Equilibria of Perceptrons for Simple Contingency Problems

The contingency between cues and outcomes is fundamentally important to theories of causal reasoning and to theories of associative learning. Researchers have computed the equilibria of Rescorla-Wagner models for a variety of contingency problems, and have used these equilibria to identify situations in which the Rescorla-Wagner model is consistent, or inconsistent, with normative models of contingency. Mathematical analyses that directly compare artificial neural networks to contingency theory have not been performed, because of the assumed equivalence between the Rescorla-Wagner learning rule and the delta rule training of artificial neural networks. However, recent results indicate that this equivalence is not as straightforward as typically assumed, suggesting a strong need for mathematical accounts of how networks deal with contingency problems. One such analysis is presented here, where it is proven that the structure of the equilibrium for a simple network trained on a basic contingency problem is quite different from the structure of the equilibrium for a Rescorla-Wagner model faced with the same problem. However, these structural differences lead to functionally equivalent behavior. The implications of this result for the relationships between associative learning, contingency theory, and connectionism are discussed.

I. Introduction

A fundamental characteristic of an adaptive agent is the ability to detect causal relations [1]. However, the real world poses constant challenges to this ability, because cues do not signal outcomes with complete certainty [2]. It has been argued that adaptive systems deal with worldly uncertainty, whether these systems are humans [3] or animals [4], [5], by becoming “intuitive statisticians.” The notion of “intuitive statistician” has been rigorously developed in a series of important papers to mean sensitivity to contingency, where contingency is defined in a normative model as a contrast between conditional probabilities [1], [6]–[11]. For instance, consider the simple situation that is detailed in the contingency table provided in Table I. The contingency between the cue and the outcome is formally defined as the difference in conditional probabilities , where [6]. More sophisticated models, such as the probabilistic contrast model (e.g., [8]) or the power PC theory [7], define more complex probabilistic contrasts that are possible when multiple cues occur and when what they signal depends upon the context in which they are considered. Table I Simple Contingency Situation in Which a Cue Can Occur or Not , and an Outcome Can Occur or Not as Well

	Indices & Order
Parameters	WFC (80%)	WFC (120%)	Fuzzy Sensitivity
	4.8296e-5	4.8296e-5	2
	3.3190e-6	3.3190e-6	5
	3.8715e-5	3.7321e-5	3
	5.2961e-6	5.2458e-6	4
	1.3389e-4	1.3380e-4	1