A structural model of intuitive probability

Though the ability of human beings to deal with probabilities has been put into question, the assessment of rarity is a crucial competence underlying much of human decision-making and is pervasive in spontaneous narrative behaviour. This paper proposes a new model of rarity and randomness assessment, designed to be cognitively plausible. Intuitive randomness is defined as a function of structural complexity. It is thus possible to assign probability to events without being obliged to consider the set of alternatives. The model is tested on Lottery sequences and compared with subjects’ preferences.

💡 Research Summary

The paper “A Structural Model of Intuitive Probability” tackles the long‑standing puzzle of how people assess rarity and probability without explicitly enumerating all possible alternatives. The authors begin by reviewing classic cognitive biases—gambler’s fallacy, conjunction fallacy, base‑rate neglect—and argue that these illustrate a mismatch between formal probability theory (which assumes knowledge of the full outcome space) and everyday human judgment, which seems to rely on the perceived structure of the event itself.

To bridge this gap, the authors first introduce Solomonoff’s algorithmic probability, where the probability of an object x is proportional to the sum over all programs that output x, leading to the approximation P(x) ≈ 2^{‑K(x)} with K(x) the Kolmogorov complexity. While mathematically elegant, this formulation predicts that highly regular (low‑complexity) objects are more probable, contrary to empirical findings that people view regular patterns as “unlikely” or “suspicious”.

The key conceptual move is the definition of unexpectedness U(x) = C_exp(x) – C_obs(x). C_exp(x) is the complexity that a rational observer would expect for a typical random generation process (e.g., copy an empty slot then instantiate each digit independently). C_obs(x) is the actual, observed complexity of the specific object as judged by a cognitive model. When the observed description is much simpler than the expected one, U becomes large and positive, signalling a feeling of surprise. The authors then map this feeling onto a subjective probability p(x) = 2^{‑U(x)}. In this framework, a five‑digit number like 33333, which can be generated by a very short description (copy then instantiate a single digit), yields a large U and thus a very low p, matching the intuition that “all the same digits” feels improbable.

To compute C_obs, the authors adopt Michael Leyton’s Generative Theory of Shape, which models perception as a hierarchical construction of objects via “fiber groups” (basic elements) and “transfer groups” (operations that move or transform those elements). The total structural complexity is expressed as C_R = C_F + C_T, where C_F is the cost of the fiber group and C_T the cost of the transfer group. Numerical values are assigned to elementary operations (copy, duplicate, increment) and to the representation of numbers (C_n = log₂(n+1)). This yields a human‑centric approximation of Kolmogorov complexity that can be calculated for concrete sequences such as lottery number sets.

The empirical test consists of a small pilot study with 26 participants approached in a café. Each participant received a French lottery ticket and was asked to pick two combinations out of a set of 14 pre‑generated options. Ten of these options were deliberately low‑complexity (e.g.,