My Ph.D. dissertation was in Bayesian Econometrics. I started working on the dissertation early in 1973, and Arnold Zellner's classic text had been available for just over a year. So, perhaps not surprisingly, one of the first things that I did was to sit down and go through his book with a fine tooth comb. It took quite a while, but it was time well spent!
The trouble started with the first equation in Chapter 2, "Principles of Bayesian Analysis With Selected Applications". There, Bayes' Theorem is stated, as follows:
"Then, according to the usual operations with pdf's we have
p(y , θ) = p(y | θ) p(θ)
= p(θ | y) p(θ) "
It didn't require a trip to the library in order to correct the last line to:
= p(θ | y) p(y),
but this illustrates what I was up against!
Missing or mis-placed parentheses; square root signs that shouldn't have been there; wrong signs;..................... all of the usual challenges.
And then there (still) are those wonderful "throw away" lines that we all hate to encounter, such as:
- "On integrating with respect to σ we obtain....".
- "On completing the square on β we have...."
- "On substituting in (3.56) and integrating with respect to the k elements of β we obtain..."
- "The integrations over σ1 and σ2 are easily performed to yield the following joint posterior pdf...."
Some of that integration is a little tricky, and requires a page or two of manipulations. (No hints are given, either.) And can you remember how to "complete the square"? You did it when you learned how to solve quadratic equations, but that was for the scalar case, not the vector/matrix case! Oh joy!
This was all pre-MCMC days, and so the implementation of the Bayesian analysis was computationally challenging, to say the least. I extended the FORTRAN programs in Appendix C of the book to the three-dimensional case. That was as far as one could go! Even plotting out posteriors meant writing specific FORTRAN code - and dealing with the darned punch-cards!
The other day I was cleaning out a filing cabinet and I found some yellowing hand-written notes. They turned out to be my notes that summarize most of the chapters in Zellner (1971), correct the early typos., and (more importantly) fill in the gaps by working through the necessary integration and the like. The notes for a few of the chapters have gone missing over the past forty years, and during the course of three international moves.
Quite a "blast from the past", though!
Arnold gave me a phenomenal amount of help during that start-up period. I thought that perhaps students of Bayesian econometrics might appreciate having access to those notes, so for what they're worth, you can find them here.
Zellner, A., 1971. An Introduction to Bayesian Inference in Econometrics. Wiley, New York.
© 2012, David E. Giles