Bayesian models of the speed-accuracy tradeoff offer a normative perspective on decision making: rather than modeling the mechanism by which evidence is accumulated (as the DDM does), they specify the optimal statistical procedure for making decisions. In this framework, the SAT arises because optimal decision rules require a specific amount of evidence before committing to a response, and gathering more evidence takes time.
The Bayesian Decision Framework
In the Bayesian formulation, the observer begins with prior probabilities P(H₁) and P(H₂) for the two hypotheses (response alternatives). Each piece of evidence (sample) updates the posterior via Bayes' rule:
LPO_n = log(P(H₁)/P(H₀)) + Σᵢ₌₁ⁿ log(L(xᵢ|H₁)/L(xᵢ|H₀))
Respond H₁ if LPO_n ≥ θ_upper
Respond H₀ if LPO_n ≤ θ_lower
Continue sampling otherwise
The thresholds θ_upper and θ_lower determine the speed-accuracy tradeoff: high thresholds require strong evidence (slow, accurate) while low thresholds accept weak evidence (fast, error-prone). This is mathematically equivalent to the sequential probability ratio test (SPRT), which Wald (1947) proved is optimal in the sense of requiring the fewest samples on average to achieve a desired error rate.
Bayesian Optimality and the DDM
A fundamental theoretical result connects Bayesian inference to the drift diffusion model. Bogacz et al. (2006) showed that when evidence arrives as continuous Gaussian samples, the SPRT is mathematically equivalent to the DDM — the log posterior odds ratio performs a random walk identical to the DDM's evidence accumulation process. The drift rate corresponds to the quality of evidence (signal-to-noise ratio), and the boundaries correspond to the posterior probability thresholds. This means the DDM implements the optimal decision procedure for two-choice tasks with Gaussian evidence.
Given the formal equivalence between Bayesian inference and diffusion, the question of optimal SAT setting becomes: what boundaries maximize reward rate? The answer depends on drift rate (task difficulty), error costs, and the inter-trial interval. For difficult tasks with short inter-trial intervals, narrow boundaries (fast, error-prone) maximize reward rate; for easy tasks with long inter-trial intervals, wide boundaries (slow, accurate) are optimal. Starns and Ratcliff (2012) found that participants approximate but do not perfectly achieve optimal boundaries.
Hierarchical Bayesian Models of SAT
Modern applications use hierarchical Bayesian methods to estimate SAT function parameters. Vandekerckhove, Tuerlinckx, and Lee (2011) developed hierarchical Bayesian implementations of the DDM that simultaneously estimate individual- and group-level parameters, properly accounting for individual differences in SAT setting. These models use MCMC sampling to estimate full posterior distributions over parameters, providing uncertainty estimates and enabling principled model comparison via Bayes factors.
The Bayesian approach to SAT has also been extended to multi-alternative decisions (using the multihypothesis SPRT), to changing environments (where the optimal policy involves time-varying boundaries), and to situations where the cost of evidence collection must be weighed against the benefit of more accurate decisions. These extensions connect the psychology of SAT to broader questions in statistical decision theory and computational neuroscience.