Computational mechanisms underlying human confidence reports by William T. Adler A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Center for Neural Science New York University January 2018 Wei Ji Ma c William T. Adler All Rights Reserved, 2018 I think it’s much more interesting to live not knowing than to have answers which might be wrong. I have approximate answers, and possible beliefs, and different degrees of certainty about different things, but I’m not absolutely sure of anything, and there are many things I don’t know anything about, such as whether it means anything to ask why we’re here, and what the question might mean. But I don’t have to know an answer. I don’t feel frightened by not knowing things, by being lost in a mysterious universe without having any purpose, which is the way it really is, as far as I can tell, possibly. Richard Feynman Dedication To Maw, who asked questions v Acknowledgements First and foremost, I want to thank Wei Ji Ma. Weiji is the best graduate school mentor that I could have asked for. I feel lucky to have joined Weiji’s lab, in part because it was so unplanned (he was not yet NYU faculty when I applied here), and in part because he allowed me to join even though I knew next to nothing about probability. Weiji has always pushed me to be extremely rigorous in forming and answering questions, and has helped me become a more precise thinker and writer. He has been a friend and an ally in political matters inside and outside of NYU. I wish him all the best in his new journey of fatherhood, and I know that he and Ting will make great parents. Rachel Denison has been an excellent collaborator on the project presented in Chapter 4. Collaborating with her has shown me how much more fun and productive it is to work with someone I admire than to work alone. Thanks to Marisa Carrasco for knowing everything there is to know about attention. Roshni Lulla and Gordon Bill were a great help on this project, collecting data and having thoughtful discussions. I want to thank my committee, Eero Simoncelli, Roozbeh Kiani, Marisa Carrasco, and my outside reader Angela Yu, for helping to shape this work. Steve Fleming and Joaquín Navajas provided helpful feedback on the work presented in Chapter 2. Thank you to the National Science Foundation and all political actors that enable taxpayer- funded research, without which most science would not be possible. This dissertation was based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1342536. I want to highlight a few of my brilliant colleagues in the Ma lab. Luigi Acerbi has been one of the most important figures of my graduate education. I learned something new vi every time I talked to Luigi, and his facility and enthusiasm for modeling were contagious. Aspen Yoo has been one of my best friends, and a great support, ever since she mysteriously appeared in the lab one day without any announcement or introduction by Weiji. I have been indebted to Bas van Opheusden ever since he stayed up with me until 3 a.m. our first year of graduate school, helping me write my first log likelihood function. Andra Mihali has always provided helpful comments on manuscripts and talks, as well as ibuprofen for headaches. Emin Orhan has provided various mathematical assistance over the years, and the analysis in Chapter 5 was very much inspired by his work. I want to thank everyone who has been with the Scientist Action and Advocacy Network (ScAAN) since the beginning. I have been so impressed by your enthusiasm for combining science with policy, and by the hard work you have put into our projects. You all have helped me to expand my conception of what a scientific career can be. Many people have provided meaningful contributions to ScAAN projects, but I especially want to thank Weiji, Jenn, Milenna, Kristina, David, Silvia, Ali, Billy, Stephen, Andra, and EG, for their consistent participation, enthusiasm, and leadership. Thanks to the department, especially Amala, for supporting us. Thank you to my New York friends in and outside of the department, especially Silvia Lopez Guzman, Corey Ziemba, and Alex Gibson. I want to thank my supportive family, especially my parents Martha and David, who taught me to love learning. Thank you to my sisters Zoe and Rachel; my soon-to-be parents- in-law Lisa and Philip; my Hags family Tania, Ariel, Charlie, and Robbie; and Jolene and DeWitt Clinton, who only occasionally spill coffee on my keyboard. Finally, I want to thank Sarah. Over the last eight years, we have watched each other grow more fully into ourselves and into our relationship. I treasure the time that we have spent together, and I am thankful that we will have so much more of it. vii Preface Each of Chapters 2 to4 represents work from a manuscript that has been previously posted as a preprint on bioRxiv, with myself as first author and Wei Ji Ma as senior author. On the manuscript associated with Chapter 4, Rachel N. Denison and I are co-first authors, and Marisa Carrasco is also an author. Chapter 5 was included in an earlier version of the manuscript associated with Chapter 3, and is in that article’s version history on bioRxiv. It has since been removed from the manuscript, and is unlikely to see publication outside of this dissertation. All human behavior data, as well as all code used for the work presented here, is available at github.com/wtadler/confidence. viii Abstract A perceptual decision is often accompanied by a subjective feeling of confidence. Because humans are able to easily report this feeling in a laboratory setting, confidence reports have long been objects of study. However, the computations underlying confidence reports are not well understood. It has been proposed that confidence in categorization tasks should be defined as the observer’s estimated probability of being correct. This definition extends Bayesian decision theory so that it describes confidence reports as well as decisions. Although this definition is elegant, the notion that confidence reports are Bayesian is a hypothesis rather than an established fact. In this dissertation, our aim is to test that hypothesis, which we call the Bayesian confidence hypothesis (BCH). We find that a proposed approach to determining the computational origins of confidence is flawed. Some authors have proposed that one way to determine whether confidence is Bayesian is to derive qualitative signatures of Bayesian confidence, and then see whether they are present in behavioral or neural data. We analyze some of these proposed signatures and find that they are less useful than they might have seemed. Specifically, they are neither necessary nor sufficient signatures of Bayesian confidence, which means that observation of (or failure to observe) these signatures provides an uncertain amount of evidence for (or against) the BCH. There has been a confusion in the literature about a second possible signature. We find no evidence that this second signature is ever expected under Bayesian confidence. Finally, the application of these signatures is a qualitative exercise because it may not always be clear whether data displays a signature, especially noisy data. Our analysis of the signatures leads us to conclude that the most powerful way to test the BCH is by using quantitative model comparison. ix We test human subjects on a set of binary categorization tasks designed to distinguish Bayesian models of confidence from other plausible models. In all experiments, the primary variable of interest to the observer was the orientation of a stimulus. In one set of experiments, we induce sensory uncertainty by manipulating properties of the stimulus, such as contrast. We find that subjects take their sensory uncertainty into account, and that confidence appears qualitatively Bayesian. Quantitatively, however, heuristic models provide a much better fit to the data. Our conclusions are robust to variants of both the experiment and the Bayesian models. In another experiment, we induce sensory uncertainty by manipulating the subjects’ attention. As in the previous set of experiments, we find that confidence reports are qualitatively Bayesian. In this experiment, we are unable to distinguish the Bayesian model from the heuristic models. Finally, we describe an exploratory analysis intended to explain why confidence reports might not be Bayesian. We trained feedforward neural networks on our tasks as if they were naïve human subjects and fit our behavioral models to the data produced by these trained networks. We find that the same heuristic models that fit our human data well also fit the network-produced data. We suggest a future research program in which neural network behavior is compared to human behavior on the basis of model rankings. x Contents Dedication v Acknowledgements vi Preface viii Abstract ix List of Figures xv List of Tables xviii 1 Introduction1 1.1 Techniques for measuring confidence ...................... 2 1.2 Models of confidence............................... 4 1.3 Dissertation outline................................ 8 2 Limitations of proposed signatures of Bayesian confidence9 2.1 Introduction.................................... 9 2.2 Binary categorization task............................ 11 xi 2.3 Derivation of Bayesian confidence........................ 13 2.4 0.75 signature: Mean Bayesian confidence is 0.75 for neutral evidence trials. 14 2.4.1 The 0.75 signature is not a necessary condition for Bayesian confidence 15 2.4.2 The 0.75 signature is not a sufficient condition for Bayesian confidence 19 2.5 Divergence signature #1: As stimulus magnitude increases, mean confidence increases on correct trials but decreases on incorrect trials .........