Archive for Kakutani’s product martingale theorem

an infinite regress of hierarchical priors

Posted in Statistics with tags , , , , on October 22, 2020 by xi'an

An interesting musing posted on X validated about the impact of perpetuating prior models on the parameters of closer priors till infinity. Using a hierarchy of exponential priors and an exponential sampling distribution. If the (temporary) top prior at level d is Exp(1), the marginal distribution of the exponential sample corresponds to a ratio of two independent products of Exp(1) random variables

X= \frac{\epsilon_{2\lfloor d/2 \rfloor}\cdots \epsilon_0}{\epsilon_{2\lfloor (d-1)/2 \rfloor+1}\cdots \epsilon_1}

And both terms converge almost surely to zero with d (by Kakutani’s product martingale theorem). Thus ending up in an indeterminate ratio. Hierarchy has to stop somewhere! (Or, assuming an expectation of one everywhere, the variability at each level has to decrease fast enough.)