Apologies, I'm skipping details, because that's how I speak with my colleagues, but I realize this is an external environment without context. No references since this is folklore (you can look at Hastie et al's Surprises in High-Dimensional Ridgeless Regression paper for the non-Bayesian version, Bruno Loureiro or Andrew Gordon Wilson probably have a paper with something similar).
Concentrating a density around a zero set means that I raise it to the power of 1/gamma (appropriately normalizing) and then take gamma to zero. If the likelihood was Gaussian, this would be equivalent to taking the variance to zero (yielding a point mass). But in overparameterized settings, this concentrates on a submanifold describing the set of interpolating solutions. In least-squares linear regression, that is the solution space. Reducing the variance on a Gaussian prior is treated as an asymptotic expansion by Laplace's method. If you choose the variance to decrease (inversely proportional to the parameter size, for example), then the marginal likelihood will increase monotonically with model size.
By any measure of model size, I mean that you can pick your favourite among the common ones, such as information metrics (e.g. mutual information / KL), statistical metrics (e.g. marginal likelihood), test error. You should be able to show the same phenomenon happening for all of them, so it isn't a quirk of marginal likelihood. It is concentration of measure working in your favor to reduce the variance in the estimator.
Okay, and that's all in-sample, which is the entire point, it won't necessarily hold out of sample.
E.g. over-fitting.
No, I am talking about out of sample error and estimates thereof. It is "overfitting" to data, but it also has lower out of sample error than the case where you do not "overfit".
This is why the notion of overfitting is not nearly as cut and dry as a basic ML course would have you believe. Just because you fit data exactly does not mean that your estimator has high error on out of sample data. A trivial counterexample is a spiking model that spikes to fit to the data but otherwise follows the correct trend outside of the dataset. The bias variance tradeoff gets thrown out at enormous scale and overfitting is not a meaningful concept. What matters is regularization and robustness, not how well you fit the data.
The reason why bias variance tradeoff and considerations of model size are a good approximation for smaller models is due to concentration of measure in the data which effectively kills any regularization in your modelling procedure. Once you enter settings where concentration of measure begins to bite in parameter space, everything changes. This isn't really that mysterious; any textbook on Gaussian processes (e.g. Rasmussen and Williams) will tell you this.
Nothing in your reply gets at the connection to out of sample data?
I don't understand what you mean. Test error is literally out of sample error. Marginal likelihood is designed to estimate out of sample error. The whole discussion is about out of sample; nothing has been about in-sample error. The in-sample error for my examples are all trivially zero, so only out of sample error is worth discussing.