likelihood-free Bayesian inference on the minimum clinically important difference

Last week, Likelihood-free Bayesian inference on the minimum clinically important difference was arXived by Nick Syring and Ryan Martin and I read it over the weekend, slowly coming to the realisation that their [meaning of] “likelihood free” was not my [meaning of] “likelihood free”, namely that it has nothing to do with ABC! The idea therein is to create a likelihood out of a loss function, in the spirit of Bassiri, Holmes and Walker, the loss being inspired here by a clinical trial concept, the minimum clinically important difference, defined as

\theta^* = \min_\theta\mathbb{P}(Y\ne\text{sign}(X-\theta))

which defines a loss function per se when considering the empirical version. In clinical trials, Y is a binary outcome and X a vector of explanatory variables. This model-free concept avoids setting a joint distribution  on the pair (X,Y), since creating a distribution on a large vector of covariates is always an issue. As a marginalia, the authors actually mention our MCMC book in connection with a logistic regression (Example 7.11) and for a while I thought we had mentioned MCID therein, realising later it was a standard description of MCMC for logistic models.

The central and interesting part of the paper is obviously defining the likelihood-free posterior as

\pi_n(\theta) \propto \exp\{-n L_n(\theta) \}\pi(\theta)

The authors manage to obtain the rate necessary for the estimation to be asymptotically consistent, which seems [to me] to mean that a better representation of the likelihood-free posterior should be

\pi_n(\theta) \propto \exp\{-n^{-2/5} L_n(\theta) \}\pi(\theta)

(even though this rescaling does not appear verbatim in the paper). This is quite an interesting application of the concept developed by Bissiri, Holmes and Walker, even though it also illustrates the difficulty of defining a specific prior, given that the minimised target above can be transformed by an arbitrary increasing function. And the mathematical difficulty in finding a rate.

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