Burn-in, bias, and the rationality of anchoring Falk Lieder, Thomas L. Griffiths, Noah D. Goodman contact: [email protected]

Introduction • To act successfully people have to perform probabilistic inference in real-time with limited computational resources.

The Optimal Time-Accuracy Tradeoff

A rational explanation of the anchoring bias

1. Mathematical Analysis:

1. Anchoring bias:

?

i

• Deciding from a single sample is rational in many problems [1], but even a single perfect sample is costly, e.g. requiring thousands of iterations of a Markov-chain-Monte-Carlo algorithm.

?

b

= =

arg min E [kai − xk + c · i|y]

People’s estimates of unknown quantities (e.g. the duration of Mars’s orbit) are biased towards salient known quantities (e.g. 365 days).

where ai ∼ Qi

2. Model:

i

Bias [Qi? ; P ]

• Therefore the mind has to tradeoff a sample’s accuracy for time. • What is the optimal time-accuracy tradeoff? How does it differ from unlimited Bayesian rationality?

i?

=

• What are the implications for human cognition?

b?

≤

Time-Accuracy Tradeoffs in Inference

log(c) − M · log(M · log(1/r)) log(r) c log (1/r)

• Numerical estimation is probabilistic inference. • People’s estimates are samples from the resource-rational approximate posterior Qi? ,y . • One free parameter: Time cost c determines i? . • Initial value of Metropolis-Hastings algorithm: s0 = anchor • Proposed adjustments St+1 ∼ Ppropose (·; st ) = N (st , σ 2 = 100).

2. Simulations:

• Posteriors were estimated from subjective confidence intervals.

1. Approach

3. Results:

• We analyse the time-accuracy tradeoff in probabilistic inference by MCMC sampling.

1. Model fit to people’s mean adjustment scores in the six numerical estimation tasks of [3]:

• We use the Metropolis-Hastings algorithm as a metaphor for the mind’s inference algorithm(s). • How many iterations should be performed, if each iteration takes a fixed amount of time?

2. Results 1. Mathematical Analysis: • The distribution Qi the Metropolis-Hastings a
Bias[Qi ; P ] ≤ M · ri , EQ (A) [EE(a)] − EP (A) [EE(a)] ≤ M · ri , i

where P is the posterior, Bias[Q; P ] is EQ [X] − EP [X] and EE(a) is EP (X) [costerror (x, a)]. 2. Simulations: Inference was simulated for 1-dimensional Gaussian posteriors. In each case the bias decayed geometrically:

The optimal number of iterations i? decreases with the relative cost of time c.

The higher the cost of time c, the larger the bias a bounded rational agent should tolerate.

3. Interpretation: 1. Satisfactory decisions are possible long before the Markov chain has converged, i.e. during the “burn-in” period. 2. Therefore rational bounded agents will often perform so few iterations that the resulting samples are substantially biased towards the initial value.

References

The model accurately predicted whether adjustments were sufficent (≥ 0.5) or insufficient (rpredicted,measured = 0.95). 2. Inferred subjective time costs c capture the effect of cognitive load in Experiment 2C of [3]: High cognitive load: Normal cognitive load:

cˆ = 0.31 cˆ = 0.18

Conclusions

[1] E. Vul, N. D. Goodman, T. L. Griffiths, and J. B. Tenenbaum. One and done? Optimal decisions from very few samples. Proc. of the 31st Annual Conf. of the Cognitive Sci. Soc., 2009.

1. Our formulation offers a new normative framework for modelling cognitive processes: resource-rationality.

[2] K.L. Mengersen and R.L. Tweedie. Rates of convergence of the Hastings and Metropolis algorithms. The Annals of Statistics, 24(1), 1996.

2. Heuristics such as anchoring-and-adjustment [4] can be understood as resource-rational approximations to Bayesian inference.

[3] N. Epley and T. Gilovich. The anchoring-and-adjustment heuristic. Psychological Science, 17(4):311–318, 2006.

3. Resource-rational inference leads naturally to a biased mind.

[4] A. Tversky and D. Kahneman. Judgment under uncertainty: Heuristics and biases. Science, 185(4157):1124–1131, 1974.

4. Our model’s quantitative prediction of how the anchoring bias increases with time pressure should be tested experimentally.