How do I find the probability of a change in a sequence using change detection pairs in that sequence? I have a sequence of satellite images of land use and I would like to be able to pinpoint the moment time a change occurred. To do this, I have a network that compares any two images and tells me the probability that there was a significant change. It works well but isn't perfect, so now I need a statistical method to combine all the results to tell me when the change occurred. For example, imagine I have imagery from five different years and got the following result from my comparison model, where 0 is "no change" and 1 is "change detected".




Year
2
3
4
5




1
0
0
1
1


2

0
1
0


3


1
1


4



0




In this case, there was a change between years 3 and 4. But due to an imperfect model, no change was detected in the comparison between years 2 and 5. How do I combine all of these data into a statistical model that would tell me what the probability of a change was between any two consecutive years? Note that it's also possible for there to be no change or multiple changes.

For each of the possible changes (and set of changes or zero change), write down an upper triangular matrix $M$ as the one you posted above ($M_0$), but how it should look in case no error occurred. For no change, it would be all zero. For a change between the ith and (i+1)th year the matrix M would contain a right upper block of ones:
$$
\begin{align}
M_{st} = 1 \quad &\mbox{if} \quad s\le i, t > i\\
M_{st} = 0 \quad &\mbox{otherwise.}
\end{align}
$$
And, presuming that two changes cannot annihilate each other, for several changes you would have a matrix with several right upper blocks of ones that are overlapping.
Then, presuming that the error probability is constant, you just check which of those correct matrices matches your measured matrix best, i.e. has the smallest number of cells with differing entries.
