Amortized Inference in Probabilistic Reasoning

Amortized Inference in Probabilistic Reasoning Samuel J. Gershman 1 (sjgershm@mit.edu) and Noah D. Goodman 2 (ngoodman@stanford.edu) 1 Department of Brain and Cognitive Sciences, MIT of Psychology, Stanford University 2 Department Abstract similar or related queries. For example, as you view an im- age, your head and eyes are continuously moving, generating an infinitude of slightly different queries. For these queries, it may be inaccurate to reuse a stored inference without modifi- cation. This raises the problem of amortized inference: how to flexibly reuse inferences so as to answer a variety of re- lated queries. Recently, Stuhlm¨uller et al. (2013) addressed this problem by using stored samples to estimate local condi- tional distributions, and then approximating answers to more complex queries by composing the local distributions. The work described in this paper seeks experimental evidence for a similar kind of flexible reuse in human reasoning. We presented subjects with a simple Bayesian network and asked them to answer a series of queries about it. One of these queries (the “target”) could be answered by reusing the answer to another query (the “sub-query”). We hypothesized that the effects of reuse would be evident compared to an in- ference with the same structure but no re-usable sub-query. Further, we hypothesized that this effect would be present only if the target was presented after the sub-query. Accord- ingly, we manipulated (between subjects) whether the target came before or after the sub-query. This design allowed us to look for two key signatures of reuse: correlations between related inferences (Experiment 1) and faster responses for in- ferences that exploit reuse (Experiment 2). Recent studies of probabilistic reasoning have postulated general-purpose inference algorithms that can be used to an- swer arbitrary queries. These algorithms are memoryless, in the sense that each query is processed independently, without reuse of earlier computation. We argue that the brain oper- ates in the setting of amortized inference, where numerous related queries must be answered (e.g., recognizing a scene from multiple viewpoints); in this setting, memoryless algo- rithms can be computationally wasteful. We propose a simple form of flexible reuse, according to which shared inferences are cached and composed together to answer new queries. We present experimental evidence that humans exploit this form of reuse: the answer to a complex query can be systematically predicted from a person’s response to a simpler query if the simpler query was presented first and entails a sub-inference (i.e., a sub-component of the more complex query). People are also faster at answering a complex query when it is preceded by a sub-inference. Our results suggest that the astonishing ef- ficiency of human probabilistic reasoning may be supported by interactions between inference and memory. Keywords: induction, Bayesian inference, memory “Cognition is recognition.” – Hofstadter (1995) Introduction One view of probabilistic reasoning holds that our brains are equipped with general-purpose inference algorithms that can be used to answer arbitrary queries (Griffiths et al., 2012; Pouget et al., 2013). An under-appreciated property of such algorithms borrowed from computer science is that they are memoryless: each query is (at least in principle) processed independently of others. While this property guarantees that inferences will not interfere with one another, it can also lead to gross computational inefficiency, since inferences are never reused; memorylessness implies that answering the same query twice requires the same amount of computation as answer two unique queries. 1 Whatever inference algorithms the brain uses, they are un- likely to be memoryless. Consider, for example, the image in Figure 1 (Gregory, 1970). Upon viewing it for the first time, most observers find it extremely difficult to identify what the image depicts. 2 However, once the image has been deciphered, all subsequent views are instantly recognized. Clearly, the visual system is not running a computationally expensive inference algorithm upon each viewing; the infer- ence is simply reused. In reality, it is rare to be faced with the exact same query multiple times. Much more pervasive is the appearance of Figure 1: What does this image depict? Amortized inference in Bayesian networks 1 To be fair, inference algorithms for dynamical systems, like Kalman filtering, involve reuse in a certain sense. However, these algorithms are not designed to reuse inferences when applied to sev- eral independent time series (even if the time series are identical). 2 Answer: a dalmatian. In this paper, we will restrict our attention to amortized in- ference for Bayesian networks. Let p(x) denote a probability distribution on variables x = {x 1 , . . . , x M }. A Bayesian net- work G is a directed acyclic graph with nodes corresponding