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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Comparing hypotheses about the weight of a coin. (a) The vertical axis shows log posterior odds in favor of $h_1$, the hypothesis that the probability of heads ($\theta $) is drawn from a uniform distribution on $[0,1]$, over $h_0$, the hypothesis that the probability of heads is $0.5$. The horizontal axis shows the number of heads, $N_H$, in a sequence of 10 flips. As $N_H$ deviates from $5$, the posterior odds in favor of $h_1$ increase. (b) The posterior odds shown in (a) are computed by averaging over the values of $\theta $ with respect to the prior, $p(\theta )$, which in this case is the uniform distribution on $[0,1]$. This averaging takes into account the fact that hypotheses with greater flexibility -- such as the free-ranging $\theta $ parameter in $h_1$ -- can produce both better and worse predictions, implementing an automatic ``Bayesian Occam's razor''. The solid line shows the probability of the sequence {\tt  HHTHTTHHHT} for different values of $\theta $, while the dotted line is the probability of any sequence of length 10 under $h_0$ (equivalent to $\theta = 0.5$). While there are some values of $\theta $ that result in a higher probability for the sequence, on average the greater flexibility of $h_1$ results in lower probabilities. Consequently, $h_0$ is favored over $h_1$ (this sequence has $N_H = 6$). In contrast, a wide range of values of $\theta $ result in higher probability for for the sequence {\tt  HHTHHHTHHH}, as shown by the dashed line. Consequently, $h_1$ is favored over $h_0$ (this sequence has $N_H = 8$).}}{20}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Graphical models showing different kinds of processes that could generate a sequence of coinflips. (a) Independent flips, with parameters $\theta $ determining the probability of heads. (b) A Markov chain, where the probability of heads depends on the result of the previous flip. Here the parameters $\theta $ define the probability of heads after a head and after a tail. (c) A hidden Markov model, in which the probability of heads depends on a latent state variable $z_i$. Transitions between values of the latent state are set by parameters $\theta $, while other parameters $\phi $ determine the probability of heads for each value of the latent state. This kind of model is commonly used in computational linguistics, where the $x_i$ might be the sequence of words in a document, and the $z_i$ the syntactic classes from which they are generated.}}{24}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Directed graphs involving three variables, $B,C,E$, relevant to elemental causal induction. $B$ represents background variables, $C$ a potential causal variable, and $E$ the effect of interest. \unhbox \voidb@x \hbox {${\rm  Graph} \ 1$}is assumed in computing \unhbox \voidb@x \hbox {$\Delta P$}\nobreakspace  {}and causal power. Computing causal support involves comparing the structure of \unhbox \voidb@x \hbox {${\rm  Graph} \ 1$}\nobreakspace  {}to that of \unhbox \voidb@x \hbox {${\rm  Graph} \ 0$}in which $C$ and $E$ are independent.}}{33}}
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The beta distribution serves as a prior on the bias $\theta $ of a coin. The mean of the distribution is $\frac  {\alpha }{\alpha + \beta }$, and the shape of the distribution depends on $\alpha + \beta $. }}{39}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces  Three hierarchical Bayesian models. (a) A model for inferring $\theta _{\text  {new}}$, the bias of a coin. $d_\text  {new}$ specifies the number of heads and tails observed when the coin is tossed. $\theta _{\text  {new}}$ is is drawn from a beta distribution with parameters $\alpha $ and $\beta $. The prior distribution on these parameters has a single hyperparameter, $\lambda $. (b) A model for inferring $e_\text  {new}$, the extension of a novel property. $d_\text  {new}$ is a sparsely observed version of $e_\text  {new}$, and $e_\text  {new}$ is assumed to be drawn from a prior distribution induced by structured representation $\@mathcal {S}$. The hyperparameter $\lambda $ specifies a prior distribution over a hypothesis space of structured representations. (c) A model that can discover the form $\@mathcal {F}$ of the structure $\@mathcal {S}$. The hyperparameter $\lambda $ now specifies a prior distribution over a hypothesis space of structural forms.}}{40}}
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\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Inferences about the distribution of features within tribes. (a) Prior distributions on $\theta $, $\qopname  \relax o{log}(\alpha +\beta )$ and $\frac  {\alpha }{\alpha +\beta }$. (b) Posterior distributions after observing 10 all-white tribes and 10 all-brown tribes. (c) Posterior distributions after observing 20 tribes. Black circles indicate obese indiviuals, and the rate of obesity varies among tribes.}}{41}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Learning a tree-structured prior for property induction. Given a collection of sparsely observed properties $d_i$ (a black circle indicates that a species has a given property), we can compute a posterior distribution on structure $S$ and posterior distributions on each extension $e_i$. Since the distribution over $S$ is difficult to display, we show a single tree with high posterior probability. Since each distribution on $e_i$ is difficult to display, we show instead the posterior probability that each species has each property (dark circles indicate probabilities close to 1).}}{47}}
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces The Metropolis-Hastings algorithm. The solid lines shown in the bottom part of the figure are three sequences of values sampled from a Markov chain. Each chain began at a different location in the space, but used the same transition kernel. The transition kernel was constructed using the procedure described in the text for the Metropolis-Hastings algorithm: the proposal distribution, $q(x^*|x)$, was a Gaussian distribution with mean $x$ and standard deviation $0.2$ (shown centered on the starting value for each chain at the bottom of the figure), and the acceptance probabilities were computed by taking $p(x)$ to be Gaussian with mean $0$ and standard deviation $1$ (plotted with a solid line in the top part of the figure). This guarantees that the stationary distribution associated with the transition kernel is $p(x)$. Thus, regardless of the initial value of each chain, the probability that the chain takes on a particular value will converge to $p(x)$ as the number of iterations increases. In this case, all three chains move to explore a similar part of the space after around 100 iterations. The histogram in the top part of the figure shows the proportion of time the three chains spend visiting each part in the space after 250 iterations (marked with the dotted line), which closely approximates $p(x)$. Samples from the Markov chains can thus be used similarly to samples from $p(x)$.}}{55}}
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\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Approaches to semantic representation. (a) In a semantic network, words are represented as nodes, and edges indicate semantic relationships. (b) In a semantic space, words are represented as points, and proximity indicates semantic association. These are the first two dimensions of a solution produced by Latent Semantic Analysis (Landauer \& Dumais, 1997). The black dot is the origin. (c) In the topic model, words are represented as belonging to a set of probabilistic topics. The matrix shown on the left indicates the probability of each word under each of three topics. The three columns on the right show the words that appear in those topics, ordered from highest to lowest probability.}}{56}}
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\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces A sample of topics from a 1700 topic solution derived from the TASA corpus. Each column contains the 20 highest probability words in a single topic, as indicated by $P(w|z)$. Words in boldface occur in different senses in neighboring topics, illustrating how the model deals with polysemy and homonymy. These topics were discovered in a completely unsupervised fashion, using just word-document co-occurrence frequencies.}}{58}}
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\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Graphical model for Latent Dirichlet Allocation (Blei, Ng, \& Jordan, 2003). The distribution over words given topics, $\phi $, and the distribution over topics in a document, $\theta $, are generated from Dirichlet distributions with parameters $\beta $ and $\alpha $ respectively. Each word in the document is generated by first choosing a topic $z_i$ from $\theta $, and then choosing a word according to $\phi ^{(z_i)}$.}}{60}}
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\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Illustration of the Gibbs sampling algorithm for learning topics. Each word token $w_i$ appearing in the corpus has a topic assignment, $z_i$. The figure shows the assignments of all tokens of three types -- {\sc  money}, {\sc  bank}, and {\sc  stream} -- before and after running the algorithm. Each marker corresponds to a single token appearing in a particular document, and shape and color indicates assignment: topic 1 is a black circle, topic 2 is a gray square, and topic 3 is a white triangle. Before running the algorithm, assignments are relatively random, as shown in the left panel. After running the algorithm, tokens of {\sc  money} are almost exclusively assigned to topic 3, tokens of {\sc  stream} are almost exclusively assigned to topic 1, and tokens of {\sc  bank} are assigned to whichever of topic 1 and topic 3 seems to dominate a given document. The algorithm consists of iteratively choosing an assignment for each token, using a probability distribution over tokens that guarantees convergence to the posterior distribution over assignments.}}{62}}
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\bibcite{ackleyhs85}{\BCAY {Ackley, Hinton,{} \BBA{} Sejnowski}{Ackley et\nobreakspace  {}al.{}}{1985}}
\bibcite{anderson90}{\BCAY {Anderson}{Anderson}{1990}}
\bibcite{ashbyar95}{\BCAY {Ashby \BBA{} Alfonso-Reese}{Ashby \BBA{} Alfonso-Reese}{1995}}
\bibcite{atran98}{\BCAY {Atran}{Atran}{1998}}
\bibcite{bakerts07}{\BCAY {Baker, Tenenbaum,{} \BBA{} Saxe}{Baker et\nobreakspace  {}al.{}}{2007}}
\bibcite{bayes63}{\BCAY {Bayes}{Bayes}{1763/1958}}
\bibcite{bernardos94}{\BCAY {Bernardo \BBA{} Smith}{Bernardo \BBA{} Smith}{1994}}
\bibcite{bishop06}{\BCAY {Bishop}{Bishop}{2006}}
\bibcite{bleigjt04}{\BCAY {Blei, Griffiths, Jordan,{} \BBA{} Tenenbaum}{Blei et\nobreakspace  {}al.{}}{2004}}
\bibcite{bleinj03}{\BCAY {Blei, Ng,{} \BBA{} Jordan}{Blei et\nobreakspace  {}al.{}}{2003}}
\bibcite{boas83}{\BCAY {Boas}{Boas}{1983}}
\bibcite{brainardf97}{\BCAY {Brainard \BBA{} Freeman}{Brainard \BBA{} Freeman}{1997}}
\bibcite{buehnerc97}{\BCAY {Buehner \BBA{} Cheng}{Buehner \BBA{} Cheng}{1997}}
\bibcite{buehnercc02}{\BCAY {Buehner, Cheng,{} \BBA{} Clifford}{Buehner et\nobreakspace  {}al.{}}{2003}}
\bibcite{carey85}{\BCAY {Carey}{Carey}{1985}}
\bibcite{charniak93}{\BCAY {Charniak}{Charniak}{1993}}
\bibcite{chaterm06}{\BCAY {Chater \BBA{} Manning}{Chater \BBA{} Manning}{2006}}
\bibcite{cheng97}{\BCAY {Cheng}{Cheng}{1997}}
\bibcite{chomsky88}{\BCAY {Chomsky}{Chomsky}{1988}}
\bibcite{collinsl75}{\BCAY {Collins \BBA{} Loftus}{Collins \BBA{} Loftus}{1975}}
\bibcite{collinsq69}{\BCAY {Collins \BBA{} Quillian}{Collins \BBA{} Quillian}{1969}}
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\bibcite{courvilledt06}{\BCAY {Courville, Daw,{} \BBA{} Touretzky}{Courville et\nobreakspace  {}al.{}}{2006}}
\bibcite{danksgt03}{\BCAY {Danks, Griffiths,{} \BBA{} Tenenbaum}{Danks et\nobreakspace  {}al.{}}{2003}}
\bibcite{doyaipr07}{\BCAY {Doya, Ishii, Pouget,{} \BBA{} Rao}{Doya et\nobreakspace  {}al.{}}{2007}}
\bibcite{dudahs00}{\BCAY {Duda, Hart,{} \BBA{} Stork}{Duda et\nobreakspace  {}al.{}}{2000}}
\bibcite{friedmank00}{\BCAY {Friedman \BBA{} Koller}{Friedman \BBA{} Koller}{2000}}
\bibcite{gelmancsr95}{\BCAY {Gelman, Carlin, Stern,{} \BBA{} Rubin}{Gelman et\nobreakspace  {}al.{}}{1995}}
\bibcite{gemang84}{\BCAY {Geman \BBA{} Geman}{Geman \BBA{} Geman}{1984}}
\bibcite{ghahramani04}{\BCAY {Ghahramani}{Ghahramani}{2004}}
\bibcite{gigerenzerspdbk89}{\BCAY {Gigerenzer et\nobreakspace  {}al.{}}{Gigerenzer et\nobreakspace  {}al.{}}{1989}}
\bibcite{gilksrs96}{\BCAY {Gilks, Richardson,{} \BBA{} Spiegelhalter}{Gilks et\nobreakspace  {}al.{}}{1996}}
\bibcite{glymour01}{\BCAY {Glymour}{Glymour}{2001}}
\bibcite{glymourc99}{\BCAY {Glymour \BBA{} Cooper}{Glymour \BBA{} Cooper}{1999}}
\bibcite{goldstein03}{\BCAY {Goldstein}{Goldstein}{2003}}
\bibcite{good80}{\BCAY {Good}{Good}{1980}}
\bibcite{gopnikm97}{\BCAY {Gopnik \BBA{} Meltzoff}{Gopnik \BBA{} Meltzoff}{1997}}
\bibcite{gopniktIP}{\BCAY {Gopnik \BBA{} Tenenbaum}{Gopnik \BBA{} Tenenbaum}{in press}}
\bibcite{griffiths05}{\BCAY {Griffiths}{Griffiths}{2005}}
\bibcite{griffithsbt04}{\BCAY {Griffiths, Baraff,{} \BBA{} Tenenbaum}{Griffiths et\nobreakspace  {}al.{}}{2004}}
\bibcite{griffithsg05}{\BCAY {Griffiths \BBA{} Ghahramani}{Griffiths \BBA{} Ghahramani}{2005}}
\bibcite{griffithss02}{\BCAY {Griffiths \BBA{} Steyvers}{Griffiths \BBA{} Steyvers}{2002}}
\bibcite{griffithss03nips}{\BCAY {Griffiths \BBA{} Steyvers}{Griffiths \BBA{} Steyvers}{2003}}
\bibcite{griffithss04}{\BCAY {Griffiths \BBA{} Steyvers}{Griffiths \BBA{} Steyvers}{2004}}
\bibcite{griffithssbt04}{\BCAY {Griffiths, Steyvers, Blei,{} \BBA{} Tenenbaum}{Griffiths et\nobreakspace  {}al.{}}{2005}}
\bibcite{griffithsst05}{\BCAY {Griffiths, Steyvers,{} \BBA{} Tenenbaum}{Griffiths et\nobreakspace  {}al.{}}{in press}}
\bibcite{griffithst05}{\BCAY {Griffiths \BBA{} Tenenbaum}{Griffiths \BBA{} Tenenbaum}{2005}}
\bibcite{griffithst06coincidence}{\BCAY {Griffiths \BBA{} Tenenbaum}{Griffiths \BBA{} Tenenbaum}{2007\BCnt {1}}}
\bibcite{griffithst06}{\BCAY {Griffiths \BBA{} Tenenbaum}{Griffiths \BBA{} Tenenbaum}{2007\BCnt {2}}}
\bibcite{hacking75}{\BCAY {Hacking}{Hacking}{1975}}
\bibcite{hagmayerslw06}{\BCAY {Hagmayer, Sloman, Lagnado,{} \BBA{} Waldmann}{Hagmayer et\nobreakspace  {}al.{}}{in press}}
\bibcite{hastings70}{\BCAY {Hastings}{Hastings}{1970}}
\bibcite{heckerman98}{\BCAY {Heckerman}{Heckerman}{1998}}
\bibcite{heibeck87}{\BCAY {Heibeck \BBA{} Markman}{Heibeck \BBA{} Markman}{1987}}
\bibcite{hofmann99}{\BCAY {Hofmann}{Hofmann}{1999}}
\bibcite{huelsenbeckr01}{\BCAY {Huelsenbeck \BBA{} Ronquist}{Huelsenbeck \BBA{} Ronquist}{2001}}
\bibcite{jeffreysb92}{\BCAY {Jeffreys \BBA{} Berger}{Jeffreys \BBA{} Berger}{1992}}
\bibcite{jenkinsw65}{\BCAY {Jenkins \BBA{} Ward}{Jenkins \BBA{} Ward}{1965}}
\bibcite{jurafskym00}{\BCAY {Jurafsky \BBA{} Martin}{Jurafsky \BBA{} Martin}{2000}}
\bibcite{kassr95}{\BCAY {Kass \BBA{} Raftery}{Kass \BBA{} Raftery}{1995}}
\bibcite{kemppt04}{\BCAY {Kemp, Perfors,{} \BBA{} Tenenbaum}{Kemp et\nobreakspace  {}al.{}}{2004}}
\bibcite{kemppt06}{\BCAY {Kemp, Perfors,{} \BBA{} Tenenbaum}{Kemp et\nobreakspace  {}al.{}}{in press}}
\bibcite{kempt03}{\BCAY {Kemp \BBA{} Tenenbaum}{Kemp \BBA{} Tenenbaum}{2003}}
\bibcite{korbn03}{\BCAY {Korb \BBA{} Nicholson}{Korb \BBA{} Nicholson}{2003}}
\bibcite{kordingw06}{\BCAY {Kording \BBA{} Wolpert}{Kording \BBA{} Wolpert}{2006}}
\bibcite{lagnados04}{\BCAY {Lagnado \BBA{} Sloman}{Lagnado \BBA{} Sloman}{2004}}
\bibcite{landauerd97}{\BCAY {Landauer \BBA{} Dumais}{Landauer \BBA{} Dumais}{1997}}
\bibcite{lee06}{\BCAY {Lee}{Lee}{2006}}
\bibcite{lundb96}{\BCAY {Lund \BBA{} Burgess}{Lund \BBA{} Burgess}{1996}}
\bibcite{mackay03}{\BCAY {Mackay}{Mackay}{2003}}
\bibcite{mannings99}{\BCAY {Manning \BBA{} Sch\"utze}{Manning \BBA{} Sch\"utze}{1999}}
\bibcite{mansinghkaktg06}{\BCAY {Mansinghka, Kemp, Tenenbaum,{} \BBA{} Griffiths}{Mansinghka et\nobreakspace  {}al.{}}{2006}}
\bibcite{marr82}{\BCAY {Marr}{Marr}{1982}}
\bibcite{medins78}{\BCAY {Medin \BBA{} Schaffer}{Medin \BBA{} Schaffer}{1978}}
\bibcite{metropolisrrtt53}{\BCAY {Metropolis, Rosenbluth, Rosenbluth, Teller,{} \BBA{} Teller}{Metropolis et\nobreakspace  {}al.{}}{1953}}
\bibcite{minkal02}{\BCAY {Minka \BBA{} Lafferty}{Minka \BBA{} Lafferty}{2002}}
\bibcite{myungfb00}{\BCAY {Myung, Forster,{} \BBA{} Browne}{Myung et\nobreakspace  {}al.{}}{2000}}
\bibcite{myungp97}{\BCAY {Myung \BBA{} Pitt}{Myung \BBA{} Pitt}{1997}}
\bibcite{neal92}{\BCAY {Neal}{Neal}{1992}}
\bibcite{neal93}{\BCAY {Neal}{Neal}{1993}}
\bibcite{neal98}{\BCAY {Neal}{Neal}{1998}}
\bibcite{nelsonms98}{\BCAY {Nelson, McEvoy,{} \BBA{} Schreiber}{Nelson et\nobreakspace  {}al.{}}{1998}}
\bibcite{newmanb99}{\BCAY {Newman \BBA{} Barkema}{Newman \BBA{} Barkema}{1999}}
\bibcite{nisbettkjz83}{\BCAY {Nisbett, Krantz, Jepson,{} \BBA{} Kunda}{Nisbett et\nobreakspace  {}al.{}}{1983}}
\bibcite{norris97}{\BCAY {Norris}{Norris}{1997}}
\bibcite{nosofsky86}{\BCAY {Nosofsky}{Nosofsky}{1986}}
\bibcite{nosofsky98}{\BCAY {Nosofsky}{Nosofsky}{1998}}
\bibcite{oaksfordc01}{\BCAY {Oaksford \BBA{} Chater}{Oaksford \BBA{} Chater}{2001}}
\bibcite{oshersonswls90}{\BCAY {Osherson, Smith, Wilkie, Lopez,{} \BBA{} Shafir}{Osherson et\nobreakspace  {}al.{}}{1990}}
\bibcite{pearl88}{\BCAY {Pearl}{Pearl}{1988}}
\bibcite{pearl00}{\BCAY {Pearl}{Pearl}{2000}}
\bibcite{pitman93}{\BCAY {Pitman}{Pitman}{1993}}
\bibcite{reed72}{\BCAY {Reed}{Reed}{1972}}
\bibcite{rice90}{\BCAY {Rice}{Rice}{1995}}
\bibcite{rips75}{\BCAY {Rips}{Rips}{1975}}
\bibcite{russelln02}{\BCAY {Russell \BBA{} Norvig}{Russell \BBA{} Norvig}{2002}}
\bibcite{sloman05}{\BCAY {Sloman}{Sloman}{2005}}
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\bibcite{spirtesgs93}{\BCAY {Spirtes, Glymour,{} \BBA{} Schienes}{Spirtes et\nobreakspace  {}al.{}}{1993}}
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\bibcite{wellmang92}{\BCAY {Wellman \BBA{} Gelman}{Wellman \BBA{} Gelman}{1992}}
\bibcite{xutIP}{\BCAY {Xu \BBA{} Tenenbaum}{Xu \BBA{} Tenenbaum}{in press}}
\bibcite{yuillek06}{\BCAY {Yuille \BBA{} Kersten}{Yuille \BBA{} Kersten}{2006}}