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Probabilistic Modeling in Physics

Claus Beisbart

in Probabilities in Physics

How can probabilistic models from physics represent a target, and how can one understand the probabilities that figure in such models? The aim of this chapter is to answer these questions by ... More

How can probabilistic models from physics represent a target, and how can one understand the probabilities that figure in such models? The aim of this chapter is to answer these questions by analyzing random models of Brownian motion and point process models of the galaxy distribution as examples. This chapter defends the view that such models represent because we may learn from them by setting our degrees of belief following the probabilities suggested by the model. This account is not incompatible with an objectivist view of the pertinent probabilities, but stock objectivist interpretations, e.g., frequentism or Lewis’ Humean account of probabilities have problems to provide a suitable objectivist methodology for statistical inference from data. This point is made by contrasting Bayesian statistics with error statistics.Less

The Framework

Timothy J. O’Donnell and Noah D. Goodman

in Productivity and Reuse In Language: A Theory of Linguistic Computation and Storage

This chapter consists of four parts. The first section discusses the ideas behind the modeling framework adopted in the book---structured probabilistic generative models. The second section discusses ... More

This chapter consists of four parts. The first section discusses the ideas behind the modeling framework adopted in the book---structured probabilistic generative models. The second section discusses some theoretical and methodological issues in the interpretation of this approach to modeling. The third section develops the modelling framework from the perspective of probabilistic programming using the Church programming language. The Church formalization makes explicit the relationship between the model and two important technical ideas from computer science (and, specifically, the theory of programming languages). The final section of the chapter provides more discussion of the four classes of model evaluated in the book.Less

Two Proposals for Causal Grammars

Thomas L. Griffiths and Joshua B. Tenenbaum

in Causal Learning: Psychology, Philosophy, and Computation

A causal theory can be thought of as a grammar that generates events, and that can be used to parse events to identify underlying causal structure. This chapter considers what the components of such ... More

A causal theory can be thought of as a grammar that generates events, and that can be used to parse events to identify underlying causal structure. This chapter considers what the components of such a grammar might be — the analogues of syntactic categories and the rules that relate them in a linguistic grammar. It presents two proposals for causal grammars. The first asserts that the variables which describe events can be organized into causal categories, and allows relationships between those categories to be expressed. The second uses a probabilistic variant of first-order logic in order to describe the ontology and causal laws expressed in an intuitive theory. This chapter illustrates how both kinds of grammar can guide causal learning.Less

Introduction

Timothy J. O’Donnell

in Productivity and Reuse In Language: A Theory of Linguistic Computation and Storage

This chapter introduces the problem of productivity: How do language learners determine which potential generalizations can actually be used to create novel expressions in their language and which ... More

This chapter introduces the problem of productivity: How do language learners determine which potential generalizations can actually be used to create novel expressions in their language and which only occur as parts of stored items? The first part of the chapter reviews historical approaches to this question, discussing previous unsuccessful attempts to reduce the problem of what is stored and what is computed to other properties. The second part of the chapter outlines a new theory of storage and computation based on the idea that the problem can be solved via a probabilistic inference which optimizes a tradeoff between fewer, simpler stored items, and simpler derivations of linguistic expressions. This inference-based model is contrasted with four other models to which it will be compared throughout the book: (i) the full-parsing model where all structure is always computed, (ii) the full-listing model where all structure is stored after the first time it is computed and (iii) two variants of the exemplar-based model which hypothesizes all possible mixtures of computation and storage in the derivation of every expression.Less

A primer on probabilistic inference

Thomas L. Griffiths and Alan Yuille

in The Probabilistic Mind:: Prospects for Bayesian cognitive science

This chapter provides the technical introduction to Bayesian methods. Probabilistic models of cognition are often referred to as Bayesian models, reflecting the central role that Bayesian inference ... More

This chapter provides the technical introduction to Bayesian methods. Probabilistic models of cognition are often referred to as Bayesian models, reflecting the central role that Bayesian inference plays in reasoning under uncertainty. It introduces the basic ideas of Bayesian inference and discusses how it can be used in different contexts. Probabilistic models provide a unique opportunity to develop a rational account of human cognition that combines statistical learning with structured representations. It recommends the EM algorithm and Markov chain Monte Carlo to estimate the parameters of models that incorporate latent variables, and to work with complicated probability distributions of the kind that often arise in Bayesian inference.Less

Epilogue

Luc Bovens and Stephan Hartmann

in Bayesian Epistemology

Presents some general reflections on the role and the challenges of probabilistic modelling in philosophy.

Probabilities, Odds, and Risks: Predispositions and Provocations as Causes

Peter V. Rabins

in The Why of Things: Causality in Science, Medicine, and Life

This chapter describes the probabilistic model, in which causes are conceptualized as events that affect the likelihood that another event will occur. In this model, causes act as influences, risk ... More

This chapter describes the probabilistic model, in which causes are conceptualized as events that affect the likelihood that another event will occur. In this model, causes act as influences, risk factors, predispositions, modifiers, and buffers. The complexity of the probabilistic concept of cause begins with the definition of the word “probability.” Its primary meaning relates to predictability or prognostication, that is, the likelihood of specific future outcomes or effects. The implication of this definition is that there is uncertainty as to the outcome. This chapter provides a history overview of the concept of probabilistic cause and considers the characteristics of probabilistic reasoning that are relevant to causality, along with the challenges of causal probabilistic logic, limitations of the probabilistic model, and whether probabilistic reasoning is different from categorical reasoning. It also outlines the criteria for choosing the appropriate model of causality.Less

Probabilistic Mental Models

Gerd Gigerenzer

in Adaptive Thinking: Rationality in the Real World

The cognitive bias research claims that people are naturally prone to making mistakes in reasoning and memory, including the mistake of over-estimating their knowledge. This chapter proposes a new ... More

The cognitive bias research claims that people are naturally prone to making mistakes in reasoning and memory, including the mistake of over-estimating their knowledge. This chapter proposes a new theoretical model for confidence in knowledge based on the more charitable assumption that people are good judges of the reliability of their knowledge, provided that the knowledge is representatively sampled from a specified reference class. It claims that this model both predicts new experimental results and explains a wide range of extant experimental findings on confidence, including some perplexing inconsistencies. It consists of three parts: an exposition of the proposed theory of probabilistic mental models (PMM theory); a report of experimental tests confirming these predictions; and an explanation of apparent anomalies in previous experimental results by means of PMMs.Less

Intuitive Theories as Grammars for Causal Inference

Joshua B. Tenenbaum, Thomas L. Griffiths, and Sourabh Niyogi

in Causal Learning: Psychology, Philosophy, and Computation

This chapter presents a framework for understanding the structure, function, and acquisition of causal theories from a rational computational perspective. Using a “reverse engineering” approach, it ... More

This chapter presents a framework for understanding the structure, function, and acquisition of causal theories from a rational computational perspective. Using a “reverse engineering” approach, it considers the computational problems that intuitive theories help to solve, focusing on their role in learning and reasoning about causal systems, and then using Bayesian statistics to describe the ideal solutions to these problems. The resulting framework highlights an analogy between causal theories and linguistic grammars: just as grammars generate sentences and guide inferences about their interpretation, causal theories specify a generative process for events, and guide causal inference.Less

Essentials to Understand Probabilistic Graphical Models: A Tutorial about Inference and Learning

Christine Sinoquet

in Probabilistic Graphical Models for Genetics, Genomics, and Postgenomics

The aim of this chapter is to offer an advanced tutorial to scientists with no background or no deep background on probabilistic graphical models. To readers more familiar with these models, this ... More

The aim of this chapter is to offer an advanced tutorial to scientists with no background or no deep background on probabilistic graphical models. To readers more familiar with these models, this chapter is to be used as a compendium of definitions and general methods, to browse through at will. Intentionally self-contained, this chapter first begins with reminders of essential definitions such as the distinction between marginal independence and conditional independence. Then the chapter briefly surveys the most popular classes of probabilistic graphical models: Markov chains, Bayesian networks, and Markov random fields. Next probabilistic inference is explained and illustrated in the Bayesian network context. Finally parameter and structure learning are presented.Less

Formalization of the Models and Inference

Timothy J. O’Donnell and Noah D. Goodman

in Productivity and Reuse In Language: A Theory of Linguistic Computation and Storage

This chapter the mathematical details of the models studied in this book. It also discusses the inference algorithms used for each of the models and various other issues of practical concern for the ... More

This chapter the mathematical details of the models studied in this book. It also discusses the inference algorithms used for each of the models and various other issues of practical concern for the simulations that we report later.Less

Conclusion

Timothy J. O’Donnell

in Productivity and Reuse In Language: A Theory of Linguistic Computation and Storage

This chapter provides a short overview of the entire book.

Probabilistic regression and Bayes nets

Thomas P. Trappenberg

in Fundamentals of Machine Learning

This chapter revises regression with the inclusion of uncertainty in the data in probabilistic models. It shows how modern probabilistic machine learning can be formulated. First, a simple stochastic ... More

This chapter revises regression with the inclusion of uncertainty in the data in probabilistic models. It shows how modern probabilistic machine learning can be formulated. First, a simple stochastic generalization of the linear regression example is offered to introduce the formalism. This leads to the important maximum likelihood principle on which learning will be based. This concept is then generalized to non-linear problems in higher dimensions and the chapter relates this to Bayes nets. The chapter ends with a discussion about how such a probabilistic approach is related to deep learning.Less

Introduction

Alison Gopni and Laura Schulz

in Causal Learning: Psychology, Philosophy, and Computation

This chapter provides a simple, clear, and (hopefully) amusing introduction to causal model and Bayes nets theories in computer science, the interventionist account of causation in philosophy, and ... More

This chapter provides a simple, clear, and (hopefully) amusing introduction to causal model and Bayes nets theories in computer science, the interventionist account of causation in philosophy, and the psychology of causal learning in both adults and children. It takes the form of a fictional e-mail exchange between a developmental psychologist and a philosopher/computer scientist in which each partner explains the background of their field to the other. In two attachments, the fictional authors review the literature on causal Bayes nets and on the psychology of causal inference.Less

Learning From Doing: Intervention and Causal Inference

Laura Schulz, Tamar Kushnir, and Alison Gopnik

in Causal Learning: Psychology, Philosophy, and Computation

This chapter starts from the premise that much of children's knowledge takes the form of abstract, coherent, causal claims that are learned from, and defeasible by, evidence. This view is consistent ... More

This chapter starts from the premise that much of children's knowledge takes the form of abstract, coherent, causal claims that are learned from, and defeasible by, evidence. This view is consistent with an interventionist view of causal knowledge, formalized in computational models using causal Bayes net representations. The chapter reviews empirical studies suggesting that, consistent with this account, preschoolers use patterns of evidence to: a) create novel, effective interventions; b) infer the structure of causal relationships, including relationships involving unobserved causes; c) accurately predict distinct outcomes from observed evidence and evidence generated by interventions; d) integrate novel evidence with prior beliefs; and e) distinguish informative interventions from confounded ones.Less

Gaussian Processes and the Null-Category Noise Model

Lawrence Neil D. and Jordan Michael I.

in Semi-Supervised Learning

This chapter presents an augmentation of the standard probabilistic classification model which incorporates a null-category. Given a suitable probabilistic model for the model category, the chapter ... More

This chapter presents an augmentation of the standard probabilistic classification model which incorporates a null-category. Given a suitable probabilistic model for the model category, the chapter obtains a probabilistic counterpart of the margin. By combining this noise model with a Gaussian process classifier (GPC), it obtains a classification methodology that is simultaneously discriminative, semi-supervised, and Bayesian. The approach incorporates the cluster assumption without explicitly modeling the data density and without requiring specialized kernels. GPCs aim to predict the posterior probability of the class label yi given a covariate vector xi. Under the standard assumptions generally invoked by GPC practitioners, this posterior probability is unaffected by unlabeled data points, providing no role for unlabeled data.Less