Posterior Meta-Replay for Continual Learning
Christian HenningMaria R. CerveraFrancesco D’AngeloJohannes von OswaldRegina TraberBenjamin EhretSeijin KobayashiBenjamin F. GreweJoão Sacramento
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Learning a sequence of tasks without access to i.i.d. observations is a widely studied form of continual learning (CL) that remains challenging. In principle, Bayesian learning directly applies to this setting, since recursive and one-off Bayesian updates yield the same result. In practice, however, recursive updating often leads to poor trade-off solutions across tasks because approximate inference is necessary for most models of interest. Here, we describe an alternative Bayesian approach where task-conditioned parameter distributions are continually inferred from data. We offer a practical deep learning implementation of our framework based on probabilistic task-conditioned hypernetworks, an approach we term posterior meta-replay. Experiments on standard benchmarks show that our probabilistic hypernetworks compress sequences of posterior parameter distributions with virtually no forgetting. We obtain considerable performance gains compared to existing Bayesian CL methods, and identify task inference as our major limiting factor. This limitation has several causes that are independent of the considered sequential setting, opening up new avenues for progress in CL.Here we apply Bayesian system identification methods to infer stimulus-neuron and neuron-neuron dependencies. Rather than reporting only the most likely parameters, the posterior distribution obtained in the Bayesian approach informs us about the range of parameter values that are consistent with the observed data and the assumptions made. In other words, Bayesian receptive fields always come with error bars. In fact, we obtain the full posterior covariance, indicating conditional (in-)dependence between the weights of both, receptive fields and neural couplings. Since the amount of data from neural recordings is limited, such uncertainty information is as important as the usual point estimate of the receptive field itself.
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Abstract Bayesian decision theory is a mathematical framework that models reasoning and decision‐making under uncertain conditions. The past few decades have witnessed an explosion of Bayesian modeling within cognitive science. Bayesian models are explanatorily successful for an array of psychological domains. This article gives an opinionated survey of foundational issues raised by Bayesian cognitive science, focusing primarily on Bayesian modeling of perception and motor control. Issues discussed include the normative basis of Bayesian decision theory; explanatory achievements of Bayesian cognitive science; intractability of Bayesian computation; realist versus instrumentalist interpretation of Bayesian models; and neural implementation of Bayesian inference. This article is categorized under: Philosophy > Foundations of Cognitive Science
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Bayesian inference is conditional on the space of models assumed by the analyst. The posterior distribution indicates only which of the available parameter values are less bad than the others, without indicating whether the best available parameter values really fit the data well. A posterior predictive check is important to assess whether the posterior predictions of the least bad parameters are discrepant from the actual data in systematic ways. Gelman and Shalizi (2012a) assert that the posterior predictive check, whether done qualitatively or quantitatively, is non‐Bayesian. I suggest that the qualitative posterior predictive check might be Bayesian, and the quantitative posterior predictive check should be Bayesian. In particular, I show that the ‘Bayesian p ‐value’, from which an analyst attempts to reject a model without recourse to an alternative model, is ambiguous and inconclusive. Instead, the posterior predictive check, whether qualitative or quantitative, should be consummated with Bayesian estimation of an expanded model. The conclusion agrees with Gelman and Shalizi regarding the importance of the posterior predictive check for breaking out of an initially assumed space of models. Philosophically, the conclusion allows the liberation to be completely Bayesian instead of relying on a non‐Bayesian deus ex machina . Practically, the conclusion cautions against use of the Bayesian p ‐value in favour of direct model expansion and Bayesian evaluation.
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Inferring Subjective Prior Knowledge: An Integrative Bayesian Approach Sean Tauber (sean.tauber@uci.edu) Mark Steyvers (mark.steyvers@uci.edu) Department of Cognitive Sciences, University of California, Irvine Irvine, CA 92697 USA Abstract The standard approach to Bayesian models of Cognition (also known as rational models) requires researchers to make strong assumptions about people’s prior beliefs. For example, it is often assumed that people’s subjective knowledge is best represented by “true” environmental data. We show that an integrative Bayesian approach—combining Bayesian cognitive models with Bayesian data analysis—allows us to relax this assumption. We demonstrate how this approach can be used to estimate people’s subjective prior beliefs based on their responses in a prediction task. Keywords: Bayesian modeling; rational analysis; cognitive models; Bayesian data analysis; Bayesian inference; knowledge representation; prior knowledge Introduction In the standard approach to Bayesian models of Cognition (also referred to as rational models), researchers make strong assumptions about people’s prior beliefs in order to make predictions about their behavior. These models are used to simulate the expected behavior—such as decisions, judgments or predictions—of someone whose computational-level solution to a cognitive task is well described by the model. Analysis of Bayesian models of cognition usually involves a qualitative comparison between human responses and simulated model predictions. For an overview of Bayesian models of cognition see Oaksford and Chater (1998); but also see Mozer, Pashler, and Homaei (2008); and Jones and Love (2011) for a critique. As an alternative to the standard approach, we present an integrative Bayesian approach that allows us to relax the assumptions about people’s prior beliefs. This approach is motivated by previous efforts to infer subjective mental representations (Lewandowsky, Griffiths, & Kalish, 2009; Sanborn & Griffiths, 2008; Sanborn, Griffiths, & Shiffrin, 2010) and more specifically to combine Bayesian models of cognition and Bayesian data analysis (Huszar, Noppeney & Lengyel, 2010; Lee & Sarnecka, 2008). The integrative approach allows us to use people’s responses on a cognitive task to infer posterior distributions over the psychological variables in a Bayesian model of cognition. It also allows us to estimate probabilistic representations of people’s subjective prior beliefs. We recently applied this approach to a Bayesian cognitive model of reconstructive memory (Hemmer, Tauber, & Steyvers, in prep). We estimated individuals’ subjective prior beliefs about the distribution of people’s heights based on their responses in a memory task. The technical requirements for integrated Bayesian inference were simplified because the posterior distribution, based on inference in the cognitive model, had a simple Gaussian form. This made it straight forward to define individuals’ responses as Gaussian distributed random variables in an integrated Bayesian model. In this study, we develop a method for applying integrated Bayesian inference that does not require the posterior of the cognitive model to have a simple parametric form. We apply this method to a Bayesian cognitive model for predictions that was developed by Griffiths and Tenenbaum (2006). Their Bayesian model of cognition was a computational-level description of how people combine prior knowledge with new information to make predictions about real-world phenomena. They asked participants to make a series of predictions about duration or extent that were similar to the following examples: If you were assessing the prospects of a 60-year-old man, how much longer would you expect him to live? If you were an executive evaluating the performance of a movie that had made $40 million at the box office so far, what would you estimate for its total gross? All of the questions used by Griffiths and Tenenbaum (2006) were based on real-world phenomena such as, life spans, box office grosses for movies, movie runtimes, poem lengths and waiting times. Their assumption was that people make predictions about these phenomena based on prior beliefs that reflect their true extents or durations in the real world. Although it is possible that people’s beliefs about these phenomena are tuned to the environment, this assumption cannot be used to explain how people make similar sorts of predictions about counterfactual phenomena that have no true statistics in the environment. For example, consider the following question: Suppose it is the year 2075 and medical science has advanced significantly. You meet a man that is 60 years old. To what age will this man live? There is no “true” answer to this question and therefore no environmental data is available. This creates a problem for a Bayesian model of cognition that requires environmental data in order to make predictions.
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Abstract The contrast sensitivity function (CSF) is crucial in predicting functional vision both in research and clinical areas. Recently, a group of novel strategies, multi-dimensional adaptive methods, were proposed and allowed more rapid measurements when compared to usual methods such as Ψ or staircase. Our study further presents a multi-dimensional Bayesian framework to estimate parameters of the CSF from experimental data obtained by classical sampling. We extensively simulated the framework’s performance as well as validated the results in a psychophysical experiment. The results showed that the Bayesian framework significantly improves the accuracy and precision of parameter estimates from usual strategies, and requires about the same number of observations as the novel methods to obtain reliable inferences. Additionally, the improvement with the Bayesian framework was maintained when the prior poorly matched the observer’s CSFs. The results indicated that the Bayesian framework is flexible and sufficiently precise for estimating CSFs.
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