From brian@stat.cmu.edu Wed Nov 20 17:21:20 2002 Date: Wed, 20 Nov 2002 17:17:59 -0500 (EST) From: Brian Junker To: 36-711 Fall 2002 , Bryce Corrigan , Catherine E Ott , jiji@andrew.cmu.edu, Jeffery Liebner , Jeffery Palmer , khuang@andrew.cmu.edu, Lynne Steuerle , Mingyu Cao , Margaret Kenney , Mihaela Serban , rmostafa@andrew.cmu.edu, Stefano Cabras , tgeylani@andrew.cmu.edu, xuerui@andrew.cmu.edu, yihengl@andrew.cmu.edu Subject: end of rejection/SIR example, FYI All, Attached is the graph that I created at the end of class. The upper 2-D histogram is from the first 1000 observations of the sample of size 10000 that I started near the end of class. The lower 2-D histogram is from a SIR resample of 1000 cases from the full 1000, with importance weights that switched the prior on the mean, theta, from N(1,1) to N(5,1). The SIR sample is smoother and better approximates the true posterior, now that we are subsampling at a factor of 1/10 of the original sample. Since the MLE for theta was 1.3, approximately, we expect that the new N(5,1) prior agrees less well with the data. In the lower figure, the posterior mode for theta may have moved up slightly (theta is on the x-axis, s^2 on the y-axis), as we would expect, but also there is more spread in the theta direction. Note especially the longer tail to higher values of theta; effectively the new prior is trying to "pull" the whole distribution upward. Also there appears to be some more spread in the direction of larger s^2; there is less tension between the prior mean 5 for theta and the data mean 1.3 if the spread s^2 is itself larger, and that kind of combination of parameters is what is being suggested by this positive skewing in the s^2 direction. Best, -BJ [ps - remember, no class Fri; see you in class Mon] [ Part 2, "" Application/POSTSCRIPT 43KB. ] [ Unable to print this part. ]