so my understanding is that it's a general term for when you model both the mean (w/ something more than just an intercept) and covariance terms of a multivariate normal in a GP setting -- it's usually used in spatial autocorrelation models but I've heard it used for temporal autocorrelation settings too. AFAIK, though, complicated covariance functions and complicated mean functions are non-identifiable, so you have to pick one or other (some might even say trying to put a trend on the mean is non-identifiable with the the covariance function -- which it sort of is, in that a GP can pick up on a trend in-sample no problem -- but I think if there is a trend identifying it can help tighten up variance, and so is worth trying to include + it's helpful for interpretability).

yep, there are lots of possible autocorrelation functions, and you can additively or multiplicatively compose them to represent multiple simultaneous dynamics, e.g. see https://www.cs.toronto.edu/~duvenaud/cookbook/

here's a nice little graphical primer on GPs that I like: https://distill.pub/2019/visual-exploration-gaussian-processes/ or here for more squiggly animations: http://www.infinitecuriosity.org/vizgp/

I don't think there's much formal testing, per se, but you can do model comparison with or without different components as 'tests' for their inclusion (and maybe in an initial exploratory data viz setting just plot the output of acf() in R a la https://www.datacamp.com/community/tutorials/autocorrelation-r)

yup, this would be folded into the dispersion term + inherent variability of the poisson distribution, in this framework. It might be a slight jostle is perfectly consistent with the variance you already see in the poisson (or any extra variance implied by overdispersion)

eh, more poisson bc it's count data, ie non-negative integer data with no (or large / unknown) upper bound, and that's the go-to for that haha. And the log link also bc it's the canonical link function for the rate param (to keep everything positive, since rates are strictly positive). But it is good to think about the sorts of process that can give rise to difft distributions!

in a Bayesian framework you're less focused on rejecting a null model by seeing if some test statistic falls in the tails of its sampling distribution under the null, and more comparing different models you've specified, one of which you're free to call a "null" model. So in the end you can say "the alternative model is favored by the data with probability > 0.999" or something, or else just look at the relevant parameter in the alternative model and say 99.999 percent of the probability mass falls above 0

yup! aka also the likelihood term in the numerator of Bayes theorem, \(Pr(X|\theta)\)

yeah, you could have a mixture ('spike-and-slab') prior like this, but fitting it would be obnoxious cos you'd need a jump move to move between them. Might be easier to just equivalently fit the two separate models and update discrete uniform model priors, though then you run into issues with marginal likelihood estimation, which can also be tricky!

yep, in the sense of having different priors. A model with a point mass prior on 0 isn't much different from a model with e.g. a laplace(0, 10000000) prior or whatever

you'd have different priors / models that you're comparing, here

nah, there's no guarantee like that here -- the marginal likelihood is bounded between (0,inf), and the marginal log-likelihood between (-inf, inf). Also you can have as many models as you want here, and if working in the BF world just take the best ML and compare it to the second & third best etc.

you could do this, but it would seem more straightforward to compare a model with \(Pr(B=0) = 1\) vs.\( Pr(B≠0) = 1\), with the former dropping the parameter from the model specification, and the latter just putting any continuous prior on B, rather than a point mass on some other value

yeah, you can set two different priors and then use BF etc. to compare them -- one that's strongly informative around 0 (or truncated to this "null region"), one that's more weakly informative, and one with a point mass at 0. Alternatively, just use your weakly informative prior and ask how much of the posterior mass falls outside some bounds around 0, or is to one side of 0, etc.

Should also note that since models are their priors, model comparison can sorta be thought of as prior comparison. But you're in pretty grave danger of overfitting if you use BF to just search through different priors, because the ultimate marginal likelihood will be found when the prior's a point mass on the MLE.