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First, the matrix has to be invertible in order to derive the transition matrix which is not the case in our application since not all LNLs in the healthy state and the dying state can be distinguished. Moreover, the typical assumption is that a given state can only have a fixed number of transitions to other states, which is not true for our application either.

These two reasons make the sum-product algorithm inefficient, since the computational time does not grow linearly but exponentially with the number of nodes. Hence, we need to consider a different approach: The observed variables ({ arvec{B}}) are not independent but, according to the true labellings ({ arvec{Y}}), essentially pairwise dependent. Hence, we are dealing with a maximum likelihood estimator for the transition matrix ({ arvec{A}}). We will describe the data model in a moment, but first we need to introduce some notation. We denote the number of distinct labels in { arvec{Y}} by {nav}, i.e. {nav} is the true number of labels that we observe and that can be determined from the observations. Instead of subdividing all LNLs into h or d states, we only make use of them in their binary form. Hence, the number of LNLs that are subdivided into two states is {nL} = 5 (11), (12) and the number of possible states in the transition matrix is {nW} = 6 (13). Note that if we would consider only one state / LNL then the resulting matrix would contain an observation for the corresponding label but not a label that occurs more than once. We assume that the observation matrix is given by the sensitivity and specificity described in Eq. (11). d2c66b5586

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