*usage correct? Alanis Morisette and subsequent brouhaha has made me permanently paranoid about this.

]]>In contrast, NHST is generally not coherent in this way. Estimation may be performed using ML (or several variants), some minimum loss function, etc. Inference is then performed separately using analytically derived sampling distributions, bootstrapped sampling distributions, model comparison with some loss function, etc.

With Bayes, you specify one thing: The posterior distribution. From that, you get your estimates and your inference. You specify the model probabilistically, you obtain expected values or modal estimates of the posterior, you make your inference from the posterior. Parameter estimates? Bayes theorem. Model comparison? Bayes theorem. Inference? Bayes theorem.

This can be good or bad; on one hand, that means you can’t hotswap any one of these components for something else, whereas within the frequentist paradigm you can swap out how you estimate and how you make inferences pretty willy-nilly. On the other hand, this is a good thing, because it’s all driven by one statistical kernel: Probability theory.

With regard to this particular paper, I believe Efron was saying that the Bayesian estimate would be very different from a bootstrapped estimate, largely due to overfitting. But he didn’t really discuss priors. Lindley was just saying, I think, that if you think some estimate is ridiculous, then your prior should express that beforehand, but because the prior wasn’t even mentioned, it’s using an incoherent version of bayes (likelihood-only, no priors really), then calling it incoherent. “Coherent bayes”, in this case, would have and use a prior.

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