The brain must make inferences about, and decisions concerning, a highly complex and unpredictable world, based on sparse evidence. An “ideal” normative approach to such challenges is often modeled in terms of Bayesian probabilistic inference. But for real-world problems of perception, motor control, categorization, language understanding, or commonsense reasoning, exact probabilistic calculations are computationally intractable. Instead, we suggest that the brain solves these hard probability problems approximately, by considering one, or a few, samples from the relevant distributions. Here we provide a gentle introduction to the various sampling algorithms that have been considered as the approximation used by the brain. We broadly summarize these algorithms according to their level of knowledge and their assumptions regarding the target distribution, noting their strengths and weaknesses, their previous applications to behavioural phenomena, as well as their psychological plausibility.

People must often make inferences about, and decisions concerning, a highly complex and unpredictable world, on the basis of sparse evidence. An “ideal” normative approach to such challenges is often modeled in terms of Bayesian probabilistic inference. But for real-world problems of perception, motor control, categorization, language comprehension, or common-sense reasoning, exact probabilistic calculations are computationally intractable. Instead, we suggest that the brain solves these hard probability problems approximately, by considering one, or a few, samples from the relevant distributions. By virtue of being an approximation, the sampling approach inevitably leads to systematic biases. Thus, if we assume that the brain carries over the same sampling approach to easy probability problems, where the “ideal” solution can readily be calculated, then a brain designed for probabilistic inference should be expected to display characteristic errors. We argue that many of the “heuristics and biases” found in human judgment and decision-making research can be reinterpreted as side effects of the sampling approach to probabilistic reasoning.

Chapter 9 develops the traditional conception of economics as an inexact science that investigates deductively the implications of assumptions that are known to be true statements of tendencies, but that are only approximately true as generalizations concerning behavior. I consider interpretations of the problematic notion of inexactness as probabilistic, as approximate, as qualified with ceteris paribus clauses, and as stating tendencies that state the contribution that causes make to outcomes and hence how things would be in the absence of interferences. The chapter argues for an account that combines a view of inexactness in terms of tendencies with an account in terms of implicit qualification. Chapter 9 explains how statements of tendencies can be true.

In this chapter, we give a comprehensive overview of the large variety of approximation results for neural networks. Approximation rates for classical function spaces as well as the benefits of deep neural networks over shallow ones for specifically structured function classes are discussed. While the main body of existing results is for general feedforward architectures, we also review approximation results for convolutional, residual and recurrent neural networks.

In this chapter, we define infinite orbits and show how to approximate such orbits. We prove that for every function that has no fixed point there is an approximate infinite orbit with unbounded variation, and we use this result to show that a certain class of quitting games admits undiscounted $\ep$-equilibria.

Chapter 3 focuses on the uses of general extenders that are addressee-oriented and express an interpersonal function in interaction. The underlying concept is described as intersubjectivity, which is tied to an awareness of the addressee’s needs. Participants in an interaction are taken to be cooperative fellow speakers, adhering toGrice's Quality and Quantity maxims. The use of adjunctive forms to indicate common ground can also create a sense of solidarity, indicating similarity, and hence also signaling positive politeness. In other situations, speakers can use disjunctive forms to signal negative politeness, that is, a concern with potentially imposing on the addressee. When general extenders are used as part of these politeness strategies, they are often described as hedges, used to indicate possible inaccuracy or imposition and a desire to avoid such things, resulting in an association with approximation.

In the problems we have considered so far, we have either ignored the actual material consumption/production (sequential environments) or assumed that materials are consumed/produced at fixed proportions (network environments). There are problems, however, where the proportions in which materials are consumed can vary provided that some specifications are satisfied. This problem, which is termed multiperiod blending or simply blending, is fundamentally different from the ones discussed thus far because it leads to nonlinear models. There are two types of blending problems: (1) different streams/inputs are blended before they are processed/converted (process blending); and (2) streams/inputs are blended to produce final products (product blending). In Section 11.1, we introduce some preliminary concepts and a formal problem statement for product blending. In Section 11.2, we present two alternative formulations for product blending, and in Section 11.3, we present two approximate linear reformulations. We close, in Section 11.4, with a discussion of models for process blending. We focus on the equations necessary to account for the key new features of blending problems: (1) the selection of input materials and their blending in variable proportions, and (2) the requirement to satisfy given property specifications.

Chapter 5 compares the phraseology of usage to exposure. It shows that more than half of patterns extracted from a student’s usage corpus also occur in her exposure corpus. At the same time the figure drops significantly if these patterns are compared to a different student’s exposure corpus supporting the assumption of representativeness. The chapter then proceeds to compare usage patterns to exposure qualitatively focusing on the processes of variation and change. It finds support for the process of approximation through which a more or less fixed pattern loosens and becomes variable on the semantic or grammatical axis presumably due to frequency effects and the properties of human memory. The chapter also proposes a reverse process, fixing, through which the pattern extends and develops verbatim associations through repeated usage. Both processes are suggested to occur within meaning-shifts units and thus be characteristic of co-selection.

Chapter 7 summarizes the findings and offers a bigger picture with regard to (1) the idiom principle in L2 acquisition and use, (2) the model of a unit of meaning and (3) the processes behind the phraseological tendency of language. It argues that the idiom principle is available to L2 users to a larger degree than is often thought. It then proposes an ‘atomic’ model of a unit of meaning, shows how the processes of fixing and approximation fit into the larger processes of delexicalization and meaning-shift, further develops the idea of a continuum of delexicalization suggested in Chapter 2 as well as explains the connection between these ideas and the concepts of relexicalization and re-metaphorization. The chapter ends with a discussion of limitations and promising directions of future research.

Chapter 3 offers an interdisciplinary overview of research on multi-word units in second language (L2) processing and use. It is motivated by a long-standing puzzle in the field of second language acquisition suggesting that while the idiom principle is the main mechanism of language production in native speakers, non-native speakers cannot benefit from it to a similar degree. The chapter shows that researchers are not unanimous in assessing the degree to which learners operate on the idiom principle and raises a few problems which might be obscuring such operation in commonly used research designs, such as a focus on specific multi-word units rather than co-selection as such, questionable representativeness of reference corpora and inability of statistical measures to capture abstracted associations. The chapter concludes by offering an alternative interpretation based on the concept of approximation adopted from studies of English as a lingua franca.