The complexity of the cognitive system makes powerful metaphors such as the proba- bilistic mind and the Bayesian brain appealing on the one hand, but limited on the other. The trick is to not only harness their productivity, but also recognize their limits. One problem confronting the notion of the probabilistic mind and the accompanying `quiet probabilistic revolution' is the apparent intractability of rational probabilistic calculation (Chater et al., 2006b, p. 293). Rational probabilistic models, however, are not typically interpreted as algorithmic or mechanistic theories but functional level theories used to establish connections between observed behavior, a rational principle of inductive inference, and the structure of the environment. These correspondences tell us when the cognitive system is performing well and, to varying degrees, are used to suggest that human behavior is consistent with rational principles of inductive inference. From an algorithmic standpoint, how should these empirical findings be interpreted? The distinction between functional and algorithmic level theories has its roots in what is now termed the rational analysis of cognition, an adaptationist program which aims to understand the structure and function of the cognition system as an adaptive response to the challenges posed by the environment (Marr, 1982; Shepard, 1987; Anderson, 1990; Oaksford & Chater, 1998). While working on a purely functional level, the tractability problem is in one sense irrelevant given that no commitment is made to a mechanistic level interpretation, but in another sense, unsatisfactory. Indeed, a principle objective of the rational analysis of cognition is to narrow down candidate algorithmic level theories by establishing empirically determined perform- ance criteria. If the grand prize in cognitive science is uncovering both why minds do what they do and how they do it, then the productivity and scope of the metaphor would ideally extend to the process level. Can the notion of the probabilistic mind be seamlessly extended to the algorithmic level, or there exist unmovable barriers to reconciling rational probabilistic models with plausible mechanisms of mind? We will examine these questions by considering 09-Charter&Oaksford-Chap09 11/3/07 12:22 PM Page 190 190 BAYESIAN BRAINS AND COGNITIVE MECHANISMS: HARMONY OR DISSONANCE? the metaphor of the probabilistic mind from an alternative adaptationist perspective, and one that views much of human inductive inference as relying on an adaptive tool- box of simple heuristics (Gigerenzer et al., 1999; Gigerenzer & Selten, 2001). Unlike the notion of the probabilistic mind, the metaphor of the adaptive toolbox is rooted to an algorithmic level hypothesis, which proposes that adaptive behavior, and inductive inference in particular, is in large part the result of an interaction between processing simplicity and ecological context. This view leads to the notion of ecological rationality. Here, the cognitive system is viewed as adapted to the relevant aspects of its environ- ment to the extent that it achieves good enough solutions using the limited resources it has available, rather than attempting to find optimal ones. On this view, organisms do not optimize but satisfy (Simon, 1996), which makes the notion of adaptive suc- cess for the organism - its ecological rationality - inseparably tied to an algorithmic level analysis. What barriers, if any, stand between a synthesis of the study of ecological rationality and functional level probabilistic models? First, we consider the role of rationality and optimality in framing the study of cognition, and examine how these concepts repre- sent key points of divergence between the study of ecological rationality and rational analysis. Second, we examine the consequences of the intractability of optimal proba- bilistic calculation, and propose that the statistical problem known as the bias/variance dilemma arises as a consequence, and represents a significant and often overlooked dimension of the functional problem facing the cognitive system (Geman et al., 1992). The bias/variance dilemma brings into focus a connection between estimation error, ecological context, and the properties of learning algorithms. Therefore, in addition, it has the potential to bridge functional level models, simple heuristics, and the notion of ecological rationality. Ultimately, the adaptationist perspective should encompass both functional and algorithmic level analyses. Our guiding concern will be the understand- ing how these two levels of analysis can be aligned.