A Normative Theory of Decision Making from Multiple Stimuli: The Contextual Diffusion Decision Model

Abstract

The dynamics of simple two-alternative forced-choice (2AFC) decisions are well-modeled by a class of random walk models (e.g. Laming, 1968; Ratcliff, 1978; Usher & McClelland, 2001; Bogacz et al., 2006). However, in real-life, even simple decisions involve dynamically changing influence of additional information. In this work, we describe a computational theory of decision making from multiple sources of information, grounded in Bayesian inference and consistent with a simple neural network. This Contextual Diffusion Decision Model (CDDM) is a formal generalization of the Diffusion Decision Model (DDM), a popular existing model of fixed-context decision making (Ratcliff, 1978), and shares with it both a mechanistic and a probabilistic motivation. Just as the DDM is a model for a variety of simple two-alternative forced-choice (2AFC) decision making tasks, we demonstrate that the CDDM supports a variety of simple context-dependent tasks of longstanding interest in psychology, including the Flanker (Eriksen & Eriksen, 1974), AX-CPT (Servan-Schreiber et al., 1996), Stop-Signal (Logan & Cowan, 1984), Cueing (Posner, 1980), and Prospective Memory paradigms (Einstein & McDaniel, 2005). Further, we use the CDDM to perform a number of normative rational analyses exploring optimal response and memory allocation policies. Finally, we show how the use of a consistent model across tasks allows us to recover consistent qualitative data patterns in multiple tasks, using the same model parameters.

Figures (16)

• Figure 1: Contextual decision making: A binary context stimulus appears at time $t^{\textup{on}}_c$ and disappears at time $t^{\textup{off}}_c$. A binary target stimulus appears at time $t^{\textup{on}}_c \ge t^{\textup{on}}_c$ and disappears at time $t^{\textup{off}}_g \ge t^{\textup{off}}_c$. The decision maker selects a binary response based on the inferred context and target stimuli. Note that while the figure illustrates the stimuli being fully temporally disjoint, our proposal also applies to the case where they overlap partially or fully in time.
• Figure 2: Temporally-separated context and target stimuli illustrated using the example of the AX-CPT task. First, a context stimulus is presented, and the decision maker infers the context. Subsequently, no stimuli are present for a period, and the memory of the context decays. Finally, the target stimulus appears, and the decision maker recalls the inferred context from the memory that is used, together with information from the target, to make a decision.
• Figure 3: Neural network reduction.Left: the full network, consisting of four units each taking one input and all-to-all inhibition. Right: if inhibition is much higher between than across pairs, the system can be asymptotically approximated as a two-unit system.
• Figure 4: Decision boundaries for the Flanker task in $\vec{z}$ space. When the priors on the context and the target are correlated, the evidence from the context $z_c$ determines the required evidence for the target $z_g$ (left panel). When the prior probabilities are uncorrelated, then context plays no role in the decision and the model reduces to the standard DDM with fixed boundaries (right panel).
• Figure 5: Average trajectories in the external context model with congruence bias on a congruent (left) and incongruent (right) trial. This model is equivalent to the congruence-bias model of the Flanker task Yu2009, in which statistical dependence between flankers and target produces dynamic biases in the decision variable on both congruent and incongruent trials, though it can just as well be applied to similar paradigms such as the Stroop task. In terms of log-likelihoods, the external context model decomposes into a linear log-likelihood random walk on the target and a saturating bias term. On a congruent trial, the influence of the flankers accelerates the progress of the decision variable. On an incongruent trial, the decision variable moves away from the true response early on due to the initial influence of the context, but eventually reverses under the influence of the target.