x;  ) for data x to be collected 2.A prior distribution for , characterised by the density h 0 (  ) 3.A set of possible actions: A 1,…,A k 4.An expression for the gain from each action for each true value of  : G 1 (  ),…, G k (  )

22. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 722 Simple example Success rate on standard drug = 0.5 Success rate on your new drug =  If  > 0.5, you gain $50M Prior for  was beta (2,2) Pilot study for 10 patients gave 3 successes Posterior for  is beta (5,9)

23. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 723 Actions and gains Action 0: No further development Gain is G 0 (  ) = 0 for all  Action 1: Develop drug further, at a cost of $6M Gain is

24. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 724 Decision For a Bayesian,  is an observation of the random variable  Following the pilot study,  ~ beta (5,9) The expected gains (utilities) of the two actions are Slide 7.16

25. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 725 General principle 1.Identify all possible actions A 1,.., A k 2.Quantify the gain function from each action for each value of  G i (  ), i = 1,…,k 3.Use expert opinion, plus experimental data x, to find the posterior density h(  | x) 4.Compute the expected gain (utility) for each action 5.Take the action with the largest utility

26. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 726 7.5 Choosing a design In Section 7.4, prior for  was beta (2,2) Backtrack to before pilot study was conducted Q:If each patient in the pilot study costs $10,000 ($0.01M), how many patients should be included?

27. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 727 Design n Treat n patients Take Action 1 if s  k n where s is number of successes, and k n is an appropriate critical value Given , S ~ B(n,  ) and

28. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 728 Expected gain from design n, given S = s, is Thus expected gain, given , is This is still a function of , and so expectation has to be taken with respect to the prior beta (2,2) distribution for  to give the utility, E 0 {G n (  )}, n = 0, 1, 2, ….

29. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 729 For each n, search for best k n - then search for the best n Utility = E 0 {G n (  )} Optimum: n = 47, k n = 18, Utility = 20.5

30. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 730 Introductory texts on Bayesian methods Lee (1989)Bernardo and Smith (1994) Gelman et al. (1995) … and their application to medical data and clinical trials Parmigiani (2002) Spiegelhalter, Abrams and Myles (2004) Recent texts emphasising MCMC and R Hoff (2009)Bolstad (2009) Introductory texts on decision theory Lindley (1971)Berger (1985) Smith (1988)Bather (2000)