Human Inferences about Sequences: A Minimal Transition Probability Model
Fig 1
Three different hypothesis spaces
(A) Sequences can be characterized by a hierarchy of statistics. We consider here binary sequences with only two items: X and Y. The simplest statistic considers stimuli in isolation, based on the frequency of items, p(X) and p(Y). A second level considers pairs of items irrespective of their order, distinguishing pairs of identical versus different items (XX and YY vs. XY and YX). The relevant statistic is the frequency of alternations, or conversely, the frequency of repetitions: p(alt.) = 1 – p(rep.). A third level considers ordered pairs, distinguishing X1Y2 from Y1X2. The relevant statistics are the two transition probabilities between consecutive items: p(Y2|X1) and p(X2|Y1). For brevity, we generally omit the subscripts. For binary sequences, the space of transition probabilities is 2-dimensional. In this space, the diagonals are special cases where transition probabilities coincide with the frequency of items and frequency of alternations. Out of the diagonals, there is no linear mapping between transition probabilities and the frequency of items (shown in red/blue and iso-contours) or the frequency of alternations (shown with transparency and iso-contours). (B) Example sequences generated from distinct statistics. From top to bottom: The sequences (1) and (2) differ in their frequency of X but not in their frequency of alternations. To generate such sequences, one can select the next stimulus by flipping a biased coin. The sequences (3) and (4) differ in their frequency of alternations, but not in their frequency of X. To generate such a sequence, one can start the sequence arbitrarily with X or Y, and then decide whether to repeat the same item or not by flipping a biased coin. The sequence (5) is biased both in its frequency of alternations and its frequency of items. It cannot be generated with a single biased coin, but instead two biased coins are required, one to decide which item should follow an X and the other to decide which item should follow a Y. The sequence (6) is a purely random sequence, with no bias in either transition probabilities, and hence, no bias in item nor alternation frequencies. It can be generated by flipping a fair coin.