Bayesian Information Criterion - Non-physical model selection Following the work of Bai & Perron (1998) and others in detecting structural changes in time series, I am trying to select the breakpoints by using the Bayesian Information Criterion. 
Basically, the response $y$ is modelled as a sequence of $n$ segments (linear regression), $n$ being unknown as well as the breakpoint locations.
One compute the sum of squared recursive residuals of $y$ for all the segments [i, j]. 
Then for each assumption regarding the number of breakpoints (1, 2, 3...), the optimal locations of the breakpoints is computed (minimizing the sum of the sum of squared recursive residuals of each segment).
Finally, the optimal model is selected (by minimizing the BIC) among the potential candidates with 0, 1, 2, 3 ... breakpoints.

As I am working with remote sensed data, the time series are noisy and irregular.
It happens that some decompositions may not be realistic (meaning that it cannot model the actual phenology) and that the above algorithm chooses a bad model.
I am able to determine if a model is physically possible using the following metrics and their combinations:


*

*the gap between two segments (it could make sense having only positive gaps),

*the slopes of the segments,

*the length of the segments, etc.


I thought that I could add a phenological penalty term to the triangular "sum of squared recursive residuals" matrix in order to guide the search of the best model with $k$ breakpoints towards one that is phenelogically possible.
However this does not seem mathematically rigorous as the BIC needs the likelihood of my model.
What could be the best formulation for this problem? 
 A: What you described sounds more like constrained changepoint detection.  This is where you impose some constraints on the resulting segmentation profile.  Specifically there have been up-down constraints (where a positive jump in mean must be followed by a negative jump):
https://arxiv.org/abs/1703.03352
https://cran.r-project.org/web/packages/PeakSegOptimal/index.html
I'm also aware of others working on different constraints but these have not been published yet.
What you want to do is to restrict the fit rather than impose the rules at the end.  This is because there may be a slightly sub-optimal fit for one $k$ that satisfies the rules but is a better fit than an alternative $k'$ where the optimal does satisfy the rules.  Thus imposing the rules at the fitting stage results in all $K$ segmentations satisfying the rules and then you can use BIC on those as usual.  The crux of the attached paper (and works in progress) is that imposing these rules makes obtaining the best segmentation that satisfies the rules difficult.
In summary, it is fine to constrain your segmentations to follow specific rules but this should be imposed in the fit and not in the penalization step.
