Marginal Log-Likelihood Increments under Dirichlet-Smoothed Markov Estimation

For a Dirichlet-smoothed transition model, the effect of adding one workflow trace to the training archive is an exact change in reference-weighted log likelihood. We derive that change and show that it is a weighted reduction of Kullback--Leibler divergence between the reference conditionals and the model. From this form we obtain an upper bound on the gain available to any acquisition, which expresses a millinat difference as a share of what is attainable, an exact covariance identity for the effect of the reference weighting, and a sign criterion for the interaction between two candidates, from which the batch objective is neither submodular nor supermodular. A case study on the BPI Challenge 2012 loan-application log measures all three and finds a positive selection result in one of the four combinations of reference weighting and budget unit. There, of two regressors fitted to identical descriptors and identical labels, the one that predicts individual increments far more accurately, median $R^2$ 0.87 against 0.62, realizes the smaller share of the attainable gain, 61 against 69 per cent, so ranking accuracy for individual traces is neither necessary nor sufficient for batch quality.