Bayesian models of inductive generalization

Neville E.Sanjana&Joshua B.Tenenbaum

Department of Brain and Cognitive Sciences

Massachusetts Institute of Technology

Cambridge,MA02139

nsanjana,jbt@https://www.sodocs.net/doc/ab11643070.html,

Abstract

We argue that human inductive generalization is best explained in a

Bayesian framework,rather than by traditional models based on simi-

larity computations.We go beyond previous work on Bayesian concept

learning by introducing an unsupervised method for constructing?ex-

ible hypothesis spaces,and we propose a version of the Bayesian Oc-

cam’s razor that trades off priors and likelihoods to prevent under-or

over-generalization in these?exible spaces.We analyze two published

data sets on inductive reasoning as well as the results of a new behavioral

study that we have carried out.

1Introduction

The problem of inductive reasoning—in particular,how we can generalize after seeing only one or a few speci?c examples of a novel concept—has troubled philosophers,psy-chologists,and computer scientists since the early days of their https://www.sodocs.net/doc/ab11643070.html,putational approaches to inductive generalization range from simple heuristics based on similarity matching to complex statistical models[5].Here we consider where human inference falls on this spectrum.Based on two classic data sets from the literature and one more comprehensive data set that we have collected,we will argue for models based on a ra-tional Bayesian learning framework[10].We also confront an issue that has often been side-stepped in previous models of concept learning:the origin of the learner’s hypothesis space.We present a simple,unsupervised clustering method for creating hypotheses spaces that,when applied to human similarity judgments and embedded in our Bayesian frame-work,consistently outperforms the best alternative models of inductive reasoning based on similarity-matching heuristics.

We focus on two related inductive generalization tasks introduced in[6],which involve reasoning about the properties of animals.The?rst task is to judge the strength of a gen-eralization from one or more speci?c kinds of mammals to a different kind of mammal: given that animals of kind and have property,how likely is it that an animal of kind also has property?For example,might be chimp,might be squirrel,and might be horse.is always a blank predicate,such as“is susceptible to the disease blick-etitis”,about which nothing is known outside of the given examples.Working with blank predicates ensures that people’s inductions are driven by their deep knowledge about the general features of animals rather than the details they might or might not know about any

one particular property.Stimuli are typically presented in the form of an argument from premises(examples)to conclusion(the generalization test item),as in

Chimps are susceptible to the disease blicketitis.

Squirrels are susceptible to the disease blicketitis.

max-similarity model based on its match to their intuitions for these particular tasks.We examine both models in our experiments.

Sloman[8]developed a feature-based model that encodes the shared features between the premise set and the conclusion set as weights in a neural network.Despite some psycho-logical plausibility,this model consistently?t the two data sets signi?cantly worse than the max-similarity model.Heit[3]outlines a Bayesian framework that provides qualitative explanations of various inductive reasoning phenomena from[6].His model does not con-strain the learner’s hypothesis space,nor does it embody a generative model of the data, so its predictions depend strictly on well-chosen prior probabilities.Without a general method for setting these prior probabilities,it does not make quantitative predictions that can be compared here.

3A Bayesian model

Tenenbaum&colleagues have previously introduced a Bayesian framework for learning concepts from examples,and applied it to learning number concepts[10],word meanings [11],as well as other domains.Formally,for the speci?c inference task,we observe posi-tive examples of the concept and want to compute the probability that a particular test stimulus belongs to the concept given the observed examples:

.These generalization probabilities are computed by averaging the predictions of a set of hypotheses weighted by their posterior probabilities:

(1)

Hypotheses pick out subsets of stimuli—candidate extensions of the concept—and is just1or0depending on whether the test stimulus falls under the subset. In the general inference task,we are interested in computing the probability that a whole test category falls under the concept:

(2)

A crucial component in modeling both tasks is the structure of the learner’s hypothesis space.

3.1Hypothesis space

Elements of the hypothesis space represent natural subsets of the objects in the domain —subsets likely to be the extension of some novel property or concept.Our goal in build-ing up is to capture as many hypotheses as possible that people might employ in concept learning,using a procedure that is ideally automatic and unsupervised.One natural way to begin is to identify hypotheses with the clusters returned by a clustering algorithm[11][7]. Here,hierarchical clustering seems particularly appropriate,as people across cultures ap-pear to organize their concepts of biological species in a hierarchical taxonomic struc-ture[1].We applied four standard agglomerative clustering algorithms[2](single-link, complete-link,average-link,and centroid)to subjects’similarity judgments for all pairs of 10animals given in[6].All four algorithms produced the same output(Figure1),sug-gesting a robust cluster structure.We de?ne the base set of clusters to consist of all 19clusters in this tree.The most straightforward way to de?ne a hypothesis space for Bayesian concept learning is to take;each hypothesis consists of one base cluster. We refer to as the“taxonomic hypothesis space”.

It is clear that alone is not suf?cient.The chance that horses can get a disease given that we know cows and squirrels can get that disease seems much higher than if we know only

Horse Cow Elephant Rhino Chimp Gorilla Mouse Squirrel Dolphin Seal Figure1:Hierarchical clustering of mammals based on similarity judgments in[6].Each node in the tree corresponds to one hypothesis in the taxonomic hypothesis space.

that chimps and squirrels can get the disease,yet the taxonomic hypotheses consistent with the example sets cow,squirrel and chimp,squirrel are the same.Bayesian generaliza-tion with a purely taxonomic hypothesis space essentially depends only on the least similar example(here,squirrel),ignoring more?ne-grained similarity structure,such as that one example in the set cow,squirrel is very similar to the target horse even if the other is not.This sense of?ne-grained similarity has a clear objective basis in biology,because a single property can apply to more than one taxonomic cluster,either by chance or through convergent evolution.If the disease in question could af?ict two distinct clusters of ani-mals,one exempli?ed by cows and the other by squirrels,then it is much more likely also to af?ict horses(since they share most taxonomic clusters with cows)than if the disease af?icted two distinct clusters exempli?ed by chimps and squirrels.Thus we consider richer hypothesis subspaces,consisting of all pairs of taxonomic clusters(i.e.,all unions of two clusters from Figure1,except those already included in),and,consisting of all triples of taxonomic clusters(except those included in lower layers).We stop with because we have no behavioral data beyond three examples.Our total hypothesis space is then the union of these three layers,.

The notion that the hypothesis space of candidate concepts might correspond to the power set of the base clusters,rather than just single clusters,is broadly applicable beyond the domain of biological properties.If the base system of clusters is suf?ciently?ne-grained, this framework can parameterize any logically possible concept.It is analogous to other general-purpose representations for concepts,such as disjunctive normal form(DNF)in PAC-Learning,or class-conditional mixture models in density-based classi?cation[5]. 3.2The Bayesian Occam’s razor:balancing priors and likelihoods

Given this hypothesis space,Bayesian generalization then requires assigning a prior

and likelihood for each hypothesis.Let be the number of base clusters, and be a hypothesis in the th layer of the hypothesis space,corresponding to a union of base clusters.A simple but reasonable prior assigns to a sequence of i.i.d. Bernoulli variables with successes and parameter,with probability

the correspondence is not exact because each hypothesis may be expressed as the union of base clusters in multiple ways,and we consider only the minimal union in de?ning .For,instantiates a preference for simpler hypotheses—that is,hy-

potheses consisting of fewer disjoint clusters(smaller).More complex hypotheses re-ceive exponentially lower probability under,and the penalty for complexity increases as becomes smaller.This prior can be applied with any set of base clusters,not just those which are taxonomically structured.We are currently exploring a more sophisticated domain-speci?c prior for taxonomic clusters de?ned by a stochastic mutation process over the branches of the tree.

Following[10],the likelihood is calculated by assuming that the examples are a random sample(with replacement)of instances from the concept to be learned.Let

,the number of examples,and let the size of each hypothesis be simply the number of animal types it contains.Then follows the size principle,

4.1A new experiment:Varying example set composition

In order to provide a more comprehensive test of the models,we conducted a variant of the speci?c experiment using the same10animal types and the same constant test category, horses,but with example sets of different sizes and similarity structures.In both data sets 1and2,the number of examples was constant across all trials;we expected that varying the number of examples would cause dif?culty for the max-similarity model because it is not explicitly sensitive to this factor.For this purpose,we included?ve three-premise arguments,each with three examples of the same animal species(e.g.,chimp,chimp, chimp),and?ve one-premise arguments with the same?ve animals(e.g.,chimp).We also included three-premise arguments where all examples were drawn from a low-level cluster of species in Figure1(e.g.,chimp,gorilla,chimp).Because of the increasing preference for smaller hypotheses as more examples are observed,Bayes will in general make very different predictions in these three cases,but max-similarity will not.This manipulation also allowed us to distinguish the predictions of our Bayesian model from alternative Bayesian formulations[5][3]that do not include the size principle,and thus do not predict differences between generalization from one example and generalization from three examples of the same kind.

We also changed the judgment task and cover story slightly,to match more closely the nat-ural problem of inductive learning from randomly sampled examples.Subjects were told that they were training to be veterinarians,by observing examples of particular animals that had been diagnosed with novel diseases.They were required to judge the probability that horses could get the same disease given the examples observed.This cover story made it clear to subjects that when multiple examples of the same animal type were presented,these instances referred to distinct individual animals.Figure3(row3)shows the model’s pre-dicted generalization probabilities along with the data from our experiment:mean ratings of generalization from24subjects on28example sets,using either,or examples and the same test species(horses)across all arguments.Again we show predictions for the best values of the free parameters and.All three models?t best at different parameter values than in data sets1and2,perhaps due to the task differences or the greater range of

Figure2:Human generalization

to the conclusion category horse

when given one or three examples

of a single premise type.

of the Bayesian model,

data.Most notably,we found a difference between generalization from one example and generalization from three exam-ples of the same kind,in the direction predicted by our Bayesian model.Generalization to the test category of horses was greater from singleton examples(e.g.,chimp)than from three examples of the same kind(e.g.,chimp,chimp,chimp),as shown in Figure2.This effect was relatively small but it was observed for all?ve animal types tested and it was

statistically signi?cant()in a25(number of examples animal type)ANOV A. The max-similarity model,however,predicts no effect here,as do Bayesian accounts that do not include the size principle[5][3].

It is also of interest to ask whether these models are suf?ciently robust as to make reason-able predictions across all three experiments using a single parameter setting,or to make good predictions on held-out data when their free parameter is tuned on the remaining data. On these criteria,our Bayesian model maintains its advantage over max-similarity.At the single value of,Bayes achieves correlations of,and on the three data sets,respectively,compared to,and for max-similarity at its single best parameter value().Using Monte Carlo cross validation[9]to estimate (1000runs for each data set,80%-20%training-test splits),Bayes obtains average test-set correlations of and on the three data sets,respectively,compared to

and for max-similarity using the same method to tune.

5Conclusion

Our Bayesian model offers a moderate but consistent quantitative advantage over the best similarity-based models of generalization,and also predicts qualitative effects of varying sample size that contradict alternative approaches.More importantly,our Bayesian ap-proach has a principled rational foundation,and we have introduced a framework for un-supervised construction of hypothesis spaces that could be applied in many other domains. In contrast,the similarity-based approach requires arbitrary assumptions about the form of the similarity measure:it must include both“similarity”and“coverage”terms,and it must be based on max-similarity rather than sum-similarity.These choices have no a priori justi?cation and run counter to how similarity models have been applied in other domains, leading us to conclude that rational statistical principles offer the best hope for explaining how people can generalize so well from so little data.Still,the consistently good perfor-mance of the max-similarity model raises an important question for future study:whether a relatively small number of simple heuristics might provide the algorithmic machinery implementing approximate rational inference in the brain.

We would also like to understand how people’s subjective hypothesis spaces have their ori-gin in the objective structure of their environment.Two plausible sources for the taxonomic hypothesis space used here can both be ruled out.The actual biological taxonomy for these 10animals,based on their evolutionary history,looks quite different from the subjective taxonomy used here.Substituting the true taxonomic clusters from biology for the base clusters of our model’s hypothesis space leads to dramatically worse predictions of peo-ple’s generalization behavior.Taxonomies constructed from linguistic co-occurrences,by applying the same agglomerative clustering algorithms to similarity scores output from the LSA algorithm[4],also lead to much worse predictions.Perhaps the most likely possibil-ity has not yet been tested.It may well be that by clustering on simple perceptual features (e.g.,size,shape,hairiness,speed,etc.),weighted appropriately,we can reproduce the tax-onomy constructed here from people’s similarity judgments.However,that only seems to push the problem back,to the question of what de?nes the appropriate features and fea-ture weights.We do not offer a solution here,but merely point to this question as perhaps the most salient open problem in trying to understand the computational basis of human inductive inference.

Acknowledgments

Tom Grif?ths provided valuable help with statistical analysis.Supported by grants from NTT Communication Science Laboratories and MERL and an HHMI fellowship to NES.

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column shows the results for a particular model.Each row is a different inductive generalization experiment,where indicates the number of examples(premises)in the stimuli.

Tesla Model S底盘全透视..

水平对置、后置后驱、低重心、前双横臂后多连杆、全铝合金车架、5门5座，你以为笔者说的是保时捷新车型吗？那笔者再补充多几个关键词好了，后置的水平对置双电刷电动机、0油耗、藏在地板下的笔记本电池组，同时拥有这些标签的，便是Tesla第二款车型Model S。Model S是五门五座纯电动豪华轿车，布局设计及车身体积与保时捷Panamera相当，并且是目前电动车续航里程的纪录保持者（480公里）。虽然现在纯电动在我国远未至于普及，但是在香港地区却是已经有Tesla的展厅，在该展厅内更是摆放了一台没有车身和内饰，只有整个底盘部分的Model S供人直观了解Model S的技术核心。 图：Tesla Model S。

图：拆除车壳之后，Model S的骨架一目了然。

图：这套是Model S的个性化定制系统，可以让买家选择自己喜爱的车身颜色、内饰配色和轮圈款式，然后预览一下效果。可以看到Model S共分为普通版、Sign at ure版和Performance版，后面两个型号标配的是中间的21寸轮圈，而普通版则是两边的19寸款式。Signature版是限量型号，在美国已全部售罄，香港也只有少量配额。 图：笔者也尝试一下拼出自己心目中的Model S，碳纤维饰条当然是最爱啦。

图：参观了一下工作车间，不少Roadster在等着检查保养呢，据代理介绍，不同于传统的汽车，电动车的保养项目要少很多，至少不用更换机油和火花塞嘛，换言之电动车的维护成本要比燃油汽车要低。 Tesla于2010年5月进军香港市场，并于翌年2011年9月成立服务中心。由于香港政府对新能源车的高度支持，香港的电动车市场发展比起大陆地区要好得多。例如Tesla的第一款车型Roadster（详见《无声的革命者——Tesla Roadster Sport 》），在香港获得豁免资格，让车主可以节省将近100万港元的税款。在这样的优惠政策之下，Tesla Roadster尽管净车价达100万港元，但50台的配额已经基本售罄。而Model S目前在香港已经开始接受报名预定，确定车型颜色和配置之后约两个月左右可以交车。

特斯拉整体介绍

Tesla Model S 特斯拉Model S是一款纯电动车型，外观造型方面，该车定位一款四门Coupe车型，动感的车身线条使人过目不忘。此外在前脸造型方面，该车也采用了自己的设计语言。另值得一提的是，特斯拉Model S的镀铬门把手在触摸之后可以自动弹出，充满科技感的设计从拉开车门时便开始体现。该车在2011年年中正式进入量产阶段，预计在2012年年内将有5000台量产车投放市场。 目录 1概述 2售价 3内饰 4动力 5车型 6技术规格 7性能表现 8荣誉 9对比测试 10车型参数 1概述

Tesla Model S是一款由Tesla汽车公司制造的全尺寸高性能电动轿车，预计于2012年年中投入销售，而它的竞争对手则直指宝马5系。该款车的设计者Franz von Holzhausen，曾在马自达北美分公司担任设计师。在Tesla汽车公司中，Model S拥有独一无二的底盘、车身、发动机以及能量储备系统。Model S的第一次亮相是在2009年四月的一期《大卫深夜秀》节目中 4 Tesla Model S 。 2售价 Model S的电池规格分为三种，分别可以驱动车辆行驶260公里、370公里和480公里。而配备这三种电池的Model S的售价则分别为57400美元、67400美元和77400美元。下线的首批1000辆签名款车型将配有可以行驶480公里的蓄电池。尽管官方尚未公布该签名款车型的具体售价，但据推测，价格将会保持在50000美元左右。 Tesla汽车公司称其将会对市场出租可以提供480公里行驶距离的电池。而从Model S中取得的收益将为第三代汽车的发展提供资金保障。 3内饰

详解特斯拉Model S

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TESLA特斯拉解析

TESLA 硅谷工程师、资深车迷、创业家马丁·艾伯哈德（Martin Eberhard）在寻找创业项目时发现，美国很多停放丰田混合动力汽车普锐斯的私家车道上经常还会出现些超级跑车的身影。他认为，这些人不是为了省油才买普锐斯，普锐斯只是这群人表达对环境问题的方式。于是，他有了将跑车和新能源结合的想法，而客户群就是这群有环保意识的高收入人士和社会名流。 2003年7月1日，马丁·艾伯哈德与长期商业伙伴马克·塔彭宁（Marc Tarpenning）合伙成立特斯拉（TESLA）汽车公司，并将总部设在美国加州的硅谷地区。成立后，特斯拉开始寻找高效电动跑车所需投资和材料。

由于马丁·艾伯哈德毫无这方面的制造经验，最终找到AC Propulsion公司。当时，对AC Propulsion公司电动汽车技术产生兴趣的还有艾龙·穆思科（Elon Musk）。在AC Propulsion公司CEO汤姆·盖奇（Tom Gage）的引见下，穆思科认识了艾伯哈德的团队。2004年2月会面之后，穆思科向TESLA投资630万美元，但条件是出任公司董事长、拥有所有事务的最终决定权，而艾伯哈德作为创始人任TESLA的CEO。 在有了技术方案、启动资金后，TESLA开始开发高端电动汽车，他们选择英国莲花汽车的Elise作为开发的基础。没有别的原因，只是因为莲花是唯一一家把TESLA放在眼里的跑车生产商。

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深度揭秘特斯拉Model S底盘：电池组电机四驱

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由于电池组横贯于位于车辆底部，这使得Model S的重心得以降低，平衡了配重，从而提升了操控性。根据官方数据，Model S的前后配重比为48：52。 在Model S刚上市时，按照电池划分共有3款型号，分别是85kWh、60kWh，以及40kWh。2013年，由于40kWh车型销量惨淡，特斯拉决定停止销售。不久前，特斯拉又推出了70Kwh车型，来取代之前的60kWh版本。 值得一提的是，当年60kWh的车型与40kWh的车型，电池组其实是一样的；两者的区别在于，特斯拉将40kWh的电池进行了软件限制，从而在一个可容纳60kWh电量的电池组中，只有40kWh的电量可用。 而85kWh电池与60kWh电池的区别，主要是电池组中装配的电池单元数量。85kWh的电池组电压为400V，由一共16个电池包组成，每个电池包装配了444颗电池单元，所以这个电池组一共有7104颗电池组成。60kWh，则是由14个电池包，共计6216颗电池单元组成。这里所说的电池单元，是由松下提供的 NCR-18650A型电池。 18650是可充电锂离子电池的一种型号，它的命名来源于这种电池的尺寸 --18mm*65mm，但由于还要加入保护电路，所以电池的实际尺寸要略微大几零点几毫米。18650电池的主要用途，是笔记本电脑的电池，它有很多生产厂商；而特斯拉则选用了松下提供的18650电池，但要注意特斯拉使用的电池与笔记本中的电池还是有差别。18650只是一个统称。

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