There are other congregate families. One is the normal-normal pair. If your data come from a normal distribution with known standard deviation, sigma, but unknown mean, mu, and if your prior on the mean, mu, has a normal distribution with self-elicited mean, nu, and self-elicited standard deviation, tau, then your posterior density for the mean, after seeing a sample of size n with sample mean x-bar, is also normal. As a practical matter, one often does not know sigma, the standard deviation of the normal from which the data come. In that case, you could use a more advanced conjugate family. But there are cases in which it is reasonable to treat the sigma as known. One example is an analytical chemist whose balance produces measurements that are normally distributed with mean equal to the true mass of the sample and standard deviation that has been estimated by the manufacturer balance and confirmed against calibration standards provided by the National Institute of Standards and Technology. For the normal-normal conjugate families, assume the prior on the unknown mean, mu, is normal with mean, nu, and standard deviation tau. Also assume that the data X1, X2... Xn are independent and come from a normal with standard deviation sigma then the posterior distribution from mu, after seeing the data, is normal with mean equal to a weighted average of the prior mean and the sample mean. And the posterior standard deviation is the square root of sigma squared times tau squared all over sigma squared plus n times tau squared. Suppose an analytical chemist wants to measure the mass of a sample of ammonium nitrate. Her balance has a known standard deviation of 0.2 milligrams. By looking at the sample, she thinks this mass is about 10 milligrams and based on her previous experience in estimating masses by "i" her uncertainty in that guess which is the standard deviation of her prior is 2. So she decides that her prior for the mass of the sample is a normal distribution with mean, 10 milligrams, and standard deviation, 2 milligrams. Now she collects five measurements on the sample and finds that the average of those is 10.5. By conjugacy of the normal-normal family, our posterior belief about the mass of the sample has the normal distribution. The new mean of that posterior normal is found by plugging into the formula. She gets 10.499. Similarly, her posterior standard deviation also changes, by plugging into the formula for the posterior standard deviation. She finds that her uncertainty has reduced from two milligrams to 0.089 milligrams. So the Bayesian analytical chemist has learned a lot in her assay of the ammonium nitrate. Her posterior mean has shifted quite a bit and her uncertainty has dropped by a lot. That's exactly what an analytical chemist wants. This is the last of the three examples of conjugate families. There are many more, but they do not suffice for every situation one might have. We learned several things in this lecture. First, we learned the new pair of conjugate families and the relevant updating formulae. Also, we worked a realistic example problem that can arise in practical situations.