Probabilistic language understanding An introduction to the Rational Speech Act framework By Gregory Scontras, Michael Henry Tessler, and Michael Franke

Fixing free parameters

Chapter 5: Vagueness

Sometimes our words themselves are imprecise, vague, and heavily dependent on context to fix their interpretations. Compositionality assumes semantic atoms with invariant meanings; context-dependent word interpretations pose a serious challenge to compositionality. Take the case of gradable adjectives: “expensive for a sweater” means something quite different from “expensive for a laptop.” What, then, do we make of the contribution from the word “expensive”? Semanticists settle on the least common denominator: a threshold semantics by which the adjective asserts that holders of the relevant property surpass some point on the relevant scale (i.e., expensive means something like “more expensive than some contextually-determined threshold on prices). Whereas semanticists punt on the mechanism by which context fixes these aspects of meaning, the RSA framework is well-suited to meet the challenge.

Application 1: Gradable adjectives and vagueness resolution

Lassiter & Goodman propose we parameterize the meaning function for sentences containing gradable adjectives so that their interpretations are underspecified (reft:lassitergoodman2013, reft:LassiterGoodman2015:Adjectival-vagu). This interpretation-fixing parameter, the gradable threshold value \(\theta\) (i.e., a degree), is something that conversational participants can use their prior knowledge to actively reason about and set. As with the ambiguity-resolving variable in the previous chapter, \(\theta\) gets lifted to the level of the pragmatic listener, who jointly infers the gradable threshold (e.g., the point at which elements of the relevant domain count as expensive) and the true state (e.g., the indicated element’s price).

The model depends crucially on our prior knowledge of the world state. Let’s start with a toy prior for the prices of books.

var book = {
  "prices": [2, 6, 10, 14, 18, 22, 26, 30],
  "probabilities": [1, 2, 3, 4, 4, 3, 2, 1]
};

var statePrior = function() {
  return categorical(book.probabilities, book.prices);
};

Exercise: Visualize the statePrior.

Next, we create a prior for the degree threshold \(\theta\). Since we’re talking about expensive books, \(\theta\) will be the price cutoff to count as expensive. But we want to be able to use expensive to describe anything with a price, so we’ll set the thetaPrior to be uniform over the possible prices in our world.

var book = {
  "prices": [2, 6, 10, 14, 18, 22, 26, 30],
  "probabilities": [1, 2, 3, 4, 4, 3, 2, 1]
};

var statePrior = function() {
  return categorical(book.probabilities, book.prices);
};

var thetaPrior = function() {
    return uniformDraw(book.prices);
}

Exercise: Visualize the thetaPrior.

We introduce two possible utterances: saying that a book is expensive, or saying nothing at all (a “null utterance”). The semantics of the expensive utterance checks the relevant item’s price against the price cutoff. The “null utterance” represents the speaker’s choice to stay silent: silence is easier to produce than saying “expensive” and it is never (literally) false. (Whence, that rational speakers prefer silence over speech which is not informative; see below.) The model of the literal listener implemented here can be written as (where \(s\) is general notation for the world state (here: price of a book), and \(\theta\) is the threshold to be inferred later on):

\[P_{L_0}(s \mid u, \theta) = P(s \mid [\![u]\!]^\theta)\] 
var book = {
  "prices": [2, 6, 10, 14, 18, 22, 26, 30],
  "probabilities": [1, 2, 3, 4, 4, 3, 2, 1]
};

var statePrior = function() {
  return categorical(book.probabilities, book.prices);
};

var thetaPrior = function() {
    return uniformDraw(book.prices);
};

var utterances = ["expensive", ""];
var cost = {
  "expensive": 1,
  "": 0
};
var utterancePrior = function() {
  return uniformDraw(utterances);
};

var meaning = function(utterance, price, theta) {
  utterance == "expensive" ? price >= theta : true;
};

var literalListener = cache(function(utterance, theta) {
  return Infer({method: "enumerate"}, function() {
    var price = statePrior();
    condition(meaning(utterance, price, theta))
    return price;
  })
})

Exercise: Check \(L_0\)’s predictions for various price cutoffs.

We get a full RSA model once we add the pragmatic speaker \(L_1\) and pragmatic listener \(L_1\) models. The pragmatic speaker is assumed to have a fixed threshold \(\theta\), the pragmatic listener hears the gradable adjective and jointly infers the relevant item price and cutoff to count as expensive.

\[P_{S_1}(u \mid s, \theta) \propto \exp ( \alpha \ (\log P_{L_0}(s \mid u, \theta) - C(u)))\] \[P_{L_1}( s, \theta \mid u) \propto P(s) \ P(\theta) \ P_{S_1}( u \mid s, \theta)\] 
var book = {
  "prices": [2, 6, 10, 14, 18, 22, 26, 30],
  "probabilities": [1, 2, 3, 4, 4, 3, 2, 1]
};

var statePrior = function() {
  return categorical(book.probabilities, book.prices);
};

var thetaPrior = function() {
    return uniformDraw(book.prices);
};

var alpha = 2; // optimality parameter

var utterances = ["expensive", ""];
var cost = {
  "expensive": 1,
  "": 0
};
var utterancePrior = function() {
  return uniformDraw(utterances);
};

var meaning = function(utterance, price, theta) {
  utterance == "expensive" ? price >= theta : true;
};

var literalListener = cache(function(utterance, theta) {
  return Infer({method: "enumerate"}, function() {
    var price = statePrior();
    condition(meaning(utterance, price, theta))
    return price;
  });
});

var speaker = cache(function(price, theta) {
  return Infer({method: "enumerate"}, function() {
    var utterance = utterancePrior();
    factor( alpha * (literalListener(utterance, theta).score(price)
                    - cost[utterance]));
    return utterance;
  });
});

var pragmaticListener = function(utterance) {
  return Infer({method: "enumerate"}, function() {
    var price = statePrior();
    var theta = thetaPrior();
    factor(speaker(price, theta).score(utterance));
    return { price: price, theta: theta };
  });
};

var expensiveBook = pragmaticListener("expensive");
viz.auto(marginalize(expensiveBook, "price"));
viz.auto(marginalize(expensiveBook, "theta"));

Exercises:

  1. What happens when you make the "expensive" utterance more costly? Why?
  2. Try altering the statePrior and see what happens to \(L_1\)’s inference.

For a better model, rather than assuming prior knowledge (e.g., knowledge about domain-specific prices), we can measure it, then feed these measurements into the model as facts about the world. Doing so allows the model to make actual predictions about the behavior we expect to observe from listeners. Here, we draw on data measuring prior knowledge that was gathered by Lassiter & Goodman (unpublished work).

///fold:
// "price" refers to the midpoint of the bin that participants marked a slider for
// "probability" refers to the average of participants' responses, after normalizing responses for each person for each item
var coffee = {
  "prices": [2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230, 234, 238, 242, 246, 250, 254, 258, 262, 266, 270],
  "probabilities": [0.00394425443541116, 0.00566290387321734, 0.00825676850536135, 0.0128170636252316, 0.016825062585583, 0.0252551834871954, 0.028468923991368, 0.0287662881619652, 0.0282906782393822, 0.0298574213919862, 0.0289804109562303, 0.0294934077950995, 0.0282267499972963, 0.027150810280172, 0.0263590035097067, 0.0281826763029633, 0.0279055635543465, 0.0278411275661919, 0.0278729305091271, 0.0260779123427457, 0.025504052957576, 0.0244726498427457, 0.0245886104865853, 0.0244622094919612, 0.0245026711395991, 0.0215246465554349, 0.018590665797836, 0.0178546907780941, 0.0171993717256383, 0.0162494284404094, 0.0160224797974072, 0.0153063570140879, 0.0146233597944038, 0.0137848051705444, 0.0143326737593802, 0.0134490255028661, 0.0135790032064503, 0.0138226943563312, 0.0131925119561275, 0.0126370445876965, 0.0106890591752964, 0.0106020628239155, 0.0101185949533398, 0.00911819843666944, 0.0106104999317876, 0.0109322919246626, 0.0107129834142186, 0.00837168284710854, 0.0080750721559225, 0.00804082264332751, 0.0067313619453682, 0.00627460791368519, 0.00589113687110393, 0.00540523790626922, 0.00552727879179031, 0.00550309276614945, 0.00523220007943591, 0.00511881990396121, 0.00512531318796172, 0.00518250532794681, 0.00363644079549071, 0.00358121922955672, 0.00340485508432679, 0.00300204070158435, 0.00285900767263514, 0.00307193123371249, 0.00265238954722026, 0.00259519926379722]
};
var headphones = {
  "prices": [3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327],
  "probabilities": [0.0100932350462178, 0.0170242011533158, 0.0239368832461044, 0.0287274552724317, 0.0335581038692007, 0.0340978261675896, 0.0343161232972866, 0.0360390185911639, 0.036657423324831, 0.0363628055611882, 0.0363696076759004, 0.0349562074307856, 0.034074002420704, 0.0319526478383908, 0.0310094760253021, 0.0310633591664631, 0.0306213012447122, 0.0267011137913525, 0.0258670033914898, 0.0258502350122376, 0.025023519418377, 0.023483875551852, 0.0220028560428636, 0.0214970268323533, 0.0208370797134518, 0.0179036695749784, 0.0171528323730662, 0.0146789727474624, 0.0151970966046565, 0.0144002087359724, 0.0133747582315602, 0.0123806405169515, 0.0120269613954518, 0.0114214380586013, 0.0119920388378585, 0.011383038780713, 0.0113361342839428, 0.0109751373856222, 0.00862772988748631, 0.00849834370871022, 0.00929436665963529, 0.00977137935316548, 0.00855300304060609, 0.00704341088484788, 0.00760242506144268, 0.0066676293070587, 0.00664151547908964, 0.00631842654192989, 0.00633866335810551, 0.00617281009506307, 0.00588389192288987, 0.00480853638779448, 0.00438645078845164, 0.00381072308793509, 0.00323537982338538]
};
var laptop = {
  "prices": [25, 75, 125, 175, 225, 275, 325, 375, 425, 475, 525, 575, 625, 675, 725, 775, 825, 875, 925, 975, 1025, 1075, 1125, 1175, 1225, 1275, 1325, 1375, 1425, 1475, 1525, 1575, 1625, 1675, 1725, 1775, 1825, 1875, 1925, 1975, 2025, 2075, 2125, 2175, 2225, 2275, 2325, 2375, 2425, 2475],
  "probabilities": [0.00143805356403561, 0.00271785465081783, 0.00606278580557322, 0.00969025393738191, 0.0148967448541453, 0.0198125393878057, 0.0241329166751748, 0.0295996079491818, 0.0330742235328304, 0.0361264464785417, 0.0401533300800443, 0.0409948060761354, 0.0413284234227614, 0.0403647740154393, 0.0401794950623575, 0.0389208352425967, 0.0381211282626673, 0.0380271843819898, 0.0337130710333348, 0.0320869893997556, 0.0303954589742173, 0.0290371975286599, 0.026737673556754, 0.0259385656770147, 0.0246868317621353, 0.0229019589247208, 0.0214921049215865, 0.0206506505418053, 0.0194727089077923, 0.0192606778139999, 0.016502461425033, 0.015779852769427, 0.014573698680051, 0.0141934767007475, 0.0128416097234241, 0.0124983882714321, 0.011580082968956, 0.0111057181921517, 0.0107390517842162, 0.0097877804629936, 0.00869628232285698, 0.00796157687645315, 0.00781439122831209, 0.0074127159955819, 0.00686298004874477, 0.00667898701991275, 0.00628316286073362, 0.00606433685126466, 0.00552579230999036, 0.00508236108646155]
};
var sweater = {
  "prices": [1.5, 4.5, 7.5, 10.5, 13.5, 16.5, 19.5, 22.5, 25.5, 28.5, 31.5, 34.5, 37.5, 40.5, 43.5, 46.5, 49.5, 52.5, 55.5, 58.5, 61.5, 64.5, 67.5, 70.5, 73.5, 76.5, 79.5, 82.5, 85.5, 88.5, 91.5, 94.5, 97.5, 100.5, 103.5, 106.5, 109.5, 112.5, 115.5, 118.5, 121.5, 124.5, 127.5, 130.5, 133.5, 136.5, 139.5, 142.5, 145.5, 148.5, 151.5, 154.5, 157.5, 160.5, 163.5, 166.5, 169.5, 172.5, 175.5, 178.5, 181.5, 184.5, 187.5, 190.5, 193.5, 196.5, 199.5, 202.5, 205.5, 208.5, 211.5, 214.5, 217.5, 220.5, 223.5, 226.5, 229.5, 232.5, 235.5, 238.5],
  "probabilities": [0.00482838499944466, 0.00832934578733181, 0.0112952500492109, 0.0173774790108894, 0.0232006658974883, 0.0258422772579257, 0.0278986695293033, 0.0295289411585088, 0.0306833716679902, 0.0318318597751272, 0.0337834568516467, 0.0339921053872795, 0.0344439315108449, 0.033934432265521, 0.032462878943956, 0.031189466255733, 0.0308771801297135, 0.028870440122745, 0.0268081450723193, 0.0255858806436071, 0.0251329374896422, 0.0228478370318916, 0.0204321414255458, 0.0198885290723185, 0.018227461808914, 0.0175834655975716, 0.0162564776853142, 0.0157731201439098, 0.0152929182194243, 0.0150019091787104, 0.0149289099733278, 0.0141896849640091, 0.0139276040018639, 0.0134566451556076, 0.0121362548659773, 0.0108558348756676, 0.010613387912742, 0.010091436302254, 0.0098485635160309, 0.00934441306894098, 0.0085581357559821, 0.00679765980394746, 0.00693786098620408, 0.00696322949112243, 0.0070065216834756, 0.00642780104088309, 0.0064321559380608, 0.00656481702666307, 0.0062031359060876, 0.00687959185242292, 0.00510183619524203, 0.00519749642756681, 0.00498195289503402, 0.00463848039694862, 0.00446827225938831, 0.00434243838506282, 0.00454705086043589, 0.0045914966088052, 0.00451510979961212, 0.00443992782954235, 0.00329590488378491, 0.00349426470729333, 0.00329078051712042, 0.00323849270094039, 0.00302968185434419, 0.00294213735024183, 0.00335510797297302, 0.00328341117067163, 0.00329874147186505, 0.00306305447627786, 0.00262071902879654, 0.00274925007756808, 0.00246374710845232, 0.00262910011008071, 0.00248819809733968, 0.00211124548886266, 0.00204178897873852, 0.00208550762922333, 0.00204890779502054, 0.00228129283166782]
};
var watch = {
  "prices": [25, 75, 125, 175, 225, 275, 325, 375, 425, 475, 525, 575, 625, 675, 725, 775, 825, 875, 925, 975, 1025, 1075, 1125, 1175, 1225, 1275, 1325, 1375, 1425, 1475, 1525, 1575, 1625, 1675, 1725, 1775, 1825, 1875, 1925, 1975, 2025, 2075, 2125, 2175, 2225, 2275, 2325, 2375, 2425, 2475, 2525, 2575, 2625, 2675, 2725, 2775, 2825, 2875, 2925, 2975],
  "probabilities": [0.040844560268751, 0.0587099798246933, 0.0656194599591356, 0.0667642412698035, 0.0615953803048016, 0.0510809063784378, 0.0467203673419258, 0.0446735950187136, 0.040047421916613, 0.0350583957334483, 0.0297508215717606, 0.0256829651118227, 0.024135920250668, 0.0228891907259206, 0.021706684520276, 0.0186449440066946, 0.0187249266247728, 0.0179250744798993, 0.0173698811746238, 0.0165581725818319, 0.0160745066032247, 0.0127927305129066, 0.0113730680265067, 0.0109485307623827, 0.00923468422650943, 0.00899007751887508, 0.00880520147998275, 0.00838023585866885, 0.00841052411004918, 0.00828830635037619, 0.00834008093757411, 0.00750681534099784, 0.00724072133740109, 0.00717291664158004, 0.00682823777708754, 0.00646995193940331, 0.00697139732982518, 0.00711846547272734, 0.00698781312802354, 0.00732316558583701, 0.00594973158122097, 0.00557461443747403, 0.00541637601910211, 0.00518850469148531, 0.00572025848989677, 0.0051443557601358, 0.00510282169734075, 0.00493720252580643, 0.00560198932991028, 0.00519158715054485, 0.00473398797752786, 0.00540907722833213, 0.00494653421540979, 0.00495500420164643, 0.00494083025189895, 0.00481566268206312, 0.00442965937328148, 0.00441189688100535, 0.00415116538135834, 0.00361842012002631]
};
var data = {
  "coffee maker": coffee,
  "headphones": headphones,
  "laptop": laptop,
  "sweater": sweater,
  "watch": watch
};

var statePrior_lookup = function(item) {
  var prices = data[item].prices;
  var probabilities = data[item].probabilities;
  return function() {
    return categorical(probabilities, prices);
  };
};

var thetaPrior_lookup = function(item) {
  var thetas = data[item].prices;
  return function() {
    return uniformDraw(thetas);
  };
};
///

var alpha = 2; // optimality parameter

var utterances = ["expensive", ""];
var cost = {
  "expensive": 1,
  "": 0
};
var utterancePrior = function() {
  return uniformDraw(utterances);
};

var meaning = function(utterance, price, theta) {
  utterance == "expensive" ? price >= theta : true;
};

var literalListener = cache(function(utterance, theta, item) {
  var statePrior = statePrior_lookup(item);
  return Infer({method: "enumerate"}, function() {
    var price = statePrior();
    condition(meaning(utterance, price, theta));
    return price;
  });
});

var speaker = cache(function(price, theta, item) {
  return Infer({method: "enumerate"}, function() {
    var utterance = utterancePrior();
    factor( alpha * (literalListener(utterance, theta, item).score(price)
                    - cost[utterance]));
    return utterance;
  });
});

var pragmaticListener = function(utterance, item) {
  // first identify the relevant priors
  var statePrior = statePrior_lookup(item);
  var thetaPrior = thetaPrior_lookup(item);
  // then run inference
  return Infer({method: "enumerate"},
  function() {
    var price = statePrior();
    var theta = thetaPrior();
    factor( speaker(price, theta, item).score(utterance) );
    return { price: price, theta: theta };
  });
};

var expensiveWatch = pragmaticListener("expensive", "watch");
print("the listener's posterior over watch prices:")
viz.density(marginalize(expensiveWatch, "price"));
print("the listener's posterior over watch price thresholds:")
viz.density(marginalize(expensiveWatch, "theta"));

var expensiveSweater= pragmaticListener("expensive", "sweater");
print("the listener's posterior over sweater prices:")
viz.density(marginalize(expensiveSweater, "price"));
print("the listener's posterior over sweater price thresholds:")
viz.density(marginalize(expensiveSweater, "theta"));

Exercises:

  1. Visualize the various state priors.
  2. Check \(L_1\)’s behavior for coffee makers and headphones and laptops.
  3. Add an \(S_2\) layer to the model and check its predictions.

Application 2: Inferring the comparison class

Implicit in the adjectives model from reft:lassitergoodman2013 is an awareness of the relevant comparison class: expensive for a watch vs. for a sweater. But what if we don’t know what the relevant comparison class is? Take the adjective tall: if I tell you John is a basketball player and he is tall, you probably infer that the comparison class is the superordinate category of all people. Similarly, if I tell you that John is a gymnast and tall, you probably infer that he is short compared to all people. But if I tell you that John is a soccer player and tall/short, you might instead infer that John is tall/short just for the subordinate category of soccer players. In an attempt to formalize the reasoning that goes into this inference, Tessler et al. (2017) augment the basic adjectives model to include uncertainty about the relevant comparison class: superordinate (e.g., compared to all people) or subordinate (e.g., compared to gymnasts or soccer players or basketball players).

This reasoning depends crucially on our prior knowledge about the relevant categories. To model this knowledge, we’ll need to intelligently simulate various categories: the heights of all people, the heights of gymnasts, the heights of soccer players, and the heights of basketball players.

// helper function
var exp = function(x){return Math.exp(x)}

// for discretization
var binParam = 5;

// information about the superordinate category prior
// e.g., the height distribution for all people
var superordinate = {mu: 0, sigma: 1};

// calculate the range in pre-defined steps;
// these values correspond to possible heights
var stateVals = _.range(superordinate.mu - 3 * superordinate.sigma,
          superordinate.mu + 3 * superordinate.sigma,
          superordinate.sigma/binParam)

// for each possible height, calculate its probability of occurrence
var stateProbs = cache(function(stateParams){
  return map(function(s){
    exp(Gaussian(stateParams).score(s))
  }, stateVals)
});

// generate a statePrior using the possible heights and their probabilities
var generateStatePrior = cache(function(stateParams) {
  return Infer({
    model: function(){
      return categorical({vs: stateVals, ps: stateProbs(stateParams)})
    }
  })
});

// information about the subordinate category priors
var subParams = {
  low: {mu: -1, sigma: 0.5}, // gymnast heights
  middle: {mu: 0, sigma: 0.5}, // soccer player heights
  high: {mu: 1, sigma: 0.5} // basketball player heights
}

display("hypothetical height prior for all people")
viz.density(generateStatePrior(superordinate))
display("hypothetical height prior for gymnasts")
viz.density(generateStatePrior(subParams["low"]))
display("hypothetical height prior for soccer players")
viz.density(generateStatePrior(subParams["middle"]))
display("hypothetical height prior for basketball players")
viz.density(generateStatePrior(subParams["high"]))

Exercise: Try altering the subParams to generate different subordinate category priors.

We can add these state priors to the basic adjectives model, together with a lifted variable concerning the comparison class. Now, the pragmatic listener \(L_1\) is told the relevant subordinate category (e.g., John is a basketball player) and hears the utterance with the scalar adjective (i.e., John is tall). On the basis of this information, \(L_1\) jointly infers the state (i.e., John’s height) and the relevant comparison class the speaker intended (e.g., tall for all people vs. tall for a basketball player).

///fold:
// helper function
var exp = function(x){return Math.exp(x)}

// for discretization
var binParam = 3;

// information about the superordinate category prior
// e.g., the height distribution for all people
var superordinate_params = {mu: 0, sigma: 1};

// calculate the range in pre-defined steps;
// these values correspond to possible heights
var stateVals = _.range(superordinate_params.mu - 3 * superordinate_params.sigma,
                        superordinate_params.mu + 3 * superordinate_params.sigma,
                        superordinate_params.sigma/binParam)

// for each possible height, calculate its probability of occurrence
var stateProbs = cache(function(stateParams){
  return map(function(s){
    exp(Gaussian(stateParams).score(s))
  }, stateVals)
});

// generate a statePrior using the possible heights and their probabilities
var generateStatePrior = cache(function(stateParams) {
  return Infer({
    model: function(){
      return categorical({vs: stateVals, ps: stateProbs(stateParams)})
    }
  })
});

// generate the uniform threshold prior
var thresholdBins ={
  positive: map(function(x){
    return  x - (1/(binParam*2));
  }, sort(stateVals)),
  negative: map(function(x){
    return  x + (1/(binParam*2));
  }, sort(stateVals))
};

var thresholdPrior = cache(function(form){
  return Infer({
    model: function() { return uniformDraw(thresholdBins[form]) }
  });
});
///

// information about the superordinate category priors
var subParams = {
  gymnasts: {mu: -1, sigma: 0.5}, // gymnast heights
  soccerPlayers: {mu: 0, sigma: 0.5}, // soccer player heights
  basketballPlayers: {mu: 1, sigma: 0.5} // basketball player heights
}

// possible utterances can be either positive (tall) or negative (short) or a null utterance
var utterances = ["tall", "short", "silence"]

// meaning function for utterances
var meaning = function(utterance, state, thresholds) {
  utterance == "tall" ? state > thresholds.tall :
  utterance == "short" ? state < thresholds.short :
  true
}

// assume a uniform prior over comparison classes
var classPrior = Infer({
  model: function(){return uniformDraw(["subordinate", "superordinate"])}
});

// set speaker optimality
var alpha = 5;

var literalListener = cache(
  function(utterance, thresholds, comparisonClass) {
    Infer({model: function(){
      var StatePrior = generateStatePrior(comparisonClass)
      var state = sample(StatePrior);
      var m = meaning(utterance, state, thresholds);
      condition(m);
      return state;
    }})
  }, 10000 // limit cache size
)

var speaker1 = cache(
  function(state, thresholds, comparisonClass) {
    Infer({model: function(){
      var utterance = uniformDraw(utterances);
      var L0 = literalListener(utterance, thresholds, comparisonClass);
      factor( alpha * L0.score(state) );
      return utterance;
    }})
  }, 10000 // limit cache size
)

var pragmaticListener = cache(function(utterance, subordinate_params) {
  Infer({model: function(){

    var statePrior = generateStatePrior(subordinate_params);
    var state = sample(statePrior);
    // separate thresholds for positive adjective and negative adjective
    var thresholds = {
      tall: sample(thresholdPrior("positive")),
      short: sample(thresholdPrior("negative"))
    }

    // uncertainty about the comparison class (superordinate vs. subordinate)
    var c = sample(classPrior)
    var comparisonClass = c == "subordinate" ? subordinate_params : superordinate_params

    var S1 = speaker1(state, thresholds, comparisonClass);
    observe(S1, utterance);

    return { comparisonClass: c, state : state }
  }})
}, 10000 // limit cache size
                             )

// the possible experiment conditions:
// you hear that someone is a member of a subordinate category
// then you are told that they are tall/short;
// the task is to figure out the implicit comparison class
var exptConditions = [
  {utt: "tall", sub: "basketballPlayers"},
  {utt: "short", sub: "basketballPlayers"},
  {utt: "tall", sub: "soccerPlayers"},
  {utt: "short", sub: "soccerPlayers"},
  {utt: "tall",  sub: "gymnasts"},
  {utt: "short", sub: "gymnasts"}
];

// generate structure predictions by mapping through the experiment conditions
var L1predictions = map(function(stim){
  var L1posterior = pragmaticListener(stim.utt, subParams[stim.sub])
  return {
    utterance: stim.utt,
    "P(superordinate comparison class)": exp(marginalize(L1posterior, "comparisonClass").score("superordinate")),
    "subordinate category": stim.sub,
    model: "L1"
  }
}, exptConditions)

display("the basketball player is tall")
display("--> height = " + expectation(marginalize(pragmaticListener("tall",{mu: 1, sigma: 0.5}), "state")))
display("the basketball player is short")
display("--> height = " + expectation(marginalize(pragmaticListener("short",{mu: 1, sigma: 0.5}), "state")))

display("probability of superordinate comparison class (i.e., tall for all people)")
viz.bar(L1predictions, {groupBy: "subordinate category"})

While these “lifted-variable” RSA models do not model semantic composition directly, they do capture its effect on utterance interpretations, which allows us to more precisely identify and investigate the factors that ought to push interpretations around. In other words, these models open up semantics to the purview of computational and experimental pragmatics; and by formalizing and thereby isolating the contributions of pragmatics, we may more accurately access the semantics.

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