Indeed, Radford, indeed, I inserted that quote because I did not agree with its message. (As most quotes I include.) But without spelling out the objection, while you did. As far as I remember, the fact that the scales of the parameters are directly dependent on the scales of the regressors (which is why I favour g-priors) does not appear clearly in the text.

]]>“It is interesting to note the strong negative correlation in these parameters. If one assigned informative independent priors on β⁰ and β¹, these prior beliefs would be counter to the correlation between the two parameters observed in the data.”

This is deeply confused, and will mislead readers of the book. In general, seeing that two parameters are correlated in the posterior, when they are independent in the prior, is NOT an indication that anything is wrong with your prior. The posterior correlation is a property of your posterior beliefs, not a property of the real world. There is no contradiction in having independent prior beliefs but dependent posterior beliefs.

In regression models, it is entirely normal for posteriors for parameters to be correlated if the associated covariates are correlated in the data. This in general says nothing at all about what the appropriate prior for the parameters would be. In designed experiments, for example, the covariate distribution (and associated correlation) is whatever the experimenter decided it would be, which need have nothing to do with the experimenter’s (or other person’s) prior for the parameters.

In this specific case, the correlation is between the intercept parameter and the slope of price versus size, and it comes about because the size is not centred at zero (and hence “correlates” with a constant). There is indeed a bit of a problem here, in that you probably don’t believe that high sensitivity of price to size goes with high overall price (which would be an implication of using independent priors), although that is certainly a possible prior belief. And there’s a general problem with the whole model in that it doesn’t directly capture the fact that negative prices are impossible, which would need to be captured by a dependent prior. But this all has nothing to do with the posterior correlation between intercept and slope.

In general, looking at your posterior distribution and then changing your prior to more closely resemble it is a double use of data, and will lead to inferences that are overly confident.

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