Probability, algorithmic complexity, and subjective randomness

We present a statistical account of human randomness judgments that uses the idea of algorithmic complexity. We show that an existing measure of the randomness of a sequence corresponds to the assumption that non-random sequences are generated by a particular probabilistic finite state automaton, and use this as the basis for an account that evaluates randomness in terms of the length of programs for machines at different levels of the Chomsky hierarchy. This approach results in a model that predicts human judgments better than the responses of other participants in the same experiment. The development of information theory prompted cognitive scientists to formally examine how humans encode experience, with a variety of schemes being used to predict subjective complexity (Leeuwenberg, 1969), memorization difficulty (Restle, 1970), and sequence completion (Simon & Kotovsky, 1963). This proliferation of similar, seemingly arbitrary theories was curtailed by Simon’s (1972) observation that the inevitable high correlation between measures of information content renders them essentially equivalent. The development of algorithmic information theory (see Li & Vitanyi, 1997, for a detailed account) has revived some of these ideas, with code lengths playing a central role in recent accounts of human concept learning (Feldman, 2000), subjective randomness (Falk & Konold, 1997), and the role of simplicity in cognition (Chater, 1999). Algorithmic information theory avoids the arbitrariness of earlier approaches by using a single universal code: the complexity of an object (called the Kolmogorov complexity after Kolmogorov, 1965) is the length of the shortest computer program that can reproduce it. Chater and Vitanyi (2003) argue that a preference for simplicity can be seen throughout cognition, from perception to language learning. Their argument is based upon the important constraints that simplicity provides for solving problems of induction, which are central to cognition. Kolmogorov complexity gives a formal means of addressing “asymptotic” questions about induction, such as why anything is learnable at all, but the constraints it imposes are too weak to support the rapid inferences that characterize human cognition. In order to explain how human beings learn so much from so little, we need to consider accounts that can express the strong prior knowledge that contributes to our inferences. The structures that people find simple form a strict (and flexible) subset of those easily expressed in a computer program. For example, the sequence of heads and tails TTHTTTHHTH appears quite complex to us, even though, as the parity of the first 10 digits of π, it is easily generated by a computer. Identifying the kinds of regularities that contribute to our sense of simplicity will be an important part of any cognitive theory, and is in fact necessary since Kolmogorov complexity is not computable (Kolmogorov, 1965). There is a crucial middle ground between Kolmogorov complexity and the arbitrary encoding schemes to which Simon (1972) objected. We will explore this middle ground using an approach that combines rational statistical inference with algorithmic information theory. This approach gives an intuitive transparency to measures of complexity by expressing them in terms of probabilities, and uses computability to establish meaningful differences between them. We will test this approach on judgments of the randomness of binary sequences, since randomness is one of the key applications of Kolmogorov complexity: Kolmogorov (1965) suggested that random sequences are irreducibly complex, a notion that has inspired several psychological theories (eg. Falk & Konold, 1997). We will analyze subjective randomness as an inference about the source of a sequence X, comparing its probability of being generated by a random source, P (X|random), with its probability of generation by a more regular process, P (X|regular). Since probabilities map directly to code lengths, P (X|regular) uniquely identifies a measure of complexity. This formulation allows us to identify the properties of an existing complexity measure (Falk & Konold, 1997), and extend it to capture more of the statistical structure detected by people. While Kolmogorov complexity is expressed in terms of programs for a universal Turing machine, many of the regularities people detect are computable by simpler devices. We will use Chomsky’s (1956) hierarchy of formal languages to organize our analysis, testing a set of nested models that can be interpreted in terms of the length of programs for automata at different levels of the hierarchy. Complexity and randomness The idea of using a code based upon the length of computer programs was independently proposed by Solomonoff (1964), Kolmogorov (1965), and Chaitin (1969), although it has come to be associated with Kolmogorov. A sequence X has Kolmogorov complexity K(X) equal to the length of the shortest program p for a (prefix) universal Turing machine U that produces X and then halts, K(X) = min p:U(p)=X l(p), (1) where l(p) is the length of p in bits. Kolmogorov complexity can be used to define algorithmic probability, with the probability of X being R(X) = 2−K(X) = max p:U(p)=X 2−l(p). (2) There is no requirement that R(X) sum to one over all sequences; many probability distributions that correspond to codes are unnormalized, assigning the missing probability to an undefined sequence. Kolmogorov complexity can be used to mathematically define the randomness of sequences, identifying a sequence X as random if l(X)−K(X) is small (Kolmogorov, 1965). While not necessarily following the form of this definition, psychologists have preserved its spirit in proposing that the perceived randomness of a sequence increases with its complexity. Falk and Konold (1997) consider a particular measure of complexity they call the “difficulty predictor” (DP ), calculated by counting the number of runs (sub-sequences containing only heads or tails), and adding twice the number of alternating subsequences. For example, the sequence TTTHHHTHTH is a run of tails, a run of heads, and an alternating sub-sequence, DP = 4. If there are several partitions into runs and alternations, DP is calculated on the partition that results in the lowest score. Falk and Konold (1997) showed that DP correlates remarkably well with subjective randomness judgments. Figure 1 shows the results of Falk and Konold (1997, Experiment 1), in which 97 participants each rated the apparent randomness of ten binary sequences of length 21, with each sequence containing between 2 and 20 alternations (transitions from heads to tails or vice versa). The mean ratings show the classic preference for overalternating sequences: the sequences perceived as most random are those with 14 alternations, while a truly random process would be most likely to produce sequences We modify DP slightly from the definition of Falk and Konold (1997), who seem to require alternating subsequences to be of even length. The equivalence results shown below also hold for their original version, but it makes the counter-intuitive interpretation of HTHTH as a run of a single head, followed by an alternating subsequence, DP = 3. Under our formulation it would be parsed as an alternating sequence, DP = 2. 2 4 6 8 10 12 14 16 18 20 Number of alternations S ub je ct iv e ra nd om ne ss Falk & Konold (1997) DP model Finite state model Figure 1: Mean randomness ratings from Falk and Konold (1987, Experiment 1), shown with the predictions of DP and the finite state model. with 10 alternations. The mean DP has a similar profile, achieving a maximum at 12 alternations and giving a correlation of r = 0.93. Subjective randomness as a statistical inference Psychologists have claimed that the way we think about chance is inconsistent with probability theory (eg. Kahneman & Tversky, 1972). For example, people are willing to say that X1=HHTHT is more random than X2=HHHHH, while they are equally likely to arise by chance: P (X1|random) = P (X2|random) = ( 12 ) . However, many of the apparently irrational aspects of human judgments can be understood by considering the possibility that people are assessing a different kind of probability – instead of P (X|random), we evaluate P (random|X) (Griffiths & Tenenbaum, 2001). The statistical basis of subjective randomness becomes clear if we view randomness judgments in terms of a signal detection task (cf. Lopes, 1982; Lopes & Oden, 1987). On seeing a stimulus X, we consider two hypotheses: X was produced by a random process, or X was produced by a regular process. Finding regularities is an important part of identifying predictable processes, a fundamental component of induction (Lopes, 1982). The decision about the source of X can be formalized as a Bayesian inference, P (random|X) P (regular|X) = P (X|random) P (X|regular) P (random)