Introduction to Bayesian statistics Three approaches to Probability  Axiomatic Probability by definition and properties  Relative Frequency Repeated trials  Degree of belief (subjective) Personal measure of uncertainty Problems  The chance that a meteor strikes earth is 1%  The probability of rain today is 30%  The chance of getting an A on the exam is 50%

Problems of statistical inference Ho: θ=1 versus Ha: θ>1 Classical approach  P-value = P(Data | θ=1)  P-value is NOT P(Null hypothesis is true)  Confidence interval [a, b] : What does it mean? But scientist wants to know:  P(θ=1 | Data)  P(Ho is true) = ? Problem  θ “not random”

Bayesian statistics Fundamental change in philosophy Θ assumed to be a random variable Allows us to assign a probability distribution for θ based on prior information 95% “confidence” interval [1.34 < θ < 2.97] means what we “want” it to mean: P(1.34 < θ < 2.97) = 95% P-values mean what we want them to mean: P(Null hypothesis is false)

Estimating P(Heads) for a biased coin Parameter p Data: 0, 0, 0, 1, 0, 1, 0, 0, 1, 0 p = 3/10 = 0.3 But what if we believe coin is biased in favor of low probabilities? How to incorporate prior beliefs into model We’ll see that p-hat =.22

Bayes Theorem

Example Population has 10% liars Lie Detector gets it “right” 90% of the time. Let A = {Actual Liar}, Let R = {Lie Detector reports you are Liar} Lie Detector reports suspect is a liar. What is probability that suspect actually is a liar?

More general form of Bayes Theorem

Example Three urns Urn A: 1 red, 1 blue Urn B: 2 reds, 1 blue Urn C: 2 reds, 3 blues Roll a fair die. If it’s 1, pick Urn A. If 2 or 3, pick Urn B. If 4, 5, 6, pick Urn C. Then choose one ball. A ball was chosen and it’s red. What’s the probability it came from Urn C?

Bayes Theorem for Statistics Let θ represent parameter(s) Let X represent data Left-hand side is a function of θ Denominator on right-hand side does not depend on θ Posterior distributionLikelihood x Prior distribution Posterior dist’n = Constant x Likelihood x Prior dist’n Equation can be understood at the level of densities Goal: Explore the posterior distribution of θ

A simple estimation example Biased coin estimation: P(Heads) = p = ? 0-1 i.i.d. Bernoulli(p) trials Let be the number of heads in n trials Likelihood is For prior distribution use uninformative prior  Uniform distribution on (0,1): f(p) = 1 So posterior distribution is proportional to f(X|p)f(p) = f(p|X)

Coin estimation (cont’d) Posterior density of the form f(p)=Cp x (1-p) n-x Beta distribution: Parameters x+1 and n-x+1 n.html n.html Data: 0, 0, 1, 0, 0, 0, 0, 1, 0, 1 n=10 and x=3 Posterior dist’n is Beta(3+1,7+1) = Beta(4,8)

Coin estimation (cont’d) Posterior dist’n: Beta(4,8) Mean: 0.33 Mode: 0.30 Median: qbeta(.025,4,8), qbeta(.975,4,8) = [.11,.61] gives 95% credible interval for p P(.11 < p <.61|X) =.95

Prior distribution Choice of beta distribution for prior

Posterior Likelihood x Prior = [ p x (1-p) n-x ] [ p a+1 (1-p) b+1 ] = p x+a+1 (1-p) n-x+b+1 Posterior distribution is Beta(x+a, n-x+b)

Prior distributions Posterior summaries:  Mean = (x+a)/(n+a+b)  Mode = (x+a-1)/(n+a+b-2)  Quantiles can be computed by integrating the beta density For this example, prior and posterior distributions have same general form Priors which have the same form as the posteriors are called conjugate priors

Data example Maternal condition placenta previa Unusual condition of pregnancy where placenta is implanted very low in uterus preventing normal delivery Is this related to the sex of the baby? Proportion of female births in general population is Early study in Germany found that in 980 placenta previa births, 437 were female (0.4459) Ho: p = versus Ha: p < 0.485

Placenta previa births Assume uniform prior Beta(1,1) Posterior is Beta(438,544) Posterior summaries  Mean = 0.446, Standard Deviation =  95% confidence interval: [ qbeta(.025,438,544), qbeta(.975,438,544) ] = [.415,.477 ]

Sensitivity of Prior Suppose we took a prior more concentrated about the null hypothesis value E.g., Prior ~ Normal(.485,.01) Posterior proportional to Constant of integration is about Mean, summary statistics, confidence intervals, etc., require numerical methods See S-script: ourses/275w05/Scripts/Bayes.ssc ourses/275w05/Scripts/Bayes.ssc