Yep, it's sad, and more sad is that, at least to my knowledge, I haven't seen this to be finally resolved, even in Bergomi series, aren't it? What confuses me exactly in the posed context of time inhomogeneous case is: OK, we need to find a good stoch. process for the dynamics of [spot(t), vol(t)] or [short rate(t)]. Fitting a vol surface (or an yield curve) gives the stoch. process marginal distributions at a discrete set of maturities. Fitting the increments gives the total law. The first fitting is calibration. Vanilas (bonds) are enough here. The second fitting is a calibration, if done to exotics, and estimation, if done to time series. In the first fitting, we have a nonlinear function of the stoch. process parameters made close to the market data. In the second one a second function of the stoch. process parameters is made close to the market data (or its history in estimation). It can happen that some parameters participate only in the first fitting others in the second (or have marginal weight in the first or second, e.g. Bergomi demonstrated the vol of vol does not in the first and does in the second, etc.). Now imagine that in this setting instead of clean parameters we put in the process time valued functions (parametrized or not). What happens is that we artificially make a brutal attempt to separate cleanly the parameters' control over marginals and increments. So the time valued function (or its parameters) will explain possibly only the marginals and any other parameters the increments. In such a context some people claim, e.g. Carmona and Nadtochiy, that we have a full overfitting and there are no other parameters left for a "free control" of increments. In general, I am fine with that. Small caveat: we have still some freedom both for adding other parameters or even in the ones in the time valued function, if parametrized. So they propose to have a forward setting by a good codebook process that starts at the fitted surface (yield curve) and then evolves dynamically. However, in my opinion that is an overfitting in the other direction, in the direction of putting all the parameters to govern the increments. Right? The truth should be in the middle but which is it?