Relevant and robust: a response to Marcus and Davis (2013).

Computational models in psychology are precise, fully explicit scientific hypotheses. Over the past 15 years, probabilistic modeling of human cognition has yielded quantitative theories of a wide variety of reasoning and learning phenomena. Recently, Marcus and Davis (2013) critique several examples of this work, using these critiques to question the basic validity of the probabilistic approach. Contra the broad rhetoric of their article, the points made by Marcus and Davis—while useful to consider—do not indicate systematic problems with the probabilistic modeling enterprise. Relevant and robust 3 Computational models in psychology are precise, fully explicit scientific hypotheses. Probabilistic models in particular formalize hypotheses about the beliefs of agents—their knowledge and assumptions about the world—using the structured collection of probabilities referred to as priors, likelihoods, etc. The probability calculus then describes inferences that can be drawn by combining these beliefs with new evidence, without the need to commit to a process-level explanation of how these inferences are performed (Marr, 1982). Over the past 15 years, probabilistic modeling of human cognition has yielded quantitative theories of a wide variety of phenomena (Tenenbaum, Kemp, Griffiths, & Goodman, 2011). Marcus and Davis (2013, henceforth, M&D) critique several examples of this work, using these critiques to question the basic validity of the probabilistic models approach, based on the existence of alternative models and potentially inconsistent data. Contra the broad rhetoric of their article, the points made by M&D—while useful to consider—do not indicate systematic problems with the probabilistic modeling enterprise. Several objections stem from a fundamental confusion about the status of optimality in probabilistic modeling, which has been discussed in responses to other critiques (see: Griffiths, Chater, Norris, & Pouget, 2012; Frank, 2013). Briefly: an optimal analysis is not the optimal analysis for a task or domain. Different probabilistic models instantiate different psychological hypotheses. Optimality provides a bridging assumption between these hypotheses and human behavior; one that can be re-examined or overturned as the data warrant. Model selection. M&D argue that individual probabilistic models require a host of potentially problematic modeling choices. Indeed, probabilistic models are created via a series of choices concerning priors, likelihoods, response functions, etc. Each of these choices embodies a proposal about cognition, and these proposals will often be wrong. The Relevant and robust 4 identification of model assumptions that result in a mismatch to empirical data allows these assumptions to be replaced or refined. Systematic iteration to achieve a better model is part of the normal progress of science. But if choices are made post-hoc, a model can be overfit to the particulars of the empirical data. M&D suggest that certain of our models suffer from this issue. For instance, they show that data on pragmatic inference (Frank & Goodman, 2012) are inconsistent with an alternative variant of the proposed model that uses a hard-max rather than a soft-max function, and ask whether the choice of soft-max was dependent on the data. The soft-max rule is foundational in economics, decision-theory, and cognitive psychology (Luce, 1959, 1977), and we first selected it for this problem based on a completely independent set of experiments (Frank, Goodman, Lai, & Tenenbaum, 2009). So it’s hard to see how a claim of overfitting is warranted here. Modelers must balance unification with exploration of model assumptions across tasks, but this issue is a general one for all computational work, and does not constitute a systematic problem with the probabilistic approach. Task selection. M&D suggested that probabilistic modelers report results on only the narrow range of tasks on which their models succeed. But their critique focused on a few high-profile, short reports that represented our first attempts to engage with important domains of cognition. Such papers necessarily have less in-depth engagement with empirical data than more extensive and mature work, though they also exemplify the applicability of probabilistic modeling to domains previously viewed as too complex for quantitative approaches. There is broader empirical adequacy to probabilistic models of cognition than M&D imply. If M&D had surveyed the literature they would have found substantial additional Relevant and robust 5 evidence for the models they reviewed—and more has accrued since their critique. For example, M&D critiqued Griffiths and Tenenbaum’s (2006) analysis of everyday predictions for failing to provide independent assessments of the contributions of priors and likelihoods, precisely what was done in several later and much longer papers (Griffiths & Tenenbaum, 2011; Lewandowsky, Griffiths, & Kalish, 2009). They similarly critiqued the particular tasks selected by Battaglia, Hamrick, and Tenenbaum (2013) without discussing the growing literature testing similar “noisy Newtonian” models on other phenomena (Gerstenberg, Goodman, Lagnado, & Tenenbaum, 2012; Gerstenberg, Goodman, Lagnado, & Tenenbaum, 2014; Sanborn, Mansinghka, & Griffiths, 2013; Smith, Dechter, Tenenbaum, & Vul, 2013; Téglás et al., 2011). Smith, Battaglia, and Vul (2013) even directly address exactly the challenge M&D posed regarding classic findings of errors in physical intuitions. In other domains, such as concept learning and inductive inference, where there is an extensive experimental tradition, probabilistic models have engaged with diverse empirical data collected by multiple labs over many years (e.g. Goodman, Tenenbaum, Feldman, & Griffiths, 2008; Kemp & Tenenbaum, 2009). M&D also insinuate empirical problems that they do not test. For instance, in criticizing the choice of dependent measure used by Frank and Goodman (2012), they posit that a forced-choice task would yield a qualitatively different pattern (discrete rather than graded responding). In fact, a forced-choice version of the task produces graded patterns of responding across a wide variety of conditions (Stiller, Goodman, & Frank, 2011, 2014; Vogel, Emilsson, Frank, Jurafsky, & Potts, 2014). Conclusions. We agree with M&D that there are real and important challenges for probabilistic models of cognition, as there will be for any approach to modeling a system as complex as the human mind. To us, the most pressing challenges include understanding the Relevant and robust 6 relationship to lower levels of psychological analysis and neural implementation, integrating additional formal tools, clarifying the philosophical status of the models, extending to new domains of cognition, and, yes: engaging with additional empirical data in the current domains while unifying specific model choices into broader principles. As M&D state, “ultimately, the Bayesian approach should be seen as a useful tool”—one that we believe has already proven its robustness and relevance by allowing us to form and test quantitatively accurate psychological hypotheses. Relevant and robust 7