Bayesian Models of Perception and Action
An Introduction
by Ma, Kording, Goldreich
 ISBN: 9780262372831  Copyright 2023
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An accessible introduction to constructing and interpreting Bayesian models of perceptual decisionmaking and action.
Many forms of perception and action can be mathematically modeled as probabilistic—or Bayesian—inference, a method used to draw conclusions from uncertain evidence. According to these models, the human mind behaves like a capable data scientist or crime scene investigator when dealing with noisy and ambiguous data. This textbook provides an approachable introduction to constructing and reasoning with probabilistic models of perceptual decisionmaking and action. Featuring extensive examples and illustrations, Bayesian Models of Perception and Action is the first textbook to teach this widely used computational framework to beginners.
•Introduces Bayesian models of perception and action, which are central to cognitive science and neuroscience
•Beginnerfriendly pedagogy includes intuitive examples, daily life illustrations, and gradual progression of complex concepts
•Broad appeal for students across psychology, neuroscience, cognitive science, linguistics, and mathematics
•Written by leaders in the field of computational approaches to mind and brain
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Contents (pg. vii)  
Acknowledgments (pg. xv)  
The Four Steps of Bayesian Modeling (pg. xvii)  
List of Acronyms (pg. xix)  
Introduction (pg. 1)  
Perception as Inference (pg. 2)  
Positioning Bayesian Models within the Space of Models (pg. 3)  
What This Book Is Not (pg. 4)  
A Disclaimer on Citations (pg. 4)  
In Conclusion (pg. 5)  
1. Uncertainty and Inference (pg. 7)  
1.1 The Goal of Perception (pg. 7)  
1.2 Hypotheses and Their Probabilities (pg. 8)  
1.3 Sensory Noise and Perceptual Ambiguity (pg. 12)  
1.4 Bayesian Inference in Visual Perception (pg. 14)  
1.5 Bayesian Inference in Auditory Perception (pg. 18)  
1.6 Historical Background: Perception as Unconscious Inference (pg. 23)  
1.7 Summary (pg. 25)  
1.8 Suggested Readings (pg. 26)  
1.9 Problems (pg. 27)  
2. Using Bayes’ Rule (pg. 31)  
2.1 Steps of Bayesian Modeling (pg. 31)  
2.2 Alternative Form of Bayes’ Rule (pg. 35)  
2.3 Areal Representation (pg. 35)  
2.4 The Prosecutor’s Fallacy (pg. 37)  
2.5 A Changing Prior: Luggage Carousel Example (pg. 38)  
2.6 A Flat Prior: Gestalt Perception Example (pg. 42)  
2.7 Optimality, Evolution, and Motivations for Bayesian Modeling (pg. 44)  
2.8 Summary (pg. 45)  
2.9 Suggested Readings (pg. 45)  
2.10 Problems (pg. 46)  
3. Bayesian Inference under Measurement Noise (pg. 53)  
3.1 The Steps of Bayesian Modeling (pg. 53)  
3.2 Step 1: The Generative Model (pg. 55)  
3.2.1 The Measurement: An Abstracted Sensory Representation (pg. 55)  
3.2.2 Graphical Model (pg. 56)  
3.2.3 The Stimulus Distribution (pg. 57)  
3.2.4 The Measurement Distribution (pg. 58)  
3.2.5 Joint Distribution (pg. 59)  
3.3 Step 2: Inference (pg. 59)  
3.3.1 The Prior Distribution (pg. 60)  
3.3.2 The Likelihood Function (pg. 60)  
3.3.3 The Posterior Distribution (pg. 63)  
3.3.4 The Posterior Mean (pg. 66)  
3.3.5 Width of the Posterior (pg. 67)  
3.3.6 The Posterior Mean Estimate (pg. 68)  
3.3.7 The MAP Estimate (pg. 68)  
3.4 Uncertainty and Confidence (pg. 69)  
3.4.1 Uncertainty (pg. 69)  
3.4.2 Bayesian Confidence (pg. 70)  
3.5 Model Mismatch in Inference (pg. 71)  
3.5.1 Prior Mismatch (pg. 72)  
3.5.2 Improper Priors (pg. 72)  
3.6 Heteroskedasticity (pg. 73)  
3.7 Magnitude Variables (pg. 73)  
3.8 Applications (pg. 75)  
3.9 Percepts (pg. 76)  
3.10 Summary (pg. 76)  
3.11 Suggested Readings (pg. 77)  
3.12 Problems (pg. 78)  
4. The Response Distribution (pg. 83)  
4.1 Inherited Variability (pg. 83)  
4.2 The Response Distribution (pg. 84)  
4.3 Belief versus Response Distributions (pg. 86)  
4.4 MaximumLikelihood Estimation (pg. 87)  
4.5 Bias and Mean Squared Error (pg. 88)  
4.5.1 An “Inverted Bias” Perspective (pg. 91)  
4.5.2 All Expenses Paid (pg. 92)  
4.6 Other Estimates (pg. 93)  
4.7 Decision Noise and Response Noise (pg. 94)  
4.8 Misconceptions (pg. 95)  
4.9 Reflections on Bayesian Models (pg. 97)  
4.10 Summary (pg. 97)  
4.11 Suggested Readings (pg. 98)  
4.12 Problems (pg. 98)  
5. Cue Combination and Evidence Accumulation (pg. 105)  
5.1 What Is Cue Combination? (pg. 105)  
5.2 Formulation of the Bayesian Model (pg. 108)  
5.2.1 Step 1: Generative Model (pg. 108)  
5.2.2 Step 2: Inference (pg. 110)  
5.2.3 Step 3: Estimate Distribution (pg. 112)  
5.3 Artificial Cue Conflict (pg. 113)  
5.3.1 Distinguishing the Distributions (pg. 114)  
5.4 Generalizations: Prior, Multiple Cues (pg. 114)  
5.5 Evidence Accumulation (pg. 115)  
5.6 Cue Combination under Ambiguity (pg. 116)  
5.7 Applications (pg. 117)  
5.8 Summary (pg. 118)  
5.9 Suggested Readings (pg. 118)  
5.10 Problems (pg. 119)  
6. Learning as Inference (pg. 125)  
6.1 The Many Forms of Learning (pg. 125)  
6.2 Learning the Probability of a Binary Event (pg. 126)  
6.2.1 Prediction (pg. 128)  
6.2.2 Update Equations (pg. 130)  
6.2.3 Uncertainty (pg. 131)  
6.2.4 Binomial Distribution (pg. 131)  
6.2.5 Nonuniform Prior (pg. 131)  
6.3 Linking Bayesian Learning to Reinforcement Learning (pg. 132)  
6.4 Learning the Precision of a Normal Distribution (pg. 134)  
6.4.1 Why Not Infer Variance? (pg. 135)  
6.4.2 Prediction (pg. 136)  
6.5 Learning the Slope of a Linear Relationship (pg. 137)  
6.6 Learning the Structure of a Causal Model (pg. 139)  
6.7 More Learning (pg. 141)  
6.8 Summary (pg. 141)  
6.9 Suggested Readings (pg. 142)  
6.10 Problems (pg. 143)  
7. Discrimination and Detection (pg. 147)  
7.1 Example Tasks (pg. 147)  
7.2 Discrimination (pg. 148)  
7.2.1 Step 1: Generative Model (pg. 148)  
7.2.2 Step 2: Inference (pg. 148)  
7.2.3 Gaussian Model (pg. 152)  
7.2.4 Decision Rule in Terms of the Measurement (pg. 152)  
7.2.5 Multiple Tasks Can Have the Same Bayesian Decision Rule (pg. 153)  
7.2.6 Step 3: Response Distribution (pg. 153)  
7.3 Detection (pg. 156)  
7.4 Confidence (pg. 156)  
7.5 Further Characterizing the Response Distribution (pg. 159)  
7.5.1 Receiver Operating Characteristic (pg. 159)  
7.5.2 Discriminability (pg. 161)  
7.6 The Relation between Bayesian Inference and Signal Detection Theory (pg. 162)  
7.7 Extensions (pg. 163)  
7.8 Summary (pg. 163)  
7.9 Suggested Readings (pg. 163)  
7.10 Problems (pg. 164)  
8. Binary Classification (pg. 169)  
8.1 Example Tasks (pg. 169)  
8.2 Generative Model (pg. 171)  
8.2.1 MirrorSymmetric ClassConditioned Stimulus Distributions (pg. 172)  
8.3 Marginalization (pg. 173)  
8.3.1 The Sum of Two Die Rolls (pg. 173)  
8.3.2 Continuous Variables (pg. 174)  
8.3.3 Conditioned Marginalization (pg. 175)  
8.3.4 Using the Generative Model (pg. 176)  
8.4 Inference (pg. 177)  
8.5 Response Distribution (pg. 179)  
8.6 Beyond MirrorImage ClassConditioned Stimulus Distributions (pg. 181)  
8.7 “Following the Arrows” (pg. 183)  
8.8 Summary (pg. 184)  
8.9 Suggested Readings (pg. 184)  
8.10 Problems (pg. 185)  
9. TopLevel Nuisance Variables and Ambiguity (pg. 191)  
9.1 Example Tasks (pg. 191)  
9.2 Size as a TopLevel Nuisance Variable in Depth Perception (pg. 192)  
9.3 Marginalization Formulation (pg. 196)  
9.4 In Color Perception (pg. 197)  
9.5 In Object Recognition (pg. 200)  
9.6 Summary (pg. 201)  
9.7 Suggested Readings (pg. 202)  
9.8 Problems (pg. 202)  
10. SameDifferent Judgment (pg. 205)  
10.1 Example Tasks (pg. 205)  
10.2 Binary Stimuli (pg. 206)  
10.2.1 Step 1: Generative Model (pg. 206)  
10.2.2 Step 2: Inference (pg. 207)  
10.2.3 Step 3: Estimate Distribution (pg. 208)  
10.3 Continuous Stimuli (pg. 210)  
10.3.1 Step 1: Generative Model (pg. 211)  
10.3.2 Step 2: Inference (pg. 211)  
10.3.3 Step 3: Response Probabilities (pg. 213)  
10.3.4 Step 2 Revisited: Inferring the Stimuli (pg. 213)  
10.4 MultipleItem Sameness Judgment (pg. 215)  
10.5 Perceptual Organization (pg. 217)  
10.5.1 Contour Integration (pg. 219)  
10.5.2 Intersecting Lines (pg. 220)  
10.6 Summary (pg. 222)  
10.7 Suggested Readings (pg. 223)  
10.8 Problems (pg. 223)  
11. Search (pg. 227)  
11.1 Many Forms of Visual Search (pg. 227)  
11.2 Target Localization: Camouflage (pg. 229)  
11.3 Target Localization with Measurement Noise (pg. 232)  
11.4 Target Detection: Camouflage (pg. 234)  
11.5 Target Detection with Measurement Noise (pg. 237)  
11.6 Applications (pg. 239)  
11.7 Summary (pg. 239)  
11.8 Suggested Readings (pg. 239)  
11.9 Problems (pg. 240)  
12. Inference in a Changing World (pg. 245)  
12.1 Tracking a Continuously Changing World State (pg. 245)  
12.1.1 Step 1: Generative Model (pg. 245)  
12.1.2 Step 2: Inference (pg. 247)  
12.2 Change Point Detection (pg. 249)  
12.2.1 Single Change Point (pg. 250)  
12.2.2 Random Change Points (pg. 252)  
12.2.3 More Realistic Change Point Detection (pg. 253)  
12.3 Summary (pg. 253)  
12.4 Suggested Readings (pg. 254)  
12.5 Problems (pg. 254)  
13. Combining Inference with Utility (pg. 257)  
13.1 Example Tasks (pg. 257)  
13.2 Deciding between Two Actions (pg. 258)  
13.3 Deciding among Several Actions (pg. 260)  
13.4 Mathematical Formulation: Expected Utility (pg. 262)  
13.4.1 Binary Classification (pg. 263)  
13.4.2 Continuous Estimation (pg. 264)  
13.5 The Cost Functions of “Pure Perception” (pg. 265)  
13.5.1 Discrete Tasks (pg. 265)  
13.5.2 Continuous Estimation (pg. 265)  
13.5.3 Perception and Action (pg. 267)  
13.6 What It Means to Decide Optimally (pg. 267)  
13.7 Complications (pg. 268)  
13.7.1 Cost Functions for Uncertain Outcomes (pg. 268)  
13.7.2 Nonlinearity between Reward and Utility (pg. 269)  
13.7.3 Distortions of Probability (pg. 270)  
13.7.4 Decision Noise (pg. 270)  
13.8 Applications (pg. 271)  
13.8.1 Visual Discrimination (pg. 271)  
13.8.2 Reaching Movements (pg. 272)  
13.8.3 Incentivized Confidence Intervals (pg. 272)  
13.8.4 Inferring Utility Functions (pg. 274)  
13.9 Summary (pg. 274)  
13.10 Suggested Readings (pg. 275)  
13.11 Problems (pg. 276)  
14. The Neural Likelihood Function (pg. 281)  
14.1 Generative Model of the Activity of a Single Neuron (pg. 281)  
14.2 Neural Likelihood Function for a Single Neuron (pg. 283)  
14.2.1 Case 1: A BellShaped Tuning Curve (pg. 285)  
14.2.2 Case 2: A Monotonic Tuning Curve (pg. 286)  
14.3 Neural Likelihood Function Based on a Population of Neurons (pg. 287)  
14.4 Toy Model (pg. 290)  
14.5 Relation between Abstract and Neural Concepts (pg. 293)  
14.6 Using the Neural Likelihood Function for Computation (pg. 294)  
14.7 The Neural Implementation of Bayesian Computation (pg. 295)  
14.8 Applications (pg. 295)  
14.9 Summary (pg. 296)  
14.10 Suggested Readings (pg. 297)  
14.11 Problems (pg. 298)  
15. Bayesian Models in Context (pg. 301)  
15.1 Bayesian versus Optimal Behavior (pg. 301)  
15.2 Overly Strong Claims of Optimality (pg. 302)  
15.3 Understanding Why Some Behaviors Are Optimal and Others Are Not (pg. 303)  
15.4 Bayesian Models Are Not Mechanistic Models (pg. 303)  
15.5 Bayesian Transfer (pg. 304)  
15.6 Probabilistic Computation and Hybrid Models (pg. 305)  
15.7 RealWorld Complexity (pg. 305)  
15.8 Learning to Be Bayesian (pg. 307)  
15.9 Suggested Readings (pg. 308)  
Appendices (pg. 311)  
A. Notation (pg. 313)  
B. Basics of Probability Theory (pg. 315)  
B.1 Objective and Subjective Probability (pg. 315)  
B.2 The Intuitive Notion of Probability (pg. 316)  
B.3 Complementary Event (pg. 316)  
B.4 Venn Diagram Representation (pg. 317)  
B.5 Random Variables and Their Distributions (pg. 317)  
B.6 Mean, Variance, and Expected Value (pg. 322)  
B.7 The Normal Distribution (pg. 323)  
B.8 The Delta Function (pg. 326)  
B.9 The Poisson Distribution (pg. 327)  
B.10 Drawing from a Probability Distribution (pg. 327)  
B.11 Distributions Involving Multiple Variables (pg. 327)  
B.12 Functions of Random Variables (pg. 333)  
B.13 Problems (pg. 340)  
C. Model Fitting and Model Comparison (pg. 343)  
C.1 What Is a Model? (pg. 343)  
C.2 Free Parameters (pg. 343)  
C.3 The Parameter Likelihood (pg. 344)  
C.4 MaximumLikelihood Estimation (pg. 344)  
C.5 Fitting Data from an Estimation Task (pg. 345)  
C.6 Absolute Goodness of Fit (pg. 354)  
C.7 Fitting Data from a Discrimination Task (pg. 354)  
C.8 Fitting Data from a Classification Task (pg. 357)  
C.9 Good Experimental Design for Bayesian Modeling (pg. 357)  
C.10 Suggested Readings (pg. 358)  
C.11 Problems (pg. 359)  
Bibliography (pg. 361)  
Index (pg. 371) 
Wei Ji Ma
Konrad Paul Kording
Daniel Goldreich
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