In this lecture, we shall study credible intervals, the Bayesians alternative to confidence intervals. Recall the confidence intervals from previous lectures. These are our frequentest way to express uncertainty about an estimate of a population mean, a population proportion or some other parameter. A confidence interval has the form of an upper and lower bound. Such that the upper and lower bounds are usually equal to the point estimate from the sample. Plus and minus the standard error times a critical value. Importantly the interpretation of, say, a 95% confidence interval on the mean. Is that 95% of similarly constructed confidence intervals will contain the true mean. And not the probability that the true mean lies between L and U is 0.95. The reason for this frequentist wording is that a frequentist may not express his uncertainty as a probability. The true mean is either within the interval, or not, with probability zero or one. And the frequentist just doesn't know which is the case. But Bayesians have no such qualms. It is fine for us to say that the probability that the true mean is contained within a given interval is 0.95. To distinguish our intervals from confidence intervals we call them credible intervals. Recall the RU-486 example. When the analyst used the beta binomial family and took the beta 1,1, the uniform distribution. As her prioron P, the probability of a child having a mother who received RU-486. As we saw, after observing four children born to mothers who received conventional therapy, her posterior on P was the beta 1,5. As you can see, the posterior probability for the beta 1,5 density, puts a lot of probability near zero and very little probability near one. For the Bayesian, her 95% credible interval is just any L and U such that the posterior probability that L < p < U is 0.95. The shortest such interval is obviously preferable. To find this interval, the Bayesian uses the formula for the area under the beta 1,5 distribution, that lies to the left of a value x. It is 1- ( 1- x )^ 5. She can use this to find the L and U which have area 0.95 under the density curve, between L and U. Unlike the symmetric confidence intervals that frequentists often obtain, the Bayesian credible interval is asymmetric. It turns out that L = 0 and U = 0.45 is the shortest interval that has probability 0.95 of containing p. What have we done? We have seen the difference in interpretations between the frequentist confidence interval and the Bayesian credible interval. Also, we have seen the general form of a credible interval. Finally, we have done a practical example constructing a 95% credible interval for the RU-486 data set.