  When humans infer underlying probabilities from stochastic observations, they
exhibit biases and variability that cannot be explained on the basis of sound,
Bayesian manipulations of probability. This is especially salient when beliefs
are updated as a function of sequential observations. We introduce a
theoretical framework in which biases and variability emerge from a trade-off
between Bayesian inference and a cognitive cost of carrying out probabilistic
computations. We consider two forms of the cost: a precision cost and an
unpredictability cost; these penalize beliefs that are less entropic and less
deterministic, respectively. We apply our framework to the case of a Bernoulli
variable: the bias of a coin is inferred from a sequence of coin flips.
Theoretical predictions are qualitatively different depending on the form of
the cost. A precision cost induces overestimation of small probabilities on
average and a limited memory of past observations, and, consequently, a
fluctuating bias. An unpredictability cost induces underestimation of small
probabilities and a fixed bias that remains appreciable even for nearly
unbiased observations. The case of a fair (equiprobable) coin, however, is
singular, with non-trivial and slow fluctuations of the inferred bias. The
proposed framework of costly Bayesian inference illustrates the richness of a
'resource-rational' (or 'bounded-rational') picture of seemingly irrational
human cognition.
