Extending our models of predication

Chapter 7: Generic language

The logical problem

Generic language (e.g., Swans are white.) is a simple and ubiquitous way to communicate generalizations about categories. Linguists, philosophers, and psychologists have scratched their collective heads for decades, trying to figure out what makes a generic sentence true or false. At first glance, generics feel like universally-quantified statements; Swans are white seems similar to All swans are white. Unlike universal quantification, however, generics are resilient to counter-examples (e.g., Swans are white even though there are black swans). Our intuitions then fall back to something more vague: Swans, in general, are white because indeed most swans are white. Thinking along these lines might lead us to characterize the truth conditions of generics in terms of the prevalence of a property in some kind. Viewed in this way, a generic statement would receive a meaning in line with other quantificational statements, as in the following code box. The trick lies in determing the appropriate threshold for a generic statement. Swans, in general, are white, but mosquitos, in general, do not carry malaria, yet people agree that Mosquitos carry malaria.

var literalMeanings = {
  some: function(prevalence){ prevalence > 0 },
  most: function(prevalence){ prevalence > 0.5 },
  all: function(prevalence){ prevalence == 1 },
  generic: function(prevalence, theta){ prevalence > theta }
}
var meaningFn = literalMeanings["generic"]
meaningFn(0.6, 0.5)

It seems doubtful that we would ever be able to capture the meaning of generics in terms of how prevalent a property is (e.g., whiteness) within the relevant kind (e.g., swans). Consider the birds: for a bird, being female practically implies you will lay eggs (the properties are present in the same proportion), yet we say things like Birds lay eggs but not Birds are female.

var theta = 0.49
var generic = function(prevalence){ prevalence > theta }
var percentage_of_birds_that_lay_eggs = 0.5;
var percentage_of_birds_that_are_female = 0.5;
var percentage_of_mosquitos_that_carry_malaria = 0.02;

display("Birds lay eggs? " + generic(percentage_of_birds_that_lay_eggs))
display("Birds are female? " + generic(percentage_of_birds_that_are_female))
display("Mosquitos carry malaria? " + generic(percentage_of_mosquitos_that_carry_malaria))

reft:tessler2016manuscript propose that the core meaning of a generic statement is in fact a threshold on prevalence, as in generic above. However, they treat that threshold as underspecified: the listener has uncertainty about the value of theta. With this underspecified threshold semantics for the generic, we can use a Bayesian model to arrive at a more precise meaning in context.

Bayesian generic language interpretation

The model takes the generic K has F (e.g., Dogs have fur) to mean that the prevalence of property F within kind K is above some threshold: (cf., Cohen 1999). But for the generic, no fixed value of the will suffice. Instead, we leave the threshold underspecified in the semantics () and infer it in context.

We can implement this reasoning with the following code. A listener hears a generic utterance and infers the prevalence of F within K by condering only those prevalence values that exceed theta.

var listener = function(utterance) {
  Infer({model: function(){
    var prevalence = sample(prevalencePrior)
    var theta = uniform(0, 1)
    condition( prevalence > theta)
    return prevalence
  }})
}

Here, we have a uniform prior over theta. The listener samples prevalence from some prior prevalencePrior, which is a prior distribution over the prevalence of the feature. The next step is to unpack what goes into our prior knowledge of prevalences for a given feature.

Prior model

Think of your favorite kind of animal. Got one in mind? What percentage of that kind of animal is female? Probably roughly 50%, regardless of the kind of animal you thought of. What percentage of that kind of animal lays eggs? That percentage depends on the kind of animal you’re considering. If you thought of a falcon, then the percentage that lays eggs is roughly 50% (only the females lay eggs). But if you thought of a bear, then 0% of them lay eggs.

We can conceive of the prior distribution over the prevalence of a feature within a kind, , as a distribution over kinds, , together with the prevalence of the feature within the kind.

var allKinds = [
  {kind: "dog", family: "mammal"},
  {kind: "falcon", family: "bird"},
  {kind: "cat", family: "mammal"},
  {kind: "gorilla", family: "mammal"},
  {kind: "robin", family: "bird"},
  {kind: "alligator", family: "reptile"},
  {kind: "giraffe", family: "mammal"},
]

var kindPrior = function(){
  uniformDraw(allKinds)
}

var prevalencePrior = Infer({model:
  function(){
    var k = kindPrior()
    var prevalence =
        k.family == "bird" ? 0.5 :  // half of birds lay eggs
        k.family == "reptile" ? 0.2 : // not sure if reptiles lay eggs
        0 // no other thing lays eggs

    return prevalence
  }
})

prevalencePrior

Exercise: What if you didn’t know that exactly 50% of birds lay eggs? Generalize the above code to sample the prevalence of laying eggs for birds, reptiles, etc. from a distribution.

A generalization of the prior model

In the above prior, we encoded the fact that people have knowledge about different types of categories (e.g., reptiles, mammals) and that this knowledge should give rise to different beliefs about the prevalence of the feature for a given kind. More generally, if speakers and listeners believe that some kinds have a causal mechanism that stably gives rise to the property, while others do not, then we would expect the prior to be structured as a mixture distribution (cf., Griffiths & Tenenbaum, 2005).

For convenience, let us denote the relevant probability as . The categories that have a stable causal mechanism produce the feature with some probability . The categories that do not have a stable causal mechanism produce the feature with some probability (perhaps this unstable mechanism is an external, environmental cause, that occurs very rarely). We would expect to be small (even zero), as certain features are completely absent in many categories (e.g., the number of lions that lay eggs). , on the other hand, could be large, giving rise to features that are often common in a kind (e.g., has four legs), but might also be substantially less than 1 for features that are non-universal in a category (e.g., has brown fur).

We formalize this idea by drawing and from Beta distributions (which has support between 0 and 1; samples from a Beta distribution correspond to probabilities). We fix the distribution for the transient cause: . The first parameter specifies the mean of the distribution, while the second parameter determines its concentration, or the inverse-variance of the distribution. (Here we use the mean-concentration parameterization of the Beta distribution, rather than the canonical pseudocount parameterization.)

///fold:
// discretized range between 0 - 1
var bins = map(function(x){
  _.round(x, 2);
},  _.range(0.01, 1, 0.02));

// function returns a discretized Beta distribution
var DiscreteBeta = cache(function(g, d){
  var a =  g * d, b = (1-g) * d;
  var betaPDF = function(x){
    return Math.pow(x, a-1)*Math.pow((1-x), b-1)
  }
  var probs = map(betaPDF, bins);
  return Categorical({vs: bins, ps: probs})
})
///

print("prevalence prior for transient causes:")
DiscreteBeta(0.01, 100)

What we plausibly can vary between contexts is the distribution of prevalences for stable causes: . Again, the first parameter specifies the mean of the distribution, while the second parameter determines its concentration. Concentration is high for properties that stably present in most kinds in exactly the same proportion (e.g. “is female”); it is lower when there is more variance (or uncertainty).

///fold:
// discretized range between 0 - 1
var bins = map(function(x){
  _.round(x, 2);
},  _.range(0.01, 1, 0.02));

// function returns a discretized Beta distribution
var DiscreteBeta = cache(function(g, d){
  var a =  g * d, b = (1-g) * d;
  var betaPDF = function(x){
    return Math.pow(x, a-1)*Math.pow((1-x), b-1)
  }
  var probs = map(betaPDF, bins);
  return Categorical({vs: bins, ps: probs})
})
///

var mean = 0.5
var concentration = 10

print("prevalence prior for stable causes:")
DiscreteBeta(mean, concentration)

We also can vary how prevalent transient vs. stable causes are. The prior that results is a mixture distribution:

The components of the prior can be probed with human intuitions through two questions. To determine the relative prominence of the stable cause, , we consider how likely it is for any member of a kind to have the relevant property. To determine the average prevalence among those kinds where the property is stably present, , we consider the proportion of the property within a single kind; looking at the variability among multiple judgments, we can estimate the contration of this proportion, . The task proceeds as follows:

We just discovered an animal on a far away island called a fep.

  1. How likely is it that there is a fep that has wings? ()
  2. Suppose there is a fep that has wings. What percentage of feps do you think have wings? ()

(Run through your intuitions with other properties like “is female” or “lays eggs”.)

The following model priorModel formalizes the above ideas computationally:

///fold:
// discretized range between 0 - 1
var bins = _.range(0.01, 1, 0.025);

// function returns a discretized Beta distribution
var DiscreteBeta = cache(function(g, d){
  var a =  g * d, b = (1-g) * d;
  var betaPDF = function(x){
    return Math.pow(x, a-1)*Math.pow((1-x), b-1)
  }
  var probs = map(betaPDF, bins);
  return Categorical({vs: bins, ps: probs})
})
///
var priorModel = function(params){
  Infer({model: function(){

    var phi = params["potential"]
    var gamma = params["prevalenceWhenPresent"]
    var delta = params["concentrationWhenPresent"]

    var StableDistribution = DiscreteBeta(gamma, delta)
    var UnstableDistribution = DiscreteBeta(0.01, 100)

    var prevalence = flip(phi) ?
      sample(StableDistribution) :
      sample(UnstableDistribution)

    return {prevalence}

  }})
}

// e.g. "Lays eggs"
viz(priorModel({
  potential: 0.3, // how prevalent the stable cause is
  prevalenceWhenPresent: 0.5, // how prevalent the feature is under the stable cause
  concentrationWhenPresent: 10   // the concentration of the stable cause
}))

Exercises:

  1. What does this picture represent? If you drew a sample from this distribution, what would that correspond to?
  2. Try to think up a property for which the three parameters above are not able to give even a remotely plausible distribution. (If you succeed, let us know; the idea is that this parameterization is sufficient to capture—in approximation—any case of relevance.)

Generic interpretation model

The generics model assumes a simple (the simplest?) meaning for a generic statement: a threshold on the prevalence of a feature in a kind.

///fold:
// discretized range between 0 - 1
var bins = map(function(x){
  _.round(x, 2);
},  _.range(0.01, 1, 0.02));

var thresholdBins = map2(function(x,y){
  var d = (y - x)/ 2;
  return x + d
}, bins.slice(0, bins.length - 1), bins.slice(1, bins.length))

// function returns a discretized Beta distribution
var DiscreteBeta = cache(function(g, d){
  var a =  g * d, b = (1-g) * d;
  var betaPDF = function(x){
    return Math.pow(x, a-1)*Math.pow((1-x), b-1)
  }
  var probs = map(betaPDF, bins);
  return Categorical({vs: bins, ps: probs})
})

var priorModel = function(params){
  Infer({model: function(){

    var potential = params["potential"]
    var g = params["prevalenceWhenPresent"]
    var d = params["concentrationWhenPresent"]

    var StableDistribution = DiscreteBeta(g, d)
    var UnstableDistribution = DiscreteBeta(0.01, 100)

    var prevalence = flip(potential) ?
      sample(StableDistribution) :
      sample(UnstableDistribution)

    return prevalence

  }})
}
///
var meaning = function(utterance, prevalence, threshold) {
  return (utterance == 'generic') ? prevalence > threshold : true
}
var thresholdPrior = function() { return uniformDraw(thresholdBins) };

var statePrior = priorModel({
  potential: 0.3, // how prevalent the stable cause is
  prevalenceWhenPresent: 0.5, // how prevalent the feature is under the stable cause
  concentrationWhenPresent: 10   // the concentration of the stable cause
})

display("prevalence prior:")
viz(statePrior)

var listener = cache(function(utterance) {
  Infer({model: function(){
    var prevalence = sample(statePrior)
    var threshold = thresholdPrior()
    var m = meaning(utterance, prevalence, threshold)
    condition(m)
    return prevalence
  }})
})

display("listener posterior:")
listener("generic")

Exercises:

  1. Come up with parameters for the prior that represent the carries malaria distribution. Test the listener’s interpretation of a generic (e.g., Wugs carry malaria).
  2. Come up with parameters for the prior that represent the lays eggs distribution. Test the listener’s interpretation of a generic (e.g., Wugs lay eggs).
  3. Come up with parameters for the prior that represent the are female distribution. Test the listener’s interpretation of a generic (e.g., Wugs are female).

So far we have a model that can interpret generic language (with a very simple semantics). We can now imagine a speaker who thinks about this type of listener and decides if a generic utterance is a good thing to say to describe the state of the world. Speaker models are interpreted as models of utterance production, or endorsement (reft:DegenGoodman2014Cogsci; reft:Franke2014). If we specify the alternative utterance to be a null utterance (i.e., silence), we model the choice between uttering the generic (i.e., endorsing its truth) or nothing at all (i.e., not endorsing its truth).

///fold:
// discretized range between 0 - 1
var bins = map(function(x){
  _.round(x, 2);
},  _.range(0.01, 1, 0.02));

var thresholdBins = map2(function(x,y){
  var d = (y - x)/ 2;
  return x + d
}, bins.slice(0, bins.length - 1), bins.slice(1, bins.length))

// function returns a discretized Beta distribution
var DiscreteBeta = cache(function(g, d){
  var a =  g * d, b = (1-g) * d;
  var betaPDF = function(x){
    return Math.pow(x, a-1)*Math.pow((1-x), b-1)
  }
  var probs = map(betaPDF, bins);
  return Categorical({vs: bins, ps: probs})
})

var priorModel = function(params){
  Infer({model: function(){

    var potential = params["potential"]
    var g = params["prevalenceWhenPresent"]
    var d = params["concentrationWhenPresent"]

    var StableDistribution = DiscreteBeta(g, d)
    var UnstableDistribution = DiscreteBeta(0.01, 100)

    var prevalence = flip(potential) ?
        sample(StableDistribution) :
    sample(UnstableDistribution)

    return prevalence
  }})
}
///

var alpha = 2;
var utterances = ["generic", "silence"];

var thresholdPrior = function() { return uniformDraw(thresholdBins) };
var utterancePrior = function() { return uniformDraw(utterances) }
var cost = {
  "generic": 1,
  "silence": 1
}

var meaning = function(utterance, prevalence, threshold) {
  return (utterance == 'generic') ? prevalence > threshold : true
}


var listener = function(utterance, statePrior) {
  Infer({model: function(){
    var prevalence = sample(statePrior)
    var threshold = thresholdPrior()
    var m = meaning(utterance, prevalence, threshold)
    condition(m)
    return prevalence
  }})
}

var speaker = function(prevalence, statePrior){
  Infer({model: function(){
    var utterance = utterancePrior();
    var L = listener(utterance, statePrior);
    factor( alpha * (L.score(prevalence) - cost[utterance]))
    return utterance
  }})
}

var observed_prevalence = 0.03

var prior = priorModel({
  potential: 0.01,
  prevalenceWhenPresent: 0.01,
  concentrationWhenPresent: 5
})

viz.density(prior)

viz(speaker(observed_prevalence, prior))

Exercises:

  1. Test Birds lay eggs vs. Birds are female. (Technical note: Due to the discretization of the state space, observed_prevalence must take odd-numbered values such as 0.03, 0.05, 0.09, … )
  2. Come up with other generic sentences. Hypothesize what the prior might be, what the prevalence might be, and test the model on it.

Extension: Generics with gradable adjectives

First, a world with entities.

var altBeta = function(g, d){
  var a =  g * d;
  var b = (1-g) * d;
  return beta(a, b)
}

var fep = function() {
  return {
    kind: "fep",
    wings: flip(0.5),
    legs: flip(0.01),
    claws: flip(0.01),
    height: altBeta(0.5, 10)
  }
}

var wug = function() {
  return {
    kind: "wug",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.3),
    height: altBeta(0.2, 10)
  }
}

var glippet = function() {
  return {
    kind: "glippet",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.2),
    height: altBeta(0.8, 10)
  }
}

var theWorld = _.flatten([repeat(10, fep), repeat(10, wug), repeat(10, glippet)])

var kinds = _.uniq(_.map(theWorld, "kind"));

print('height distribution over all creatures')
viz.density(_.map(theWorld, "height"))

var rs = map(function(k){
  print('height distribution for ' + k)
  viz.density(_.map(_.filter(theWorld,{kind: k}), "height"), {bounds:[0,1]})
}, kinds)

print('')

Now, let’s calculate prevalence distributions. These will be somewhat boring because there are only 3 kinds of creatures in this world.

/// fold:
var round = function(x){
  var rounded = Math.round(10*x)/10
  return rounded == 0 ? 0.01 : rounded
}

var makeHistogram = function(prevalences){
  return map(function(s){
    return reduce(function(x, i){
      var k = x == s ? 1 : 0
      return i + k
    }, 0.001, prevalences)
  }, stateBins)
}

var altBeta = function(g, d){
  var a =  g * d;
  var b = (1-g) * d;
  return beta(a, b)
}

var fep = function() {
  return {
    kind: "fep",
    wings: flip(0.5),
    legs: flip(0.01),
    claws: flip(0.01),
    height: altBeta(0.5, 10)
  }
}

var wug = function() {
  return {
    kind: "wug",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.3),
    height: altBeta(0.2, 10)
  }
}

var glippet = function() {
  return {
    kind: "glippet",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.2),
    height: altBeta(0.8, 10)
  }
}
///
var stateBins = [0.01,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]

var theWorld = _.flatten([repeat(10, fep), repeat(10, wug), repeat(10, glippet)])

var allKinds = _.uniq(_.map(theWorld, "kind"))

var prevalence = function(world, kind, property){
  var members = _.filter(world, {kind: kind})
  return round(listMean(_.map(members, property)))
}

var prevalencePrior = function(property, world){
  var p =  map(function(k){return prevalence(world, k, property)}, allKinds)
  return makeHistogram(p)
}

viz.bar(stateBins, prevalencePrior("legs", theWorld))

With individuals in the world, the extended model evaluates generics with gradable adjectives (e.g., giraffes are tall) by first checking to see how many of a relevent subset of the kind could truthfully be described to hold the property at the individual level, and then using this information to infer the prevalence of the property in the kind. With prevalence in hand, the model proceeds as before.

///fold:
// discretized range between 0 - 1
var stateBins = [0.01,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1]
var thresholdBins = [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9]
var alpha_1 = 5

var round = function(x){
  var rounded = Math.round(10*x)/10
  return rounded == 0 ? 0.01 : rounded
}

var makeHistogram = function(prevalences){
  return map(function(s){
    return reduce(function(x, i){
      var k = x == s ? 1 : 0
      return i + k
    }, 0.001, prevalences)
  }, stateBins)
}

var altBeta = function(g, d){
  var a =  g * d;
  var b = (1-g) * d;
  return beta(a, b)
}

var fep = function() {
  return {
    kind: "fep",
    wings: flip(0.5),
    legs: flip(0.01),
    claws: flip(0.01),
    height: round(altBeta(0.5, 10))
  }
}

var wug = function() {
  return {
    kind: "wug",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.3),
    height: round(altBeta(0.2, 10))
  }
}

var glippet = function() {
  return {
    kind: "glippet",
    wings: flip(0.5),
    legs: flip(0.99),
    claws: flip(0.2),
    height: round(altBeta(0.8, 10))
  }
}
///

var theWorld = _.flatten([repeat(10, fep), repeat(10, wug), repeat(10, glippet)])
var allKinds = _.uniq(_.map(theWorld, "kind"))

var propertyDegrees = {
  wings: "wings",
  legs: "legs",
  claws: "claws",
  tall:" height"
}


var prevalence = function(world, kind, property){
  var members = _.filter(world, {kind: kind})
  return round(listMean(_.map(members, property)))
}

var prevalencePrior = function(property, world){
  var p =  map(function(k){return prevalence(world, k, property)}, allKinds)
  return makeHistogram(p)
}

var propertyPrior = function(property){
  var p = _.map(theWorld, property)
  return makeHistogram(p)
}

var statePrior = function(probs){ return categorical(probs, stateBins) }
var thresholdPrior = function() { return uniformDraw(thresholdBins) }

var utterancePrior = function(property) {
  var utterances = property == "height" ?
      ["tall", "null"] :
  ["generic", "null"]
  return uniformDraw(utterances)
}

var meaning = function(utterance, state, threshold) {
  return utterance == "generic" ? state > threshold :
  utterance == "tall" ? state > threshold :
  true
}

var literalListener = cache(function(utterance, threshold, stateProbs) {
  Infer({method: "enumerate"}, function(){
    var state = statePrior(stateProbs)
    var m = meaning(utterance, state, threshold)
    condition(m)
    return state
  })
})

var speaker1 = cache(function(state, threshold, stateProbs, property) {
  Infer({method: "enumerate"}, function(){
    var utterance = utterancePrior(property)
    var L0 = literalListener(utterance, threshold, stateProbs)
    factor(alpha_1 * L0.score(state))
    return utterance

  })
})

var pragmaticListener = cache(function(utterance, property, world) {
  Infer({method: "enumerate"}, function(){
    var stateProbs = property == "height" ?
        propertyPrior(property) :
    prevalencePrior(property, world)
    var state = statePrior(stateProbs)
    var threshold = thresholdPrior()
    var S1 = speaker1(state, threshold, stateProbs, property)
    observe(S1, utterance)
    return state
  })
})

var worldWithTallness = map(function(individual){
  var tallDistribution = Infer({method: "enumerate"}, function(){
    var utterance = utterancePrior("height")
    factor(pragmaticListener(utterance, "height").score(individual.height))
    return utterance
  })
  return _.extend(individual,
                  {tall: Math.exp(tallDistribution.score("tall"))})
}, theWorld)

var speaker2 = function(kind, predicate){
  Infer({method: "enumerate"}, function(){
    var property = predicate.split(' ')[1]
    var degree = propertyDegrees[property]
    var world = _.isNumber(theWorld[0][degree]) ? 
        worldWithTallness : theWorld
    var prev = prevalence(world, kind, property)
    var utterance = utterancePrior(property)

    var L1 = pragmaticListener(utterance, property, world)
    factor(2*L1.score(prev))

    return utterance=="generic" ? 
      kind + "s " + predicate :
     "don't think so"
  })
}

viz.auto(speaker2("glippet", "are tall"))

References:


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