About DIC and WAIC, etc. ... that's a huge topic for me.
There's a relatively well-cited result that shows that you can arbitrarily choose any penalty in an information criterion (i.e., penalty in the form -logL + penalty), as long as it meets certain weak conditions, and it will eventually work with big enough samples. So the issues of most importance are usually how selection accuracy increases with sample size, not its asymptotic performance. The leads to a certain hesitancy for me in terms of why I should pay attention to any particular IC.
Having said that, DIC and WAIC are pretty elegant in my mind, and I've been glad to see research on them.
Predictive scenarios aren't really my focus, although I've become much more interested in them the last couple of years for various reasons. Most of my work is in what might be thought of as unsupervised models.
Yeah, I don't mean to make it sound like there's no good reason for CV... I came across as sort of harsh. I just feel like CV got taken up in application a lot because it makes intuitive sense but with relatively little in terms of theoretical scrutiny, especially relative to other alternatives. When it has been compared to alternatives, the comparisons are sort of weak in that they often involve strawman or unrepresentative assumptions. CV involves certain assumptions itself, for example, in terms of choices, that affect results in sort of unpredictable ways sometimes. I'm a big fan of bootstrapping too and have often felt that that avenue hasn't been explored as much as it could be.
I'm more giddy to respond to questions like this than I should be, given that information criteria are an area of my research.
AIC and BIC are approximations to other quantities under certain common but idealized scenarios. What's become clear over the last 20 years or so is that for most other scenarios those approximations are not nearly as good as other approximations, or just calculating the targeted values of interest.
So, for something like DL models, criteria such as AIC and BIC are less applicable. But what is confusing to me is why other information criteria aren't pursued more. There's literature on the asymptotic equivalence of cross-validation and certain information criteria, and CV has certain problems that those criteria overcome (I've been looking citations but Google thinks I'm spamming it or something).
I think part of the reason for the adoption of CV is a cultural one, to be honest, and due to historical quirks. Like bootstrapping and other empirical simulation methods, it has an appeal due to its relatively nonparametric nature. But there is a ton of mischaracterizations about CV (e.g., relative to bootstrapping) and its warts have been overlooked in general due to the position it has attained in applied settings.
There is also a huge gap between the literature on information criteria and the rest of statistics and applied data analysis. People tend to learn about AIC and BIC through certain standard sources, sources mostly pertaining to their initial derivation, or that follow the same line of reasoning. But there's other literature that derives AIC and BIC through other means, or approach them from other perspectives, and that literature tends to get ignored. As a result there's a lot of misleading statements that are made about each, even from talented individuals (e.g., that the rationale for BIC depends on Bayesianism, or that there is a true model under consideration), and a lot of advances in the area are totally ignored. It's like there's two literatures, one attended to by people who are interested in information criteria, and another by people that "just want to understand AIC and BIC." It's confusing to me really, because I generally see some kind of trickle-down in other areas of statistics, but with information criteria, it's like there's an enormous gap.
About DIC and WAIC, etc. ... that's a huge topic for me.
There's a relatively well-cited result that shows that you can arbitrarily choose any penalty in an information criterion (i.e., penalty in the form -logL + penalty), as long as it meets certain weak conditions, and it will eventually work with big enough samples. So the issues of most importance are usually how selection accuracy increases with sample size, not its asymptotic performance. The leads to a certain hesitancy for me in terms of why I should pay attention to any particular IC.
Having said that, DIC and WAIC are pretty elegant in my mind, and I've been glad to see research on them.
Predictive scenarios aren't really my focus, although I've become much more interested in them the last couple of years for various reasons. Most of my work is in what might be thought of as unsupervised models.