Concepts in a Probabilistic Language of Thought
Noah D. Goodman∗, Stanford University
Joshua B. Tenenbaum, MIT
Tobias Gerstenberg, MIT
February 15, 2014
To appear in Concepts: New Directions, Eds. Margolis and Laurence, MIT Press.

1

Introduction

Knowledge organizes our understanding of the world, determining what we expect
given what we have already seen. Our predictive representations have two key properties: they are productive, and they are graded. Productive generalization is possible
because our knowledge decomposes into concepts—elements of knowledge that are
combined and recombined to describe particular situations. Gradedness is the observable effect of accounting for uncertainty—our knowledge encodes degrees of belief
that lead to graded probabilistic predictions. To put this a different way, concepts form
a combinatorial system that enables description of many different situations; each suchsituation specifies a distribution over what we expect to see in the world, given what wehave seen. We may think of this system as a probabilistic language of thought (PLoT)
in which representations are built from language-like composition of concepts and thecontent of those representations is a probability distribution on world states. The purpose of this chapter is to formalize these ideas in computational terms, to illustrate key
properties of the PLoT approach with a concrete example, and to draw connections
with other views of conceptual structure.
People are remarkably flexible at understanding new situations, guessing at unobserved properties or events, and making predictions on the basis of sparse evidence
combined with general background knowledge. Consider the game of tug-of-war: twoteams matching their strength by pulling on either side of a rope. If a team containing the first author (NG) loses to a team containing the third author (TG), that mightprovide weak evidence that TG is the stronger of the two. If these teams contain only
two members each, we might believe more in TG’s greater strength than if the teams
contain eight members each. If TG beats NG in a one-on-one tug-of-war, and NG goeson to beat three other individuals in similar one-on-one contests, we might believe that
TG is not only stronger than NG but strong in an absolute sense, relative to the general
population, even though we have only directly observed TG participating in a single
match. However, if we later found out that NG did not try very hard in his matchagainst TG, but did try hard in his later matches, our convictions about TG’s strength∗ Address correspondence to: ngoodman@stanford.edu.

1might subside.
This reasoning is clearly statistical. We may make good guesses about these propositions, but we will be far from certain; we will be more certain of TG’s strength after
seeing him play many games. But our reasoning is also highly abstract. It is not limited to a particular set of tug-of-war contestants. We can reason in analogous ways
about matches between teams of arbitrary sizes and composition, and are unperturbed
if a new player is introduced. We can also reason about the teams as collections: if
team Alpha wins its first four matches but then loses to team Bravo, whom it has not
faced before, we judge team Bravo very likely to be stronger than average. The smaller
team Bravo is, the more likely we are to judge a particular member of team Bravo to
be stronger than the average individual. And similar patterns of reasoning apply toinferences about skill and success in other kinds of team contests: We could be talking
about math teams, or teams for doubles ping-pong, and make analogous inferences for
all the situations above.
Our reasoning also supports inferences from complex combinations of evidence
to complex conclusions. For example, suppose that participants have been paired up
into teams of two. If we learn that NG was lazy (not trying hard) whenever his teamcontested a match against TG’s team, but NG’s team nonetheless won each of thesematches, it suggests both that NG is stronger than TG and that NG is often lazy. If
we then learned that NG’s teammate is stronger than any other individual in the population, we would probably revise the former belief (about NG’s strength) but not thelatter (about his laziness). If we learned that NG’s team had won all of its two-ontwo matches but we were told nothing about NG’s teammate, it is a good bet thatthe teammate—whomever he is—is stronger than average; all the more so, if we also
learned that NG had lost several one-on-one matches while trying hard.
Finally, our reasoning in this one domain can be modularly combined with knowledge of other domains, or manipulated based on subtle details of domain knowledge.
If we observed TG lifting a number of very heavy boxes with apparent ease, we mightreasonably expect his tug-of-war team to beat most others. But this would probably
not raise our confidence that TG’s math team (or even his ping-pong team) are likely
to be unusually successful. If we know that NG is trying to ingratiate himself to TG,
perhaps to receive a favor, then we might not weight his loss very heavily in estimating
strength. Likewise if we knew that NG had received a distracting text message during
the match.
We will return to this extended example in more detail in section 3, but for now wetake it merely as an illustration, in one simple domain, of the key features of humancognition that we seek to capture in a general computational architecture. How can
we account for the wide range of flexible inferences people draw from diverse patternsof evidence such as these? What assumptions about the cognitive system are needed
to explain the productivity and gradedness of these inferences? What kind of representations are abstract enough to extend flexibly to novel situations and questions, yet
concrete enough to support detailed quantitative predictions about the world? There are
two traditional, and traditionally opposed, ways of modeling reasoning in higher-levelcognition, each with its well-known strengths and limitations. Symbolic approaches(e.g. Newell, Shaw, & Simon, 1958) can naturally formulate a wide array of inferences
but are traditionally confined to the realm of certainty. They would be challenged to2capture all the gradations of reasoning people find so intuitive and valuable in an uncertain world. Probabilistic network approaches—whether based on neural networks(Rumelhart & McClelland, 1988) or Bayesian networks (Pearl, 1988, 2000)—support
graded inferences based on sparse or noisy evidence, but only over a fixed finite setof random variables. They lack the representational power and productivity of morestructured symbolic approaches, and would be hard-pressed to formulate in a coherent
fashion all of the inferences described above—let alone the infinite number of similarinferences we could have listed but did not.
More recently, researchers have begun to move beyond the dichotomy between
statistical and symbolic models (Anderson, 1996) and have argued that much of cognition can be understood as probabilistic inference over richly structured representations (Tenenbaum, Kemp, Griffiths, & Goodman, 2011). This has led to a proliferation of structured statistical models, which have given real insight into many cognitive
domains: inductive reasoning (Kemp & Tenenbaum, 2009), causal learning (Goodman, Ullman, & Tenenbaum, 2011; Griffiths & Tenenbaum, 2009), word learning(M. C. Frank, Goodman, & Tenenbaum, 2009; Piantadosi, Tenenbaum, & Goodman,2012), and mental state inference (Baker, Saxe, & Tenenbaum, 2009), to name a few.
But the computational tools used for these different models do not yet amount to atruly general-purpose or integrated approach. They are both insufficiently flexible—
requiring new models to be described for each different situation—and idiosyncratic—
requiring different representational tools across different specific domains.
We require a new computational architecture for cognition, grounded in a new theory of concepts—a theory that does justice to two distinct but equally important roles
that concepts play in mental life. On the one hand, concepts enable predictive generalization: they summarize stable regularities, such as typical distributions of objects,
properties and events. This is the role primarily addressed by prototype (Rosch, 1999),
exemplar (Nosofsky, 1986) and other statistical accounts of concepts. On the other
hand, concepts provide the basic building blocks of compositional thought: they canbe flexibly combined with other concepts to form an infinite array of thoughts in order to reason productively about an infinity of situations. They can be composed tomake new concepts, which are building blocks of yet more complex thoughts. Indeed,
concepts get much of their meaning and their function from the role that they play inthese larger-scale systems of thought. These are the roles primarily addressed (albeit itin different ways) by classical (rule-based) theories of concepts (Bruner, Goodnow, &Austin, 1967), and by the “theory theory” (Gopnik, 2003) and other accounts basedon inferential or conceptual roles (Block, 1997). While these theories differ in crucialways, we group them under the heading of symbolic approaches, because they highlight the compositional aspects of concepts that require a powerful symbol-processing
architecture.
Our goal in this chapter is to sketch a new account of concepts that combines these
two aspects, their statistical and symbolic functions, and to show how this account can
explain more of the richness of human reasoning than has been previously captured
using traditional approaches. We can phrase our hypothesis, somewhat informally, as:Probabilistic language of thought hypothesis (informal version): Concepts have a language-like compositionality and encode probabilistic knowl-

3

edge. These features allow them to be extended productively to new situations and support flexible reasoning and learning by probabilistic inference.
This view of the nature of concepts provides a deeper marriage of statistical inference and symbolic composition. Because they are probabilistic, concepts support
graded reasoning under uncertainty. Because they are language-like, they may be flexibly recombined to productively describe new situations. For instance, we have a setof concepts, such as “strength” and “game”, for the tug-of-war reasoning domain described above that we may compose with each other and with symbols referring toentities (individuals and teams) in the domain. These combinations then describe distributions on possible world states, which we may reason about via the rules of probability. Our proposal for the PLoT can be seen as making the statistical view of concepts
more flexible and systematic by enriching it with a fine-grained notion of composition
coming from symbolic approaches. It can also be seen as making symbolic approaches
to concepts more useful for reasoning in an uncertain world, by embedding them in aprobabilistic framework for inference and decision.
The level of description intended in the PLoT hypothesis is neither the highest level,of input-output relations, nor the lower level of psychological processing. Instead, we
aim to use the PLoT to describe conceptual representations and the inferences thatthey license across situations and domains. The process by which these inferences
are implemented is not directly part of the hypothesis, though it can be very useful toconsider the possible implementations when evaluating connections between the PLoT
and other views of concepts.

2Formalizing the PLoTThe PLoT hypothesis as stated above is an evocative set of desiderata for a theoryof concepts, more than a concrete theory itself. Indeed, it is not a priori obvious that
it is possible to satisfy all these desiderata at once in a concrete computational system.
We are in need of a compositional formal system—a language—for expressing probability distributions over complex world states. Our first clue comes from the idea ofrepresenting distributions as generative processes: the series of random steps by which
the world comes to be as it is. But while generative processes are a useful way torepresent probabilistic knowledge, adopting such a representation only transforms our
problem into one of finding a compositional language for generative processes. The
solution to this problem comes from a simple idea: if you have described a deterministic process compositionally in terms of the computation steps taken from start to end,
but then inject noise at some point along the way, you get a stochastic process; thisstochastic process unfolds in the original steps, except where a random choice is made.
In this way a distribution over outputs is determined, not a single deterministic output, and this distribution inherits all the compositionality of the original deterministic
process. The stochastic λ-calculus realizes this idea formally, by extending a universal
computational system (λ-calculus) with points of primitive randomness. The probabilistic programming language Church (Goodman, Mansinghka, Roy, Bonawitz, &Tenenbaum, 2008) extends the sparse mathematical system of stochastic λ-calculus
into a more usable system for describing statistical processes.
4

We now give a brief introduction to the syntax and ideas of probabilistic modelingusing Church, sufficient to motivate a more formal version of the PLoT; further details
and many examples can be found at http://probmods.org. Church uses a syntax
inherited from the LISP family of languages (McCarthy, 1960). Thus operators precede their arguments, and are written inside grouping parentheses: e.g., (+ 1 2) encodes
the operation “add 1 and 2”. We use define to assign values to symbols in our program
and lambda for creating functions. We could, for example, create a function double thattakes one number as an input and returns its double. The code would look like this:(define double (lambda (x)(+ x x))). Applying this function to 2 would look like (double 2)
and result in the value 4.
What differentiates Church from an ordinary programming language is the inclusion of random primitives. For example, the function (flip 0.5) can be interpreted as asimple coin flip with a weight (i.e. a Bernoulli random draw) outputting either true or
false. Every time the function is called, a new random value is generated—the coin isflipped. These random primitives can be combined just as ordinary functions are—for
instance (and (flip 0.5)(flip 0.5)) is the more complex process of taking the conjunctionof two random Booleans. A Church program specifies not a single computation, but a
distribution over computations. This sampling semantics (see Goodman, Mansinghka,
et al. (2008) for more details) means that composition of probabilities is achieved by ordinary composition of functions, and it means that we may specify probabilistic models
using all the tools of representational abstraction in a modern programming language.
We will not provide a primer on the power of function abstraction and other such tools
here, but we will use them in what we hope are intuitive and illustrative ways in theexamples below.
A number of language features in Church parallel core aspects of conceptual representation. Perhaps the most familiar (and most controversial) for cognitive modeling is
the use of arbitrary symbols. In Church (as in LISP) a symbol is a basic value that has
only the property that it is equal to itself and not to any other symbol: (equal? ‘bob ‘bob)is true, while (equal? ‘bob ‘jim) is false. (The single quote syntax simply indicates thatwhat follows is a symbol). Critically, symbols can be used as unique identifiers on
which to hang some aspect of conceptual knowledge. For instance they can be used
to refer to functions, as when we used define to create the double function above andthen reused this doubling function by name. Symbols can also be used together withfunctions to represent knowledge about (an unbounded set of) objects. For instance,
the function( define eyecolor ( lambda (x) ( if ( flip ) ‘ blue ‘ brown )))takes a person x and randomly returns1 an eye color (e.g. (eyecolor ‘bob) might return‘blue). That is, it wraps up the knowledge about how eye color is generated independently of which person is asked about—a person is simply a symbol (‘bob) that is
associated with another symbol (‘blue) by the eyecolor function.
Of course the above representation of an object’s property has a flaw: if we ask
about the eye color of Bob twice we may get different answers! Church includes an
1 In Church the conditional has a traditional but cryptic syntax: (if a b c) returns b if a is true, and cotherwise. Thus (if (flip)b c) randomly returns b or c.

5operator mem that takes a function and returns a memoized version: one that makes
its random choices only once for each distinct value of the function’s arguments, andthereafter when called returns the answer stored from that first evaluation. For instance,a memoized version of the eyecolor function,( define eyecolor ( mem ( lambda (x) ( if ( flip ) ‘ blue ‘ brown ))))

could output either ‘blue or ‘brown for Bob’s eye color, but only one of these possibilities,
to be determined the first time the function is called. This ensures that (equal? (eyecolor‘bob)(eyecolor ‘bob)) is always true.
Thus symbols can be used as “indices” to recover random properties or as labels
which allow us to recover stored information. These uses are conceptually very similar,
though they have different syntax, and they can be combined. For instance, we canaccess one function in another by its name, passing along the current objects of interest:( define eyecolor( mem ( lambda (x)( if ( flip 0.1)
( if ( flip ) ‘ blue ‘ brown )( if ( flip ) ( eyecolor ( father x)) ( eyecolor ( mother x)))))))

This (false, but perhaps intuitive) model of eye color asserts that the color is sometimes simply random, but most of the time depends on the eye color of one of a person’s parents—which is accessed by calling the father or mother function from inside theeyecolor function, and so on. Symbols and symbolic reference are thus key language
constructs for forming complex concepts and situations from simple ones.
How does reasoning enter into this system? The fundamental operation of belief
updating in probabilistic modeling is conditioning. We can define conditioning within
Church via the notion of rejection sampling: if we have a distribution represented bydist (a stochastic function with no input arguments) and a predicate condition (that takes
a value and returns true or false) then we can define the distribution conditioned on thepredicate being true via the process:( define conditional( lambda ()( define sample ( dist ))( if ( condition sample ) sample ( conditional ))))

That is, we keep sampling from dist until we get a sample that satisfies condition, thenwe return this sample.2 It can be cumbersome to split our knowledge and assumption
in this way, so Church introduces a syntax for conditionals in the form of the query
function:( query... definitions...
query-expression
condition-expression )Our initial distribution is the query-expression evaluated in the context of the ...definitions...,and our predicate is the condition-expression evaluated in the same context.
2 For readers familiar with Bayesian belief updating in probabilistic models, this process can be seen as
taking a prior model specified by dist and generating a sample from the posterior corresponding to distconditioned on the evidence that condition is true.

6For instance, referring again to the eye-color example, we could ask about Bob’s
mother’s likely eye color, given that Bob has blue eyes:( query( define eyecolor ... as above... )( eyecolor ( mother ‘ bob ))( equal ? ‘ blue ( eyecolor ‘ bob )))Notice that there is a distinction between the definitions, which represent probabilisticknowledge reusable across many queries, and the query and condition expressions,which represent the particular question of interest at the moment. In this example, theparticular people need to be introduced only in the question of interest because theconceptual knowledge is defined over arbitrary symbols.
Equipped now with a compositional formal language for representing distributions and performing conditional inference, we can revisit the probabilistic language ofthought hypothesis. Within a probabilistic language like Church, knowledge is encodedin stochastic function definitions. These functions describe elements of stochastic processes that can be composed together to describe various situations, to pose various
questions and to answer those questions with reasonable probabilistic guesses. Indeed,just as concepts are the stable and reusable components of human thinking, stochastic
functions are the units of knowledge encoded in a church program. Motivated fromthis observation we can formulate a stronger PLoT hypothesis:
Probabilistic language of thought hypothesis (formal version): Concepts are stochastic functions. Hence they represent uncertainty, compose
naturally, and support probabilistic inference.
Notice that this formal version realizes the informal PLoT in a precise way, showing that the hypothesis is coherent and allowing us to ask more detailed questions aboutplausibility. For instance we can begin to ask the usual philosophical questions about
concepts of this system: What constitutes meaning? How are concepts related? How
are they acquired and used? The answers to these questions can be subtle, but they
are determined in principle from the basic claim that concepts are stochastic functions.
For instance, on the face of it, the meaning of a stochastic function is simply its definition and the relation between concepts is determined by constituency—in the example
above, the meaning of eyecolor is its definition and it is related to other concepts only
by its use of mother and father functions. However, when we consider the inferential
relationships between concepts that come from conditional inference—query—we see
additional aspects of meaning and conceptual relation. Conditioning on parentage can
influence eye color, but also vice versa; conditioning on hair color may influence judgements about eye color indirectly, and so on. In the next section we give an extended
example in the domain of simple team games, illustrating these foundational issues aswell as exploring the empirical adequacy of the PLoT hypothesis.

3

Example: Ping pong in Church

Consider the information shown in Figure 1. Most people conclude that TG is relatively strong, while BL is average-to-weak. Below we describe various patterns ofevidence that we displayed to people in the guise of a ping pong tournament. How can7

Figure 1: Screenshot of a single player tournament. The winner of each match isindicated by a laurel wreath.
we account for people’s sensitivity to the uncertain nature of the evidence in such situations? While capturing the abstract, symbolic structure that remains invariant between
this particular situation and other similar situations that could involve different players,
teams, and outcomes?
This simple sports domain is built around people, teams and games. In Church,
we can use symbols as placeholders for unspecified individuals of these types. Thismeans that we do not need to define in advance how many people participate, what
the size of the teams will be, or how many games a tournament will have. Instead, we
begin by describing a lexicon of abstract concept definitions useful for reasoning about
these games. We define an individual player’s strength, personstrength, via a functionthat draws from a Gaussian distribution (with arbitrary M = 10 and SD = 3):( define personstrength ( mem ( lambda ( person ) ( gaussian 10 3) )))Memoization ensures that the strength value assigned to a person is persistent and does
not change between games. However, we assume that players are sometimes lazy on agiven match. The chance of a person being lazy in a particular game is 10%:( define lazy ( mem ( lambda ( person game ) ( flip 0.1) )))The overall strength of a team, on a given game, is the sum of the strength of each
person in the team. If a person in the team is lazy, however, he only plays with half of
his actual strength.
( define teamstrength( mem ( lambda ( team game )( sum ( map ( lambda ( person )( if ( lazy person game )(/ ( personstrength person ) 2)
( personstrength person )))team )))))

8

Table 1: Patterns of observation for the single player tournaments. Note: An additional
set of 4 patterns was included for which the outcomes of the games were reversed. The
bottom row shows the omniscient commentator’s information in Experiment 2. For
example, in the confounded case, player B was lazy in the second game.
confoundedevidence(1,2)

strong indirect
evidence(3,4)

weak indirect
evidence(5,6)

diverseevidence
(7,8)

A > B
A > B
A > B

A > B
B > C
B > D

A > B
B < C
B < D

A > B
A > C
A > D

lazy,game: B,2

B,1

B,1

C,2

Note: A > B means that A won against B.

Finally, we specify how the winner of a game is determined. We simply say the theteam wins who has the greater overall strength:( define winner( mem ( lambda ( team1 team2 game )( if (< ( teamstrength team1 game ) ( teamstrength team2 game ))‘ team1 ‘ team2 ))))

This set of function definitions specifies a simple lexicon of concepts for reasoning
about the ping pong domain.
The way in which we can define new concepts (e.g. teamstrength) based on previously defined concepts (personstrength and lazy) illustrates one form of compositionality
in Church. The set of concept definitions refers to people (teams, etc.) without having to declare a set of possible people in advance: instead we apply generic functions
to placeholder symbols that will stand for these people. That is, the concepts may
be further composed with symbols and each other to describe specific situations. For
instance, the inference in Figure 1 can be described by:( query... CONCEPTS...
; The query:( personstrength ‘ TG ); The evidence:( and(= ‘ team1 ( winner ‘( TG ) ‘( NG ) 1) )(= ‘ team1 ( winner ‘( NG ) ‘( AS ) 2) )(= ‘ team1 ( winner ‘( NG ) ‘( BL ) 3) )))Here ...CONCEPTS... is shorthand for the definitions introduced above—a lexicon of concepts that we may use to model people’s inferences about a player’s strength not onlyin the situation depicted in Figure 1 but in a multitude of possible situations with varying teams composed of several people, playing against each other with all thinkablecombinations of game results in different tournament formats. This productive extension over different possible situations including different persons, different teams anddifferent winners of each game, renders the Church implementation a powerful modelfor human reasoning.
We wanted to explore how well our simple Church model predicts the inferences
people make, based on complex patterns of evidence in different situations (cf. Ger9

A wins
1

3

A wins57

1.5
strength

strength

1.5

1

0.5

9

11

13

15

17

19

10

12

14

16

18

20

strongindirectweakindirectdiverse

round
robin

1

0.5

0

0

A loses2

4

A loses
6

8

1.5

weakness

weakness

1.5

1

0.5

0

confoundedstrongindirectweakindirect

1

0.5

0

diverseconfounded confoundedpartner
opponent(a) Single player tournament(b) Two-player tournament

Figure 2: Mean strength estimates (grey bars) and model predictions (black bars) for
the single player (left) and two-player tournaments (right). The top row shows strength
judgments for cases in which the player won her game(s). The bottom row shows weakness judgments for cases in which the player lost. Numbers above the bars correspondto the patterns described in Tables 1 and 2. Error bars are ± 1 SEM.
stenberg & Goodman, 2012). In Experiment 1, participants’ task was to estimate anindividual player’s strength based on the outcomes of different games in a ping pong
tournament. Participants were told that they will make judgments after having seen single player and two-player tournaments. The different players in a tournament could be
identified by the color of their jersey as well as their initials. In each tournament, there
was a new set of players. Participants were given some basic information about thestrength of the players which described some of the modeling assumptions we made.
That is, participants were told that individual players have a fixed strength which does
not vary between games and that all of the players have a 10% chance of not playing
as strongly as they can in each game. This means that even if a player is strong, he cansometimes lose against a weaker player.3
Table 1 shows the patterns of evidence that were used for the single player tournaments. Table 2 shows the patterns for the two-player tournaments. In all tournaments,
participants were asked to judge the strength of player A. For the single player tournaments, we used four different patterns of evidence: confounded evidence in which A
wins repeatedly against B, strong and weak indirect evidence where A only wins one
match herself but B either continues to win or lose two games against other players
and diverse evidence in which A wins against three different players. For each of thosepatterns, we also included a pattern in which the outcomes of the games were exactly
reversed. For the two player tournaments, we used six different patterns of evidence: Insome situations A was always in the same team as B (confounded with partner) whilein other situations A repeatedly played against the same player E (confounded with
opponent). As in the single player tournaments, we also had patterns with mostly indirect evidence about the strength of A by having his partner in the first game, B, either
win or lose against the same opponents with different teammates (weak/strong indirect
3 Demos of the experiments can be accessed here:

http://web.mit.edu/tger/www/demos/BPP_demos.html

10evidence). Finally, we had one pattern of diverse evidence in which A wins with different teammates against a new set of opponents in each game and one round robin
tournament in which A wins all his games in all possible combinations of a 4-player
tournament. Further experimental details can be found in Appendix A.
In order to directly compare the model predictions with participants’ judgments
we z-scored the model predictions and each individual participant’s judgments. Furthermore, we reverse coded participants’ judgments and the model predictions for thesituations in which the outcomes of the games were reversed so that both strength and“weakness” judgments go in the same direction.
Figure 2 shows the mean strength estimates (gray bars) together with the model predictions (black bars) for the single and two-player tournaments. The top panels display
the situations in which A won his game(s). The bottom panels show the situations inwhich A lost. Our model predicts participants’ judgments in the single and two-player
tournaments very well with r = .98 and RMSE = .19. A very high median correlation
with individual participants’ judgments of r = .92 shows that the close fit is not merely
due to an aggregation effect. These results show that our model predicts participants’
inferences very accurately—a single, concise representation of the task is sufficient topredicts people’s inferences for a great diversity of patterns of evidence.
A still greater variety of evidence is available by composing the basic concepts
together in different ways: there is no reason for evidence not to directly refer to aplayer’s strength, laziness, etc. For instance:( query... CONCEPTS...
; The query:( personstrength ‘ TG ); The evidence:( and(= ‘ team1 ( winner ‘( TG ) ‘( NG ) 1) )(= ‘ team1 ( winner ‘( NG ) ‘( AS ) 2) )(= ‘ team1 ( winner ‘( NG ) ‘( BL ) 3) )( lazy ‘ NG 1) )) ; additional kinds of evidence ( Expt. 2) .

While in Experiment 1, the match results were the only source of information participants could use as a basis for their strength judgments, Experiment 2 introduced
an omniscient commentator who gave direct information about specific players. After
participants saw a tournament’s match results, an omniscient commentator, who always
told the truth, revealed that one player was lazy in a particular game. We were interested in how participants updated their beliefs about the strength of player A given thisadditional piece of evidence. Importantly, we do not need to change anything in theconcept definitions to derive predictions for these situations, since only the way they
are composed into evidence changes.
Figure 3 shows the mean strength judgments (gray bars) together with the model
predictions (black bars, see Table 1 for the different patterns of evidence). The darkgray bars indicate participants’ first judgments based on the tournament information
only. The light gray bars indicate participant’s second judgments after they received thecommentator’s information. The model predicts participants’ ratings very accurately
again with r = .97 and RMSE = 0.29. The model’s median correlation with individual
participants’ judgments is r = .86. These results show that participants, as well as ourmodel, have no difficulty in integrating different sources of evidence to form an overall
11

Table 2: Patterns of observation for the two-player tournaments. Note: An additional
set of 6 patterns was included in which the outcomes of the games were reversed.
confoundedwith partner(9,10)
AB
AB
AB

AB
BC
BD

confoundedwith opponent(11,12)

>
CD
>
EF
>
GH
weak indirect
evidence(15,16)
>
>
>

EF
EF
EF

strong indirect
evidence(13,14)

AB
AC
AD

>
>
>
diverseevidence(17,18)

EF
EG
EH

AB
BC
BD

>
<
<
round
robin
(19,20)

EF
EF
EF

AB
AC
AD

>
>
>

EF
GH
IJ

AB
AC
AD

>
>
>

CD
BD
BC

judgment of a player’s likely underlying strength. The model predicts participants’
judgments very accurately by being sensitive to the degree to which the initial strength
estimate should be updated in the light of new evidence provided by the commentator.

4

Intuitive theories

The examples above provide concrete illustrations of how to represent concepts asfunctions in a probabilistic language of thought, how a system of such concepts supports inferences that are both productive and probabilistic, and how these inferences
can capture the outputs of human reasoning at a high level of quantitative accuracy.
But while reasoning about team games like ping pong is very illustrative, it is of relatively limited scope compared to many of the concepts involved in everyday humanreasoning. In this section we discuss how the same machinery can describe abstract
concepts that are the backbone of thinking about everyday life, and which have often
not fit easily into more traditional formal frameworks.
Intuitive theories (Carey, 2009; Gopnik & Wellman, 2012; Wellman & Gelman,
1992), like their more familiar scientific counterparts, are comprised of a system of interrelated and inter-defined concepts articulating a basic ontology of entities, the properties of and relations between those entities, and the causal laws that govern how these
entities evolve over time and interact with each other. For instance intuitive physics isa system for reasoning about physical objects, and intuitive psychology for reasoning
about intentional agents. These are called “theories” because, like in scientific theories, the essential constructs of intuitive theories are typically not directly observable.
Yet intuitive theories also specify how unobservable states, properties and processes
do impact observable experience—and thus how they support competencies such asprediction, explanation, learning and reasoning.
Intuitive theories can be found in some form in young infants, and are also to someextent shared with many other species; they are arguably the earliest and oldest abstract
concepts we have (Carey, 2009). They provide the scaffolding for many of children’sconceptual achievements over the first few years of life. They also provide core building blocks for meaning in natural language, at the same time as they are enriched12

A wins

strength

1.5

1

3

5

7

IR1

1
0.5
0

A loses

weakness

1.5

2

4

6

8

IR2

confoundedstrongindirectweakindirectdiverseirrelevant

1
0.5
0

Figure 3: Mean strength estimates and model predictions. Dark grey bars = estimates
after tournament information only, light grey bars = estimates after omniscient commentator info, black bars = model predictions. Error bars are ± 1 SEM. Note: Situations IR1 and IR2 were cases in which the information given by the commentator was
irrelevant; see Appendix B for details.
and transformed fundamentally as children develop their natural language abilities.
Through adulthood, they continue to serve as the basis for our common-sense understanding of the world. Yet while these intuitive theories have long been a prime target
for exploration by developmental and cognitive psychologists, linguists, and philosophers, they have not received much treatment in the literature on formal models ofconcepts. This may be because they do not fit well into the general-purpose mathematical and computational modeling frameworks that have been available and useful for
more mundane category concepts—prototypes, exemplars, and logical accounts.
Starting around ten years ago, several authors began to consider Bayesian networks
as formal models for intuitive theories (Goodman et al., 2006; Gopnik et al., 2004;
Rehder, 2003; Tenenbaum & Griffiths, 2003), focusing on their causal aspects. These
efforts were ultimately limited by the fact that Bayesian networks, like neural networks
before them, fail to capture genuine productivity in thought. An intuitive theory ofphysics or psychology must be able to handle an infinite range of novel situations, differing in their specifics but not their abstract character, just as we illustrated above ona much smaller scale for an intuitive theory of tug-of-war or ping pong. Hierarchical Bayesian models have been proposed as one way to increase the representational
power of Bayesian networks, and they have given reasonable accounts of some aspects
13

of abstract causal theories (e.g. Tenenbaum et al., 2011). But hierarchical Bayesianmodels on their own still lack sufficient representational power to address the finegrained compositionality inherent in our intuitive theories The PLoT allows us to takea major step forward in this regard. Both Bayesian networks and hierarchical Bayesianmodels of intuitive theories can be naturally written as Church programs, preserving
their insights into causal reasoning and learning, but Church programs go much further in letting us capture the essential representations of common-sense physics andpsychology that have defied previous attempts at formalization within the probabilistic
modeling tradition.
To illustrate, consider how we might capture the core concepts of an intuitive
psychology—a probabilistic model of how agents act rationally in response to theirmental states, aiming to satisfy their desires as efficiently as possible given their beliefs. As as initial step, imagine extending the game model above to take into accountthe fact that laziness for a particular player in a given game may not simply be a random event, but an intentional choice on the part of a player—he may estimate that theother team is so weak that it is not worth his effort to try hard. We pose this as a churchmodel by imagining that a player asks himself “how should I act, such that my teamwill win?”; this translates into a query:( define lazy ( mem ( lambda ( person game )( query( define action ( flip L))
action(= ( teamof person ) ( winner ( team1of game ) ( team2of game ) game ))))))

where we have helped ourselves to some innocuous helper functions to look up theteam of a player and so on. The parameter L controls the a priori tendency to be lazy;
this gives a simple way of including a principle of efficiency: a tendency to avoid undue
effort. The condition statement of the query specifies the player’s goal—for his team
to win the game—hypothetically assuming that this goal will be achieved. The outputof the query is an action (trying hard, or not) that is a reasonable guess on the player’s
part for how that goal may be achieved. An inference about which team will win amatch now leads to a sub-inference modeling each player’s choice of whether to exert
their full effort, given the players on each team. We could further extend this model
to take into account private evidence that each player might have about the strengthsof the other players, expressing his or her process of belief-formation about the total
strengths of the two teams as an additional set of nested sub-inferences.
The pattern of using an embedded query to capture the choices of another agent is
a very general pattern for modeling intuitive psychology (Stuhlmüller & Goodman,2013). We could write down the abstract structure schematically as:( define choice ( lambda ( belief state goal ?)( query( define action ( action-prior ))action( goal ? ( belief state action )))))

where belief is taken to be the agent’s summary of the world dynamics (transitions
from states to states, given actions), and goal? is a goal predicate on states picking out
those that the agent desires. Of course many additional refinements and additions may
be needed to build an adequate model of human intuitive psychology—agents form
14

A

A

A

A
B

B

B

B

B

B

B

B

a)

b)
A

A

B

B
A

A
B

B

B

B

B

B

c)

B

d)

Figure 4: Diagrammatic illustrations of four collision events in a simple physics world.
Note: Solid arrows represent the actual trajectories of ball A before the collision andof ball B before and after the collision. Dashed arrows and faded circles represent
the counterfactual trajectory of ball B. The brown rectangle, yellow rectangle and blue
circle represent a brick, and the entry and exit of a teleport, respectively.
beliefs, experience emotions, and so on. Yet we hope that the ease of writing down evena simple version of a theory of mind that captures both the theory’s abstractness and its
potential for productive probabilistic inferences illustrates the power of the PLoT view.

5

Mental simulation and sampling

There is a central link between the sampling semantics of Church programs, mental
simulation, and the causality central to many intuitive theories. A Church program
naturally expresses the causal, generative aspect of people’s knowledge through thefunction dependencies in the program. The function dependencies dictate the causal
flow of the sampling process: functions whose outputs serve as an input to anotherfunction must be evaluated first. Each run of a Church program can be interpreted as
the dynamic generation of a possible world that is consistent with the causal laws as
specified in the program (Chater & Oaksford, 2013). Because the sampling process isstochastic, a Church program specifies a probability distribution over possible worlds,
and different modes of reasoning can be seen as different forms of mental simulation
on top of this basic sampling process. While the notion of mental representation andsimulation of possible worlds has had many advocates (Craik, 1967; Hegarty, 2004;
Johnson-Laird, 1983, 2006), the PLoT view integrates this idea naturally into a viewof mental models that is also probabilistic, causal and sufficiently expressive to capture
core intuitive theories.
We will illustrate the implications of this view via a concrete example from the15domain of intuitive physics: people’s judgments about the causal relations among the
trajectories of physical objects in motion. We are concerned with judgments of “actual
causation”: whether one specific event caused another specific event to happen. Consider a relatively simple scenario which consists of two billiard balls A and B, somesolid walls with an opening gate, a brick, and a teleport gate that can be either active
or inactive. Figure 4 shows diagrammatic illustrations of causal interactions between
A and B in this world, assuming simple Newtonian elastic collisions between moving
bodies. The question of interest is whether ball A’s collision with ball B caused ball B
to go through the red gate on the left of the screen, prevented it from going through,or did neither. The tools of probabilistic programs and sampling-based inference allow us to give a precise formal account of these causal judgments, which also accords
well with intuitions of mental simulation and gives strong quantitative fits to behavioral
experiments.
To explain these judgments, we first need to be able to represent the relevant physical knowledge at the right level of abstraction. Despite its simplicity, our domain
already affords an infinite number of interactions between A and B and we want amodel that yields a causal judgment for each possible situation. Rather than having to
specify a new model for each causal interaction of interest (as we would have to do
if we adopted a Bayesian network formulation (Pearl, 2000)), we want to represent
the general laws that govern the interactions between the objects in our world. One
way of representing people’s knowledge of physical object motion in Church is by
writing down a probabilistic and approximate version of some aspects of Newtonianmechanics. Functions in the Church program compute the inertial time-evolution andthe outcome of collisions by taking as input the mass and velocity of objects as well as
more general aspects of the world such as friction and gravity. So far these are standard,deterministic simulation routines (so we leave out details). Critically, we also assume
that some noise in each object’s momentum is inserted just after each collision, andperhaps at other times as well, resulting in trajectories that are noisy versions of their
Newtonian counterparts. Recent research has shown that people’s intuitive physicaljudgments in several domains are well described by such noisy Newtonian simulations(Battaglia, Hamrick, & Tenenbaum, 2013; Sanborn, Mansinghka, & Griffiths, 2013;
K. Smith, Dechter, Tenenbaum, & Vul, 2013; K. A. Smith & Vul, 2012). Once
we have a Church program that captures people’s intuitive physics, we can model predictions about the future (e.g., will ball B go through the gate?) as simple forward
simulations, and inferences about the past (e.g., where did ball B likely come from?)by a query of the past given the present—simulating possible histories that could haveled up to the current state.
More subtly, a Church program can also be used to evaluate counterfactuals (e.g.,would ball B have gone through the gate if the collision with A hadn’t happened?). In
line with Pearl (2000), the evaluation of counterfactuals in a Church program involves
three steps: First, we condition all the random choices in the program based on what
actually happened to estimate the unobserved values of the actual world. Second, we
realize the truth of the counterfactual antecedent (e.g. that the collision did not happen) by intervening in the program execution that generated the actual world. This
intervention breaks the normal flow of the program by setting some function inputs todesired values. For example, to model what would have happened if there had been no16

collision between A and B, we could set ball A’s velocity to zero or move ball A outsideof the scene shortly before the time of collision. Finally, to evaluate the truth of thecounterfactual, we reevaluate all the functions downstream from the point at which weintervened in the program. This process generates a sample over counterfactual worldstates, and repeatedly running this process allowing for different stochastic functions
evaluations can be used to express people’s uncertainty over what would have happened in the relevant counterfactual world. Notice that the key feature of Church thatallows this process to work is that it specifies a process for sampling particular situations and makes explicit the steps of the causal history that lead up to a situation (in theform of a program execution trace). Counterfactuals are then evaluated by a series of“simulation” steps that result in imagined counterfactual worlds.
In a series of empirical studies, we have shown that people’s quantitative judgments of actual causation are closely linked to such a probabilistic counterfactual analysis (Gerstenberg, Goodman, Lagnado, & Tenenbaum, 2012). When judging whether
one event caused another event to happen, people compare what actually happened
with what they think would have happened in the counterfactual world in which thecandidate causal event had been absent. They appear to estimate something like theprobability that the candidate cause was necessary to produce the outcome event: theprobability that the outcome, which did in fact occur, would not have occurred in acounterfactual world where the candidate cause was absent. Consider the top pair ofdiagrams shown in Figure 4. In both clips, what actually happened is identical. However, the outcome in the relevant counterfactual worlds would likely have been different. In Figure 4a, ball B would have bounced off the brick if it hadn’t collided with ball
A. In contrast, in Figure 4b, ball B would have gone through the gate even without thecollision with ball A. As predicted by our account, people’s judgments about whether
A caused B to go through the gate are significantly higher for Figure 4a compared to
Figure 4b (Gerstenberg, Goodman, Lagnado, & Tenenbaum, submitted). In the bottom
pair of cases, the contrast in the relevant counterfactual worlds was realized by comparing situations in which the teleport was either on (Figure 4c) or off (Figure 4d). Whilepeople judged that A prevented B from going through the gate in Figure 4c, they didn’t
think that A prevented B in Figure 4d. In Figure 4c, ball B would have gone through thegate via the teleport if it hadn’t collided with A. In Figure 4d, in contrast, the teleport
was off and B would not have gone through the gate even if there had been no collision with A. The fact that people’s judgments differ dramatically between situations
in which what actually happened was held constant supports the idea that causal andcounterfactual judgments are inextricably linked. It also shows that it is not possible
to give an account of people’s causal judgments just in terms of what actually happened (e.g. Wolff, 2007). Finally, it demonstrates the flexibility of people’s intuitive
theories and the critical way in which this flexibility supports mental simulation, counterfactual and causal reasoning. Once people have learned how the teleport works, they
have no trouble imagining its effects and incorporating these into their counterfactual
simulations, whose outcome in turn influences their causal judgments.
While we have focused here on the example of how we can explain people’s causal
judgments in the domain of intuitive physics, the general framework applies equallywell to any domain for which we are able to represent our knowledge in terms of a
Church program. For example, we could model people’s judgments about whether
17agent A’s argument convinced agent B to try harder as a function of what actually happened and people’s subjective degree of belief that B would still have tried harder had
A not said anything. A Church program that captures people’s intuitive understandingof psychology looks different from a Church program that captures people’s intuitive
understanding of physics, but we can understand people’s causal judgments in termsof the same process that compares the actual outcome with the outcomes of mental
simulations (sampling) of the relevant counterfactual worlds.

6

Concepts and natural language

Thus far we have sketched a notion of concepts as stochastic functions in Churchand intuitive theories as systems of interrelated concepts. We have also described how
such intuitive theories can be used to describe the complex causal knowledge that people use to reason about the world. We would additionally like a theory of concepts
to help us formulate the meanings of words, and more generally the ways that natural language can be used to convey thought. In Goodman and Lassiter (to appear)
we use Church to formulate an architecture for language understanding in which word
meanings are grounded into intuitive theories, and utterance interpretation is a rich inferential process starting with these meanings. We summarize here the parts of thisarchitecture that illuminate the role of concepts in thought.
As described above, we view intuitive theories as collections of function definitions; together they form a distribution over all the expressions that can be composedfrom these functions—this constitutes prior knowledge of the domain. We posit thatthe foundational operation of natural language interpretation is to update this prior belief distribution into a posterior distribution. Because belief update in a probabilistic
system happens by conditioning, we need utterances to lead to conditions that can beused in a query. That is, there must be a meaning function that maps from strings of words
to Boolean-valued expressions in the PLoT (i.e. expressions which can be the condition of a query). This meaning function is essentially the “narrow” language facility(Hauser, Chomsky, & Fitch, 2002), mapping from sound strings to the PLoT. In Goodman and Lassiter (to appear) we describe the meaning function in two (fairly standard)
steps: First we look up the PLoT expression for each word in a linguistic lexicon (anassociation between words an PLoT expressions), then we compose these expressions
recursively until we have built a meaning for the whole sentence. These steps may
be non-deterministic, but additional uncertainty causes no difficulties since we are already within a probabilistic framework. Thus, the meaning function lets us constructfrom a natural language expression a condition-expression that can be used in query to
update beliefs about some question—the query-expression. But notice that while they
are constructed in the same PLoT these two expressions play very different cognitiveroles: the query-expression is a question about the world; the condition-expression is aconstraint on the causal process that generates the answer to this question.
The architecture is reminiscent of Jackendoff’s “languages of thought” (Jackendoff, 1995), in which there are several modules (for example, natural language andcognition) each with their own representation language, and interfaces for translatingfrom one module to another. In our approach the representation languages are mathematically similar for natural language and cognition (based in the stochastic lambda

18calculus) and their “interface” is defined by their roles in the inference (query) of language interpretation. Despite their mathematical similarities, the different cognitiveroles of these different kinds of expressions imply two different, but interlocking, principles of compositionality in the cognitive architecture. One instance of compositionality allows us to build rich (distributions on) generative histories, while the other allows
us to build up complex conditioning statements to constrain these histories. A naive
approach would try to make these two kinds of composition directly compatible, by
requiring that each natural language constituent describe a probability distribution andrelying on linguistic composition to combine these distributions. Our approach allows these distinct modes of composition to apply separately in natural language andthought, resulting in complex interactions that can look non-compositional when only
one type of representation is considered.
Meaning conveyed by natural language is further enriched by pragmatic inference.
As described elsewhere (M. Frank & Goodman, 2012; Goodman & Lassiter, toappear; Goodman & Stuhlmüller, 2013; Stuhlmüller & Goodman, 2013) a broad
range of pragmatic inferences can be understood as the output of conditioning in an
intuitive theory of communication, and can also be formalized using the PLoT tools
described above. Adding this layer of inference results in futher complexities andcontext-sensitivities to the effective relationship between words and concepts, but isessential for understanding how we talk about our thoughts and understand what otherpeople mean to say.

7

Concept acquisition

If concepts are definitions in a library of useful (stochastic) functions, what is concept learning? Forming new concepts from examples is fundamentally a problem of
induction—in our case the problem of program induction. This can be formulated as
Bayesian inference of a set of concepts that best explain the experience we have inthe world: conditioned on generating the examples we have seen, what is the likely
new concept? Hypothesized concepts are formed in an effective language of thought
based on the concepts learned so far—all the expressions that can be formed by composing the underlying PLoT and the already-defined function symbols. We can view
these hypotheses as being generated by a higher-order “program-generating program,”a stochastic function that generates candidate stochastic functions that might explain agiven set of observed examples. Concept learning as probabilistic program induction isphilosophically and mathematically well-posed, but a great deal of research is neededboth to reduce it to useful engineering practice and to validate it as a model of humanconcept learning. Induction over such an infinite combinatorial space is simply statedas probabilistic conditioning, but such inferences are extremely challenging to implement in general. Yet recent progress has shown that this approach can be successful
in certain cases: grammar-based program induction has been used to describe category
learning (Goodman, Tenenbaum, Feldman, & Griffiths, 2008), learning of relational
concepts (Kemp, Goodman, & Tenenbaum, 2008), learning simple visual concepts(Lake, Salakhutdinov, & Tenenbaum, 2013), and learning of number concepts (Piantadosi et al., 2012).
Notice that in this notion of inductive program elaboration, each concept begins life19

as simply a new symbol to which a function will come to be attached. The impetus to
add such a placeholder symbol may come from natural language (upon hearing a newword), from the interaction of knowledge about natural kinds and specific examples(as suggested by Margolis (1998) and Carey (this volume)), or from other explanatory
pressures. Richer content and relationships to other concepts would be incorporated
into the web of function definitions inductively, as the learner encounters additional
examples and gains experience in the new domain.
Language, social context, and other factors may play a critical role in positing
both the existence and the content of new concepts. For instance, Shafto, Goodman,and Frank (2012) argue that the social context of examples (for instance, that they are
generated communicatively by a helpful person) can strongly impact the inferences
made. Similarly, language can provide a strongly constraining form of evidence forconcept formation—for instance (Piantadosi et al., 2012) use a linguistic count-list as
a key input in learning numerical concepts. The PLoT offers a powerful way to think
about these and other bootstrapping phenomena at the interface of social interaction,
language acquisition and concept acquisition, such as the contributions of syntactic andsemantic bootstrapping in learning verbs, or the contributions of pragmatic inference inlearning quantifiers. Again, further research is needed to understand how these factors
integrate with inductive learning from examples in a full theory of concept acquisition;
what is important for us here is that the PLoT provides us with a theory of concepts,
and an approach to concept learning as probabilistic inference, that is able to explore
these interactions in a productive way.
One important implication of the inductive view just indicated is that concept learning changes the effective language of thought. While this effective language has thesame mathematical expressivity as the underlying PLoT, particular thoughts may bevastly simpler (and thus more cognitively tractable) in the effective language. Changing the effective language, by adding a new concept, then affects a number of cognitive
functions. For instance, future concept induction will take place in the new effective language, which provides a different inductive bias than the original. In this way,concepts which are complex and unlikely to be constructed by a learner initially may
become simpler and more plausible later on in the process of elaborating her conceptual library. This process may be a critical driver of children’s long-term cognitive
development.

8

Summary and next steps

We have argued that concepts should be viewed as the stable representations of aprobabilistic language of thought—more formally, as functions in an enriched stochastic lambda calculus. This view allows fine-grained compositionality while supporting
reasoning by probabilistic inference. Compositionality is key to explaining the productivity of thought, while probabilistic inference explains graded and successful reasoning in an uncertain world. The PLoT hypothesis seeks to explains complex humancognition in a way that previous formal theories of concepts have not, and helps us tounderstand many topics in everyday cognition with both new qualitative insights andquantitative accuracy.
Importantly, the PLoT hypothesis builds on and unifies many attractive aspects of20

previous views on concepts. Like classical and symbolic theories the PLoT puts compositionality and symbolic scaffolding at center stage. Unlike these theories however,but very much in the spirit of prototype, exemplar, and connectionist approaches, the
PLoT explains why human reasoning is graded and why this is useful. It does so byborrowing from probability theory, and modern Bayesian modeling approaches, the basic mechanics of reasoning under uncertainty. This reliance on probabilistic inference
makes a natural connection to inferential role notions of concept meaning: it is not
merely the proximal definitions, but the complex ways that information flows underinference that matter in practice. Rather than working at the level of monolithic probability distributions, the stochastic lambda calculus and Church allow us to work fromthe point of view of generative, sampling systems. This in turn makes a key connection to mental simulation and—poetically, but perhaps also literally—the importanceof imagination in thinking.
For work on Bayesian models of cognition the PLoT view holds particular importance. Recent advances in building probabilistic models of higher-level cognition
share the basic mechanics of probabilities and many aspects of their philosophy, butthey bring a bewildering and heterogenous array of additional representational tools
and claims. The view presented here serves as a key unification by showing that allof these Bayesian models can be represented in, and hence reduced to, a simple system built from little more that function abstraction and random choice. It gives hopethat advances in probabilistic models of targeted domains are compatible with eachother and can ultimately be combined into a broader architecture for modeling humanknowledge, reasoning and learning.
Church models, and the PLoT more generally, are intended to capture the representations of knowledge people use to reason about the world, and the inferences that aresupported by this knowledge. They are not intended to convey the algorithmic processof this inference, much less the neural instantiation. Indeed, establishing connections
to these other levels of psychological analysis is one of the key future challenges for
the PLoT hypothesis; others being further broadening of scope and demonstration of
empirical adequacy within higher-level cognition. The implementations of Church, atthe engineering level, suggest one set of ideas to motivate psychological process models. Indeed, implementations of Church query work through various combinations ofcaching and Monte Carlo simulation, which provide a very different view of computation than one might expect from a course on probability: not so much arithmetic
tabulation as noisy dynamical systems tuned to result in samples from the desired distributions. Long engineering practice shows that these algorithms can give efficientsolutions to tough statistical inference problems; recent work on probabilistic programing languages (e.g. Wingate, Stuhlmüller, & Goodman, 2011) shows that they can berealized in general-purpose ways suitable to a PLoT. Recent work has provided initialconnections between such inference algorithms and human cognitive processes (e.g.
Griffiths, Vul, & Sanborn, 2012). Yet classic and ongoing work on cognitive architecture, concepts, and neural dynamics all have additional insights that must also be
understood in moving the PLoT toward the process and neural levels.

21

9

Acknowledgments

This work was supported by NSF STC award CCF-1231216, ONR awards N0001409-1-0124 and N00014-13-1-0788, and a John S. McDonnell Foundation Scholar Award.

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A

Experiment 1

A.1

Participants

30 (22 female) recruited through Amazon Mechanical Turk participated in the experiment.
The mean age was 31.3 (SD = 10.8).

A.2

Materials and Procedure

The experiment was programmed in Adobe Flash CS5. Participants viewed 20 tournaments
in total. First, one block of 8 single player tournaments and then another block of 12 two-player
tournaments. The order of the tournaments within each block was randomized. Participants
could remind themselves about the most important aspects of the experiment by moving themouse over the Info field on the top right of the screen (see Figure 1). Based on the results of thethree matches in the tournament, participants estimated the strength of the indicated player on aslider that ranged from -50 to 50. The endpoints were labelled “very weak” and “very strong”.
It took participants 7.4 (SD = 3.3) minutes to complete the experiment.

A.3

DesignTable 1 shows the patterns of evidence that were used for the single player tournaments.
Table 2 shows the patterns for the two-player tournaments. In all tournaments, participants wereasked to judge the strength of player A.
For the single player tournaments, we used four different patterns of evidence: confoundedevidence in which A wins repeatedly against B, strong and weak indirect evidence where A only
wins one match herself but B either continues to win or lose two games against other players anddiverse evidence in which A wins against three different players. For each of those patterns, we
also included a pattern in which the outcomes of the games were exactly reversed.

B

Experiment 2

B.1

Participants

20 (11 female) recruited through Amazon Mechanical Turk participated in the experiment.
The mean age was 34 (SD = 9.8).

B.2

Materials, Procedure and Design

Participants viewed 10 single player tournaments which comprised the 8 situations used inExperiment 1 plus two additional patterns (IR 1, 2). Participants first judged player A’s strength
based merely on the match results in the tournament. Afterwards, participants received information from the omniscient commentator about one player who was lazy in a particular match.
Participants then rated A’s strength for a second time, whereby the slider was initialized at thefirst judgment’s position. It took participants 9.4 (SD = 4) minutes to complete the experiment.
The bottom row of Table 1 shows what information the omniscient commentator revealedin each situation. For example, in situation 3 in which participants first saw strong indirect
evidence, the commentator then said: “In game 1, Player B was lazy.” In the additional pattern(IR 2), A wins against B, B wins against C and D wins against E. The commentator then reveals
that E was lazy in game 3. For the patterns in which A lost his game, the results of each matchas shown in Table 1 were reversed and the corresponding losing player was indicated as having
been lazy. For example, in situation 2, A lost all three games against B and the commentator
revealed that A was lazy in game 2.

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