"Burn-in, bias, and the rationality of anchoring"

Bayesian inference provides a unifying framework for learning, reasoning, and decision making. Unfortunately, exact Bayesian inference is intractable in all but the simplest models. Therefore minds and machines have to approximate Bayesian inference. Approximate inference algorithms can achieve a wide range of time-accuracy tradeoffs, but what is the optimal tradeoff? We investigate timeaccuracy tradeoffs using the Metropolis-Hastings algorithm as a metaphor for the mind’s inference algorithm(s). We characterize the optimal time-accuracy tradeoff mathematically in terms of the number of iterations and the resulting bias as functions of time cost, error cost, and the difficulty of the inference problem. We find that reasonably accurate decisions are possible long before the Markov chain has converged to the posterior distribution, i.e. during the period known as “burn-in”. Therefore the strategy that is optimal subject to the mind’s bounded processing speed and opportunity costs may perform so few iterations that the resulting samples are biased towards the initial value. The resulting cognitive process model provides a rational basis for the anchoring-and-adjustment heuristic. The model’s quantitative predictions match published data on anchoring in numerical estimation tasks. In conclusion, resource-rationality–the optimal use of finite computational resources–naturally leads to a biased mind.