Applied Bayesian Statistics STAT 388/488

Transcription

1 STAT 388/488 Dr. Earvin Balderama Department of Mathematics & Statistics Loyola University Chicago August 29, 207

2 Course Info STAT 388/

3 A motivating example (See http: //math.luc.edu/~ebalderama/bayes_resources/handouts/eddy_what_is_bayesian.pdf) Alice and Bob play a casino game; first player to 6 points wins. Before the game starts, the casino rolls a ball randomly onto a pool table (that Alice and Bob can t see) until it comes to a complete stop. It s position is marked and remains for the duration of the game. 2 Each point is awarded based on another ball being rolled randomly onto the table: If the ball stops to the left of the initial mark, Alice is awarded the point. If the ball stops to the right of the initial mark, Bob is awarded the point. Alice and Bob are told nothing except who is awarded each point. 3

4 First, some questions Let θ be the probability that Alice is awarded a point. Before the game starts, What s your best guess about θ? 4

5 First, some questions Let θ be the probability that Alice is awarded a point. Before the game starts, What s your best guess about θ? 2 What s the probability that θ is greater than a half? 4

6 First, some questions Let θ be the probability that Alice is awarded a point. Before the game starts, What s your best guess about θ? 2 What s the probability that θ is greater than a half? Suppose the game is being played, and the score is now Alice 5, Bob 3. What s your best guess about θ now? 2 What s the probability that θ is greater than a half now? 4

7 Frequentist approach The Frequentist approach requires the (theoretical) notion of long-run frequency distributions: Quantifying uncertainty in terms of repeating the sampling process many times. The parameters are fixed and unknown. The sample (data) is random. Probability statements are only made about the randomness in the data. 5

8 Frequentist approach Sample statistic A statistic is a numerical summary of a sample. For example, X is a statistic, and is an estimator of the population mean µ. Here, one would never say P(µ > 0) = Sampling distribution The distribution of a sample statistic that arises from repeating the process that generated the data many times. Here, one would never say the distribution of µ is Normal(5.3, 0.8). 6

9 Frequentist approach 95% confidence interval An interval constructed from the data that should contain the true parameter value 95% of the time if we repeated the process that generated the data many times and computed an interval each time. Here, one would never say the probability that µ is in the interval (4.2, 5.6) is p-value Probability of observing a test statistic at least as extreme as observed in the sample if we repeated the process that generated the data many times. Here, one would never say the probability that H 0 is true is

10 Frequentist approach Examples of repeatable data generation: Sometimes it s hard to imagine repeating the data generation: 8

11 Some debate about the merits of the p-value not-even-scientists-can-easily-explain-p-values/ scientists-perturbed-by-loss-of-stat-tools-to-sift-research-fudge-from-fact/ 9

12 What are Frequentist answers to these questions? Before the game starts, What s your best guess about θ? 2 What s the probability that θ is greater than a half? 0

13 What are Frequentist answers to these questions? Before the game starts, What s your best guess about θ? 2 What s the probability that θ is greater than a half? After collecting observations, What s your best guess about θ now? 2 What s the probability that θ is greater than a half now? 0

14 What are Frequentist answers to these questions? Before the game starts, What s your best guess about θ? 2 What s the probability that θ is greater than a half? After collecting observations, What s your best guess about θ now? 2 What s the probability that θ is greater than a half now? Bonus question, What s Bob s probability of winning? 0

15 Bayesian approach The Bayesian approach consists of finding the most credible values of a parameter, conditional on the data: Uncertainty is described using probability distributions that are updated as data is observed. The true parameter values are fixed and unknown, but their uncertainty is described probabilistically and so are treated as random variables. The sample (data) is considered fixed. Probability statements express degree of belief and uncertainty in the unknown parameters.

16 Bayesian learning Prior distribution, f (θ) The uncertainty distribution of θ, before observing the data. Posterior distribution, f (θ y) The uncertainty distribution of θ, after observing the data. Bayes Rule Provides the rule for updating the prior: f (θ y) = f (y θ)f (θ) f (y) Posterior Likelihood Prior 2

17 Bayesian learning Likelihood function, f (y θ) Distribution of the data given θ. This function is created by choosing a reasonable probability model for the data, then writing the probability of the data under this model. Regarded as a function of the model s parameters (Remember, the data is considered fixed!). Bayes Rule Provides the rule for updating the prior: f (θ y) = f (y θ)f (θ) f (y) Posterior Likelihood Prior 3

18 Back to example The probability of Alice being awarded a point is a random variable θ [0, ] Usually, we form a prior by assigning (varying levels of) probabilities across all possible values of θ. If we have no relevant prior information we might use an uninformative prior such as The likelihood may be The posterior then turns out to be θ Uniform(0, ) y θ Binomial(n, θ) θ y Beta(y +, n y + ) 4

19 Specifying a Beta prior distribution A more flexible prior is θ Beta(a, b), where a and b control the shape. When a = b =, this specifies the uniform prior. The posterior then turns out to be E(θ y) = θ y Beta(y + a, n y + b). y + a n + a + b, V(θ y) = (y + a)(n y + b) (n + a + b) 2 (n + a + b + ) A prior is conjugate with respect to the likelihood if the posterior distribution is in the same family as the prior. Thus, the Beta prior is a conjugate prior for the Binomial likelihood. 5

20 Back to example The score is Alice 5, Bob 3. What s your best guess about θ now? 2 What s the probability that θ is greater than a half now? Bonus question, What s Bob s expected probability of winning? 6

21 Advantages of Bayesian approach Bayesian concepts (arguably) easier to interpret than frequentist ideas. Able to incorporate scientific/expert knowledge via the prior. In some cases the computing is easier (hierarchical models). Easy to incorporate data from multiple sources. Sample size reduction via prior or adaptive trial design. Imputing missing data comes naturally. FDA document on the use of Bayesian methods: gov/regulatoryinformation/guidances/ucm07072.htm 7

22 Disadvantages of Bayesian approach Picking a prior can be subjective. Slow computation time for complex problems. Less common/familiar. Nonparametric methods are challenging. Frequentist properties are desirable. 8

23 Frequentist vs Bayesian 9