on 25-Apr-2017 (Tue)

What is new in the eleventh global edition? Of course, a large part of the changes in any edition consist of adding some updated data here and a new example there. However, we have rewritten and refreshed several basic chapters. Content remains much the same, but we think that the revised chapters are simpler and flow better.

the only possible truth conditions for a Ramsey conditional A > B make it equivalent to a type of dynamic-doxastic conditional, expressing a hypothetical disposition to revise the agent’s beliefs.

(Full Introspection of Revision) s ∼ t ⇒ bel s∗A = bel t∗A .

First, the recent rise in the popularity of rational models of cognition and Bayesian analysis (e.g., Anderson, 1990; Chater & Oaksford, 1999; Tenenbaum, Kemp, Griffiths, & Goodman, 2011) has resulted in theories of a wide range of different aspects of human cognition that are framed at the computational level.

A second reason to consider levels of analysis between the computational and algorith- mic is to clarify and organize research that has begun to explore this space. A recognition that human cognition necessarily involves making approximations or otherwise limiting rational solutions has been a common theme in psychological theory since Simon (1955) introduced the notion of “bounded rationality.”

We propose that the consideration of these constraints should be done somewhat independently of the definition of computational-level models, but that the principle of rationality provides the key to exploring the new levels of analysis that result.

should we assume them? We propose that the consideration of these constraints should be done somewhat independently of the definition of computational-level models, but that the principle of rationality provides the key to exploring the new levels of analysis that result. Distinguishing between computational problems and the resources available for solving them provides a way to build a bridge between the computational and algorithmic levels.

there is presumably only one true cognitive architecture

abstract computational architectures and define them by a set of basic operations and their costs

“rational process models” that push the notion of rationality toward the algorithmic level by postulating cognitive mechanisms that resemble the approximation algorithms that statisticians and computer scientists use to solve the problem identified by the computational level theory.

“resource-rational analysis,” which derives the strategy that makes optimal use of finite computational resources.

The focus of this level is not on computation per se, in the formal sense defined by Turing (1937) or the informal sense of carrying out calculations, but rather on the structure of the abstract problem being solved and the nature of its ideal solution.

The problem and the solution can be defined purely mathematically, with no requirement of conforming to any notion of computation.

The central question is what function an aspect of cognition serves, leading recent proponents of computational-level analysis to call this approach a “function-first” strategy (Griffiths et al., 2010).

“the cognitive system operates at all times to optimize the adaptation of the behavior of the organism”

We should expect ideal solutions

to the problems that human beings face to provide insight into human cognition only to the extent that human minds solve those problems well.

1. Precisely specify what are the goals of the cognitive system. 2. Develop a formal model of the environment to which the system is adapted. 3. Make the minimal assumptions about computational limitations. 4. Derive the optimal behavioral function, given items 1 through 3. 5. Examine the empirical literature to see if the predictions of the behavioral function are confirmed. 6. If the predictions are off, iterate.

What role should constraints play in computational-level theories?

resource-rational analysis can be used to derive the algorithm that makes opti- mal use of computational resources assumed by each of these models.

These scientists have developed a variety of strategies for approximating the resulting computations, and those strategies are a source of hypotheses about the cognitive processes by which the mind approximates the optimal solutions iden- tified by computational-level theories.

greedy maximization algorithm

With a small sample, they deviate from this ideal in systematic ways, producing biases (such as order effects) that are easy to compare against human performance. Such comparisons yield clues about the computational constraints that might be relevant to explaining human behavior.

Considering different algorithms for approximating the optimal solution can inspire abstract computational architectures that are useful for defining models that lie between the computational and algorithmic levels.

Digging deeper, we can con- sider specific schemes for generating those samples—Markov chain Monte Carlo, particle filters, or importance sampling—as providing a set of abstract computational architectures that each have their own constraints in terms of time, memory, and the conditions under which they succeed and fail.

For example, rather than merely assuming a Monte Carlo approximation, we might ask what the best way to use a set of random samples might be. In the next section, we lay out a methodology based on this principle.

Like Anderson’s schema for developing rational models of cognition, a resource-rational model can be developed in four simple steps: 1. Function: Formalize the problem that the cognitive mechanism solves and charac- terize the optimal solution (computational level of analysis). 2. Model of mental computation: Posit a family of algorithms that approximate the optimal solution and their computational costs. This can be done by defining an abstract computational architecture by a set of elementary operations (e.g., to draw one sample from the posterior distribution) and their costs (e.g., based on execution time). 3. Optimal resource allocation: Find the algorithm in this class that optimally trades off approximation accuracy against time and other resources (e.g., by maximizing expected utility per unit time or the value of computation; see below). 4. Evaluate and refine: Compare the model’s predictions to human behavior. Revise the functional characterization (Step 1), or the model of mental computation (Step 2) and the resulting algorithm (Step 3) accordingly. Alternatively, one might pro- ceed to the next level below by modeling how the basic operations might be approximated or considering additional resource constraints.

This method enables us to reverse-engineer not only the problem that a system solves (computational level of analysis) but also its computa- tional resources.

Many such “compilation” methods exist in the computer science literature that we can use to generate new levels of analysis for particular cognitive abilities. For instance, particle filtering (mentioned above) is a general approach that leads to specific algorithms carrying in the number of particles, the resampling criteria, and so on (Abbott & Griffiths, 2011).

To the degree that evolution, development, and learning have adapted the system to make optimal use of its finite computational resources, resource- rational analysis can be used to derive the system’s algorithm from assumptions about its computational resources.

(Lieder, Goodman & Huys, 2013): c ¼ arg max c2C n VOCðcÞ VOCðcÞ¼E PðBjcÞ max a E PðQ;SjBÞ Qðs; aÞ½ hi costðcÞ

Resource-rational analysis has been applied to decision-making (Lieder, Hsu, & Grif- fiths, in press; Vul, Goodman, Griffiths, & Tenenbaum, 2009), prediction (Lieder et al., 2013), and planning (Lieder et al., 2013), yielding predictions that are surprisingly differ- ent from the optimal solution to the problem defined at the computational level.

Vul et al. (2014) analyzed an abstract computational architecture whose elementary operation was to sample from the posterior distribution on whether or not an action will yield a reward. The analysis found that as the cost of sampling increases, the resource-rational algorithm changes from expected utility maximization (i.e., infinitely many samples) to probability matching (i.e., using a single sample). For a wide range of realistic computational costs, the optimal number of samples was surprisingly close to one, indicating that—in contrast to expected-utility theory—resource-rationality is consistent with probability matching (Herrnstein & Loveland, 1975)

we might assume that the abstract computational architecture has access to an

generating even a single perfect sample from the posterior distribution can be a very hard problem.

This suggests forming a further level of analysis by replacing the basic operation of drawing a perfect sample by performing one step of an approximate sampling algorithm, such as the Metropolis-Hastings algorithm

A model based on this analysis predicts a phenomenon that has been taken as evidence for human irrationality—anchoring and adjustment, in which people “anchor” on a first guess and “adjust” it insufficiently (Tversky & Kahneman, 1974). Taking into account the cost of computation reveals that this is exactly what an abstract computa- tional architecture whose elementary operations are the propose and the accept/reject step of the Metropolis-Hastings algorithm should do.

Resource-rational analysis of abstract computational architectures provides a rigorous way to derive the heuristics that represent a good compro- mise between accuracy and effort (see also Lieder et al., in press).

Second, instead of assuming that the mind makes optimal use of its computational resources, future research may investigate whether and how the mind learns to compute resource-rationally.

future studies may analyze and reverse-engineer increasingly more realistic models of mental computation, pushing rational analysis increasingly further down toward the algorithmic level. More realistic models of mental computation may provide elementary operations for searching through long-term memory, allocating attention, storing information in working memory, comparison, modification, and combination of information.

Russell’s (1997) notion of bounded optimality.

Resource rationality and boundedly rational analysis differ from computational rationality in how they characterize the constraints and the problem being solved. Concretely, they characterize the constraints by the computational costs imposed by an abstract computational architecture, and the problem being solved is construed as optimizing a function that combines accuracy and computational cost.

By contrast, computational rationality characterizes constraints by concrete psychological assumptions about an underlying cognitive architecture, and the problem being solved is construed as finding the best strategy supported by that architecture.

Whether it is asymptotic rational- ity, as assumed in rational process models, or the rational use of finite resources, as assumed in resource-rational models, we see this principle as playing a major role in the refinement of the notion of levels of analysis as the needs of cognitive science change.

This is an important point in the context of recent debates about Bayesian models of cog- nition (e.g., Griffiths et al., 2010; McClelland et al., 2010), in which the fundamental dis- agreement is not about whether human minds and brains use statistics, but what kinds of representations those statistics are computed over.

Even more impor- tant, it might give us a generic recipe for constructing connectionist and other psychologi- cal process models that approximate the ideal solution expressed in a Bayesian model, producing testable claims about cognitive processes.

Abstract computational architectures are mod- els of mental computation in the sense in which the Turing machine is a model of how computers process information.