Trend-concordance grey incidence with likelinesses closed-form small-sample inference without elicited prior densities
Grey incidence analysis ranks the factors of a system from short sequences without assuming a probability model, but the grade it returns has no sampling theory: nothing indicates whether an ordering reflects the system or the few data at hand. This paper reduces trend agreement between two sequences to the signs of their centred increments and analyses the resulting counts with Lovegrove's likelinesses, averages of probabilities over an underlying set of distributions that replace the elicitation of a prior density by a statement of what is possible. The result is a concordance incidence grade with closed-form inference: estimation, the likeliness that one factor outranks another, prediction of future concordance and a baseline for searches over many candidates, with bounded underlying sets encoding partial knowledge. Comparisons whose sign falls below the resolution of the data are treated as indeterminate; discarding them is exact when they carry no information, and otherwise the grade becomes a grey number whose width measures what the data cannot decide. The grade is symmetric in any system and resolves the conflict between the totality and pair-symmetry axioms by moving system dependence into inference. Simulations show that plug-in prediction underestimates variance by a factor of up to 1 + m/n regardless of sample size, and that a fair-coin baseline yields up to 63% spurious patterns in searches, where a correctly bounded set keeps them below 5%. Worked textbook examples and a regularisation benchmark with known truth illustrate the method.