## the worst possible proof [X’ed]

Another surreal experience thanks to X validated! A user of the forum recently asked for an explanation of the above proof in Lynch’s (2007) book, Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. No wonder this user was puzzled: the explanation makes no sense outside the univariate case… It is hard to fathom why on Earth the author would resort to this convoluted approach to conclude about the posterior conditional distribution being a normal centred at the least square estimate and with σ²X’X as precision matrix. Presumably, he has a poor opinion of the degree of matrix algebra numeracy of his readers [and thus should abstain from establishing the result]. As it seems unrealistic to postulate that the author is himself confused about matrix algebra, given his MSc in Statistics [the footnote ² seen above after “appropriately” acknowledges that “technically we cannot divide by” the matrix, but it goes on to suggest multiplying the numerator by the matrix

$(X^\text{T}X)^{-1} (X^\text{T}X)$

which does not make sense either, unless one introduces the trace tr(.) operator, presumably out of reach for most readers]. And this part of the explanation is unnecessarily confusing in that a basic matrix manipulation leads to the result. Or even simpler, a reference to Pythagoras’  theorem.

### 2 Responses to “the worst possible proof [X’ed]”

1. > Lynch joined the Department of Sociology in September after receiving his master’s degree in statistics and his Ph.D. in sociology from Duke University.

Just a guess, but what people don’t regularly use – they lose.

But for their audience they should just demonstrate Bayesian analyses through naive ABC (two stage sampling), pointing out the re-weighting of the prior with percent of times observations were (closely) matched, switch to re-weighting by likelihood when available and then nth roots of likelihood as a simple sequential importance sampler.

Its just a lot of grunt work to do it – adjusting it as one discovers how it can be miss-understood but the real barrier I think is giving up linear algebra and calculus as everyone should know and use that stuff all the time.

Keith

2. Beware: stack exchange is addictive!

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