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Bayesian Prior and Posterior Study Guide for ES205 Yu-Chi Ho Jonathan T. Lee Nov. 24, 2000 Outline Conditional Density Bayes Rule Conjugate Distribution Example Other Conjugate Distributions Application 2 Conditional Density The conditional probability density of w happening given x has occurred, assume px(x) 0: pW , X w, x pW | X w | x p X x pW , X w, x pW | X w | x p x x p X |W x | w p w w N 3 Bayes Rule Replace the joint probability density function with the bottom equation from page 3: pW | X 1 ,, X n w | x1 , xn N p X 1 ,, X n |W x1 ,, xn | w pW w p X 1 ,, X n x1 ,, xn 4 Conjugate Distribution W: parameter of interest in some system X: the independent and identical observation on the system Since we know the model of the system, the conditional density of X|W could be easily computed, e.g., p X |W x | w N 5 Conjugate Distribution (cont.) N If the prior distribution of W belong to a family, for any size n and any values of the observations in the sample, the posterior distribution of W must also belong to the same family. This family is called a conjugate family of distributions. 6 Example An urn of white and red balls with unknown w being the fraction of the balls that are red. Assume we can take n sample, X1, …, Xn, from the urn, with replacement, e.g, n i.i.d. samples. This is a Bernoulli distribution. N 7 Example (cont.) Total number of red ball out of n trials, Y = X1 + … + Xn, has the binomial distribution p X1 , , X n |W x1 , , xn | w w 1 w y Assume the prior dist. of w is beta distribution with parameters and pW W w 1 N n y 1 w 1 8 Example (cont.) The posterior distribution of W is pW | X 1 ,, X n w | x1 , xn p X 1 ,, X n |W x1 ,, xn | w pW w y 1 w 1 w n y 1 which is also a beta distribution. 9 Example (cont.) Updating formula: • • N ’ = + y Posterior (new) parameter = prior (old) parameter + # of red balls ’ = + (n – y) Posterior (new) parameter = prior (old) parameter + # of white balls 10 Other Conjugate Distributions N The observations forms a Poisson distribution with an unknown value of the mean w. The prior distribution of w is a gamma distribution with parameters and . The posterior is also a ngamma distribution with parameters i 1 xi and + n. Updating formula: ’ = + y ’ = + n 11 Other Conjugate Distributions (cont.) N The observations forms a negative binomial distribution with a specified r value and an unknown value of the mean w. The prior distribution of w is a beta distribution with parameters and . The posterior is also a beta distribution n with parameters + rn and i1 xi . Updating formula: ’ = + rn ’ = + y 12 Other Conjugate Distributions (cont.) The observations forms a normal distribution with an unknown value of the mean w and specified precision r. The prior distribution of w is a normal distribution with mean and precision . The posterior is also a normal distribution nx r with mean and precision + nr. nr Updating formula: N nx r nr nr 13 Other Conjugate Distributions (cont.) N The observations forms a normal distribution with the specified mean m and unknown precision w. The prior distribution of w is a gamma distribution with parameters and . The posterior is also a gamma distribution n n 1 with parameters and 2 i 1 xi m2 . 2 Updating formula: ’ = + n/2 n ’ = + ½ i 1 xi m2 14 Summary of the Conjugate Distributions N Observations Prior Posterior Bernoulli Beta Beta Poisson Gamma Gamma Negative binominal Normal Beta Normal Gamma Gamma Beta Normal Normal 15 Application N Estimate the state of the system based on the observations: Kalman filter. 16 References: • DeGroot, M. H., Optimal Statistical Decisions, McGraw-Hill, 1970. • Ho, Y.-C., Lecture Notes, Harvard University, 1997. • Larsen, R. J. and M. L. Marx, An Introduction to Mathematical Statistics and Its Applications, Prentice Hall, 1986. 17