///fold:
var butLast = function(xs){
return xs.slice(0, xs.length-1);
};
var KL = function(dist1, dist2){
var values = dist1.support();
return sum(map(function(value){
var scoreP = dist1.score(value);
var scoreQ = dist2.score(value);
var probP = Math.exp(scoreP);
var probQ = Math.exp(scoreQ);
return (probP == 0.0 ? 0.0 :
probQ == 0.0 ? 100 :
probP * (scoreP - scoreQ));
}, values));
};
var powerset = function(set) {
if (set.length == 0) {
return [[]];
} else {
var rest = powerset(set.slice(1));
return map(function(element) {
return [set[0]].concat(element);
}, rest).concat(rest);
}
};
///
var params = {
respondantAlpha : 1,
questionerAlpha : 1,
relevanceBeta: 1
};
var bakedGoods = ['raspPie', 'raspCake', 'lemonPie', 'lemonCake'];
var setsOfBakedGoods = map(function(v){return v.join('+');}, powerset(bakedGoods));
var pieCakeContext = {
// worlds include all possible sub-sets of 0 < k < N pies and cakes
worlds : setsOfBakedGoods,
// actions include ordering 1 pie/cake
actions: bakedGoods,
// questions include yes/no question for each baked good
questions: [{type: 'polar', text: 'lemonPie?'},
{type: 'polar', text: 'lemonCake?'},
{type: 'polar', text: 'raspPie?'},
{type: 'polar', text: 'raspCake?'}],
// assume questioner is uncertain but answerer has Delta on true world (e.g. shopkeep)
questionerBeliefs: Categorical({vs: setsOfBakedGoods}),
respondantBeliefs: Delta({v: 'lemonPie+lemonCake'}),
// raspberry pie is #1 preference (U=5), lemon cake is #2 preference (U=3).
// otherwise U=1 if whatever you order is in stock and 0 if it's not in stock
decisionProblem: function(w, a) {
return _.includes(w, a) ? (a == 'raspPie' ? 5 : a == 'lemonCake' ? 4 : 1) : 0;
},
// assume simplest meaning: 'yes' is true if item in question is available o.w. 'no' is true
meaning: function(world, question, response) {
return (response == 'yes' ? _.includes(world, butLast(question.text)) :
response == 'no' ? !_.includes(world, butLast(question.text)) :
console.error('unknown response', response));
}
};
// R0 chooses among responses licensed by the question
var getLicensedResponses = function(question) {
if(question.type == 'polar') {
return ['yes', 'no'];
} else {
return console.error('non-polar questions not yet supported');
}
};
// response cost is proportional to length in words
var cost = function(response) {
return response.split(' ').length;
};
// weight possible actions proportional
var relevanceProjection = function(beliefs, context) {
return Infer({method: 'enumerate'}, function() {
var action = uniformDraw(context.actions);
var world = sample(beliefs);
var decisionProblem = context.decisionProblem;
factor(Math.exp(decisionProblem(world, action)));
return action;
});
};
// gives updated beliefs about world state after hearing response to question
var updateBeliefs = function(beliefs, question, response, context) {
var meaning = context.meaning;
return Infer({method: 'enumerate'}, function() {
var world = sample(beliefs);
condition(meaning(world, question, response));
return world;
});
};
// agents weigh (1) general epistemic goal, (2) action-oriented goal, and (3) production cost
var utility = function(type, utterance, beliefs1, beliefs2, context) {
// respondant wants to *minimize* KL b/w beliefs (bringing closer to own belief)
// questioner wants to *maximize* KL b/w old & new beliefs (asking informative Q)
var signFlip = type == 'R0' ? -1 : 1;
var epistemicUtility = KL(beliefs1, beliefs2) * signFlip;
var actionUtility = KL(relevanceProjection(beliefs1, context),
relevanceProjection(beliefs2, context)) * signFlip;
console.log(type + ' considering utterance: ', utterance);
console.log('epistemic term:', epistemicUtility);
console.log('action-specific term:', actionUtility);
console.log('cost term:', cost(utterance));
console.log('--------------');
return ((1 - params.relevanceBeta) * epistemicUtility
+ params.relevanceBeta * actionUtility
- cost(utterance));
};
var R0 = cache(function(question, context) {
return Infer({method: 'enumerate'}, function(){
var response = uniformDraw(getLicensedResponses(question));
var ownBeliefs = context.respondantBeliefs;
var otherBeliefs = updateBeliefs(context.questionerBeliefs, question, response, context);
factor(params.respondantAlpha * utility('R0', response, ownBeliefs, otherBeliefs, context));
return response;
});
});
var Q1 = cache(function(context) {
return Infer({method: 'enumerate'}, function(){
var question = uniformDraw(context.questions);
var expectedUtility = expectation(context.questionerBeliefs, function(trueWorld) {
console.log(trueWorld)
var possibleResponses = R0(question, extend(context, {
respondantBeliefs: Delta({v: trueWorld})
}));
return expectation(possibleResponses, function(response) {
var currBeliefs = context.questionerBeliefs;
var updatedBeliefs = updateBeliefs(currBeliefs, question, response, context);
return utility('Q1', question.text, updatedBeliefs, currBeliefs, context);
});
});
factor(params.questionerAlpha * expectedUtility);
return question;
});
});
console.log('====================');
console.log('R0 test 1: "do you have lemon pie?"');
console.log('====================');
var lemonPieQuestion = pieCakeContext.questions[0];
console.log(R0(lemonPieQuestion, pieCakeContext));
console.log('====================');
console.log('R0 test 2: "do you have raspberry pie?"');
console.log('====================');
var raspberryPieQuestion = pieCakeContext.questions[2];
console.log(R0(raspberryPieQuestion, pieCakeContext));
console.log('====================');
console.log('Q1 test');
console.log('====================');
console.log(Q1(pieCakeContext));
Answerer should respond to ‘what time do you close?’ differently than ‘what is the price of a fifth of Jim Beam?’
///fold:
var toPairs = function(obj){
return _.sortBy(_.toPairs(obj));
}
var toObject = function(pairs){
return _.zipObject(pairs);
}
var makeObjectScorer = function(dist) {
var newDist = Infer({method: 'enumerate'}, function(){
return toPairs(sample(dist));
});
return function(value) {
return newDist.score(toPairs(value));
}
}
var KL = function(erpTrue, erpApprox){
var values = erpTrue.support([]);
return sum(map(
function(value){
var scoreP = erpTrue.score(value);
var scoreQ = erpApprox.score(value);
var probP = Math.exp(scoreP);
return probP == 0.0 ? 0.0 : probP * (scoreP - scoreQ);
}, values));
};
///
// ---------------
// | World knowledge |
// ---------------
var prices = [4, 5, 6];
var closingTimes = [8, 9];
var worldPrior = function(){
return {price: uniformDraw(prices),
time: uniformDraw(closingTimes)}
};
// -------------------
// | Question knowledge |
// -------------------
var timeQuestion = "What time do you close tonight?";
var priceQuestion = "What is the price of a fifth of Jim Beam?";
var timeQuestionMeaning = function(w) {return w.time}
var priceQuestionMeaning = function(w) {return w.price}
var questions = [timeQuestion, priceQuestion];
var questionPrior = function(){return uniformDraw(questions)};
// -----------------
// | Answer knowledge |
// -----------------
var priceAnswers = map(function(p) {return "the whiskey costs $" + p;}, prices);
var timeAnswers = map(function(t) {return "we close at " + t;}, closingTimes);
// evenly distribute probability across response types
var answerPrior = function(){
return flip(.5) ? uniformDraw(priceAnswers) : uniformDraw(timeAnswers)
};
var timeAnswerMeaning = function(time){
return function(world){return world.time == time};
};
var priceAnswerMeaning = function(price){
return function(world){return world.price == price};
};
// -----------
// | Interpreter |
// -----------
var meaning = function(utterance){
return ((utterance === timeQuestion) ? timeQuestionMeaning :
(utterance === priceQuestion) ? priceQuestionMeaning :
_.includes(timeAnswers, utterance) ? timeAnswerMeaning(last(utterance.split(' '))*1.0) :
_.includes(priceAnswers, utterance) ? priceAnswerMeaning(last(utterance.split('$'))*1.0) :
console.error('unknown utterance!', utterance));
};
var interpreter = cache(function(answer){
return Infer({method: 'enumerate'}, function(){
var world = worldPrior();
var answerMeaning = meaning(answer);
condition(answerMeaning(world));
return world;
});
});
var makeTruthfulAnswerPrior = function(trueWorld) {
return Infer({method: 'enumerate', model: function(){
var answer = answerPrior();
var interpScorer = makeObjectScorer(interpreter(answer));
factor(interpScorer(trueWorld))
return answer;
}});
};
// ------
// | QUDs |
// ------
var qudPrice = function(world){return world.price;};
var qudTime = function(world){return world.time;};
var qudPrior = function(){
return (flip(.5) ? "qudTime" : "qudPrice");
};
var QUDPosteriorFromMeaning = function(question) {
var correspondingQUD = question === 'timeQuestion' ? 'qudTime' : 'qudPrice';
return Delta({v: correspondingQUD})
};
var QUDPosteriorFromInference = function(question){
return Infer({method: 'enumerate'}, function() {
var qudName = qudPrior();
var q_erp = questioner('explicit', qudName);
observe(q_erp, question);
return qudName;
});
};
var nameToQUD = function(qudName){
return (qudName == "qudTime" ? qudTime :
qudName == "qudPrice" ? qudPrice :
qudName == "identity" ? function(v) {return v} :
console.error('unknown qud name', qudName));
};
// -------
// | Models |
// -------
var getProjectedWorldPrior = function(qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
return qud(worldPrior());
});
};
var rationality = 1;
var questioner = function(type, qudName) {
return Infer({method: 'enumerate'}, function() {
var question = questionPrior();
var possibleAnswer = Infer({method: 'enumerate'}, function(){
var trueWorld = worldPrior();
return sample(answerer(type, question, trueWorld));
});
var infGain = function(answer){
var prior = getProjectedWorldPrior(qudName);
var posterior = Infer({method: 'enumerate'}, function(){
var world = worldPrior();
observe(answerer(type, question, world), answer);
return nameToQUD(qudName)(world);
});
return KL(posterior, prior);
};
factor(expectation(possibleAnswer, infGain) * rationality);
return question;
});
};
var answerer = function(type, question, trueWorld) {
var truthfulAnswerPrior = makeTruthfulAnswerPrior(trueWorld);
var qudPosterior = (type === 'pragmatic' ? QUDPosteriorFromInference(question) :
type === 'explicit' ? QUDPosteriorFromMeaning(question) :
type === 'literal' ? Delta({v: 'identity'}) :
console.error('type not recognized'));
return Infer({method: 'enumerate'}, function(){
var qud = nameToQUD(sample(qudPosterior));
var answer = sample(truthfulAnswerPrior);
var utility = expectation(interpreter(answer), function(possibleWorld) {
return _.isEqual(qud(possibleWorld), qud(trueWorld))
});
factor(Math.log(utility) * rationality);
return answer;
});
};
var world = {price: 4, time: 8};
print('A0:');
viz.table(answerer('literal', timeQuestion, world))
viz.table(answerer('literal', priceQuestion, world))
print('A1:');
viz.table(answerer('explicit', timeQuestion, world))
viz.table(answerer('explicit', priceQuestion, world))
print('A2:')
viz.table(answerer('pragmatic', timeQuestion, world))
viz.table(answerer('pragmatic', priceQuestion, world))
Depending on the context, liquor merchants will be more or less likely to give over-informative answers. Our model accounts for this via inference about the underlying goal, given the context.
///fold:
var KL = function(erpTrue, erpApprox){
var values = erpTrue.support([]);
return sum(map(
function(value){
var scoreP = erpTrue.score(value);
var scoreQ = erpApprox.score(value);
var probP = Math.exp(scoreP);
return probP == 0.0 ? 0.0 : probP * (scoreP - scoreQ);
}, values));
};
///
// ---------------
// | World knowledge |
// ---------------
var buyWhiskeyContext = "I'd like to buy some whiskey.";
var spendFiveDollarsContext = "I only have $5 to spend.";
var prices = [1,2,3,4, 5, 6,7,8,9,10];
var isNumeric = function(x){
return _.includes(prices, x);
};
var worldPrior = function(){
return uniformDraw(prices);
};
// -------------------
// | Question knowledge |
// -------------------
var isMoreThanFiveQuestion = "Does Jim Beam cost more than $5?";
var isMoreThanFiveQuestionMeaning = function(world){
return world > 5;
};
var questions = [isMoreThanFiveQuestion];
var questionPrior = function(){
return uniformDraw(questions);
};
// -----------------
// | Answer knowledge |
// -----------------
var literalAnswers = ["yes", "no"];
var priceAnswers = map(function(p) {return "the whiskey costs $" + p;}, prices);
// evenly distribute probability across response types
var answerPrior = function(){
return flip(0.5) ? uniformDraw(literalAnswers) : uniformDraw(priceAnswers);
};
var numericAnswerMeaning = function(number){
return function(world){
return world == number;
};
};
var booleanAnswerMeaning = function(bool){
return function(world){
return (world > 5) == bool;
};
};
// -----------
// | Interpreter |
// -----------
var meaning = function(utterance){
return ((utterance === "yes") ? booleanAnswerMeaning(true) :
(utterance === "no") ? booleanAnswerMeaning(false) :
(utterance === isMoreThanFiveQuestion) ? isMoreThanFiveQuestionMeaning :
_.includes(priceAnswers, utterance) ? numericAnswerMeaning(last(utterance.split('$'))*1.0) :
console.error('unknown utterance!', utterance));
};
var interpreter = cache(function(answer){
return Infer({method: 'enumerate'}, function(){
var world = worldPrior();
var answerMeaning = meaning(answer);
condition(answerMeaning(world));
return world;
});
});
var makeTruthfulAnswerPrior = function(trueWorld) {
return Infer({method: 'enumerate', model: function(){
var answer = answerPrior();
factor(interpreter(answer).score(trueWorld));
return answer;
}});
};
// ------
// | QUDs |
// ------
var qudPrice = function(world){return world;};
var qudPriceGreaterThan5 = function(world){return world > 5;};
var qudPrior = function(context){
var p = ((context === buyWhiskeyContext) ? 0.5 :
(context === spendFiveDollarsContext) ? 0.99 :
console.error('unknown context'));
return (flip(p) ? "qudPriceGreaterThan5" : "qudPrice");
};
var QUDPosteriorFromMeaning = function(question) {
var questionMeaning = (question === isMoreThanFiveQuestion ?
"qudPriceGreaterThan5" :
console.error('unknown question'));
return Delta({v: questionMeaning})
};
var QUDPosteriorFromInference = function(question, context){
return Infer({method: 'enumerate'}, function() {
var qudName = qudPrior(context);
var q_erp = questioner('explicit', qudName);
observe(q_erp, question);
return qudName;
});
};
var nameToQUD = function(qudName){
return (qudName == "qudPriceGreaterThan5" ? qudPriceGreaterThan5 :
qudName == "qudPrice" ? qudPrice :
qudName == "identity" ? function(v) {return v} :
console.error('unknown qud name', qudName));
};
// -------
// | Models |
// -------
var getProjectedWorldPrior = function(qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
return qud(worldPrior());
});
};
var rationality = 1;
var questioner = function(type, qudName) {
return Infer({method: 'enumerate'}, function() {
var question = questionPrior();
var possibleAnswer = Infer({method: 'enumerate'}, function(){
var trueWorld = worldPrior();
return sample(answerer(type, question, trueWorld));
});
var infGain = function(answer){
var prior = getProjectedWorldPrior(qudName);
var posterior = Infer({method: 'enumerate'}, function(){
var world = worldPrior();
observe(answerer(type, question, world), answer);
return nameToQUD(qudName)(world);
});
return KL(posterior, prior);
};
factor(expectation(possibleAnswer, infGain) * rationality);
return question;
});
};
var answerer = function(type, question, trueWorld, context) {
var truthfulAnswerPrior = makeTruthfulAnswerPrior(trueWorld);
var qudPosterior = (type === 'pragmatic' ? QUDPosteriorFromInference(question, context) :
type === 'explicit' ? QUDPosteriorFromMeaning(question) :
type === 'literal' ? Delta({v: 'identity'}) :
console.error('type not recognized'));
return Infer({method: 'enumerate'}, function(){
var qud = nameToQUD(sample(qudPosterior));
var answer = sample(truthfulAnswerPrior);
var utility = expectation(interpreter(answer), function(possibleWorld) {
return qud(possibleWorld) === qud(trueWorld);
});
factor(Math.log(utility) * rationality);
return answer;
});
};
var world = 4;
print('A0:');
viz.table(answerer('literal', isMoreThanFiveQuestion, world, buyWhiskeyContext))
viz.table(answerer('literal', isMoreThanFiveQuestion, world, spendFiveDollarsContext))
print('A1:');
viz.table(answerer('explicit', isMoreThanFiveQuestion, world, buyWhiskeyContext))
viz.table(answerer('explicit', isMoreThanFiveQuestion, world, spendFiveDollarsContext))
print('A2:')
viz.table(answerer('pragmatic', isMoreThanFiveQuestion, world, buyWhiskeyContext))
viz.table(answerer('pragmatic', isMoreThanFiveQuestion, world, spendFiveDollarsContext))
Our final example, using an experiment by Clark (1979) testing how merchants responded to different ways of asking about the credit cards they accept, demonstrates how our pragmatic answerer can use Gricean reasoning to infer underlying goals directly from the question utterance (via a model of the questioner) instead of using broader context clues.
///fold:
var KL = function(erpTrue, erpApprox){
var values = erpTrue.support([]);
return sum(map(
function(value){
var scoreP = erpTrue.score(value);
var scoreQ = erpApprox.score(value);
var probP = Math.exp(scoreP);
return probP == 0.0 ? 0.0 : probP * (scoreP - scoreQ);
}, values));
};
var powerset = function(set) {
if (set.length == 0)
return [[]];
else {
var rest = powerset(set.slice(1));
return map(
function(element) {
return [set[0]].concat(element);
},
rest).concat(rest);
}
}
var butLast = function(xs){
return xs.slice(0, xs.length-1);
};
var cardLikelihoods = {
'Visa' : 0.5,
'MasterCard' : 0.5,
'AmericanExpress' : 0.5,
'Diners' : 0.5,
'CarteBlanche' : 0.5
};
var getCardFromQuestion = function(q) {
return butLast(last(q.split(' ')))
}
///
// --------------------------------------------------------------------
// ---------------
// | World knowledge |
// ---------------
var cardTypes = ['Visa','MasterCard', 'AmericanExpress', 'Diners', 'CarteBlanche'];
var cardPowerSet = powerset(cardTypes);
// Reflect real probabilities of acceptance from Clark
var worldPrior = function(){
return {
'Visa' : flip(cardLikelihoods['Visa']),
'MasterCard' : flip(cardLikelihoods['MasterCard']),
'AmericanExpress' : flip(cardLikelihoods['AmericanExpress']),
'Diners' : flip(cardLikelihoods['Diners']),
'CarteBlanche' : flip(cardLikelihoods['CarteBlanche'])
};
};
// -------------------
// | Question knowledge |
// -------------------
var cardQuestions = map(function(s) {return "Do you accept " + s + "?"}, cardTypes);
var cardQuestionMeaning = function(card) {
return function(world) {return world[card]}
}
var creditCardsQuestion = "Do you accept credit cards?";
var creditCardsQuestionMeaning = function(world){
return _.some(_.values(world));
};
var questions = cardQuestions.concat([creditCardsQuestion])
var questionPrior = function(){return uniformDraw(questions)};
// -----------------
// | Answer knowledge |
// -----------------
var cardAnswerSpace = powerset(cardTypes);
var booleanAnswerSpace = ["yes", "no"];
// 'yes' 'no' or some combination of cards
var answerPrior = function(){
return uniformDraw(booleanAnswerSpace.concat(cardAnswerSpace))
};
var cardAnswerMeaning = function(cardList){
return function(questionMeaning){
return function(world){
return _.every(map(function(card) {return world[card]}, cardList));
};
};
};
var booleanAnswerMeaning = function(bool){
return function(questionMeaning){
return function(world){return questionMeaning(world) == bool}
}
}
var cost = function(answer) {
return _.includes(booleanAnswerSpace, answer) ? 0 : answer.length;
}
// -----------
// | Interpreter |
// -----------
var meaning = function(utterance){
return (_.includes(cardAnswerSpace, utterance) ? cardAnswerMeaning(utterance) :
_.includes(cardQuestions, utterance) ? cardQuestionMeaning(getCardFromQuestion(utterance)) :
utterance === creditCardsQuestion ? creditCardsQuestionMeaning :
utterance === "yes" ? booleanAnswerMeaning(true) :
utterance === "no" ? booleanAnswerMeaning(false) :
console.error('unknown utterance in meaning!', utterance));
};
var interpreter = cache(function(question, answer){
return Infer({method: 'enumerate', model: function(){
var world = worldPrior();
var answerMeaning = meaning(answer);
var questionMeaning = meaning(question);
condition(answerMeaning(questionMeaning)(world));
return world;
}});
});
var makeTruthfulAnswerPrior = dp.cache(function(question, trueWorld) {
return Infer({method: 'enumerate', model: function(){
var answer = answerPrior();
factor(interpreter(question, answer).score(trueWorld));
return answer;
}});
});
// ------
// | QUDs |
// ------
var qudPrior = function(){
var filteredPowerSet = filter(function(set){return !_.isEmpty(set)}, cardPowerSet);
return "QUD" + uniformDraw(filteredPowerSet);
};
var nameToQUD = function(qudName) {
var cardList = qudName.slice(3).split(",");
return function(world){
return _.some(map(function(card){return world[card]}, cardList));
};
};
var QUDPosteriorFromInference = function(question){
return Infer({method: 'enumerate'}, function() {
var qudName = qudPrior();
observe(questioner('explicit', qudName), question);
return qudName;
});
};
var QUDPosteriorFromMeaning = function(question){
var correspondingQUD = (_.includes(cardQuestions, question) ?
"QUD" + getCardFromQuestion(question) :
"QUD" + cardTypes.join(','));
return Delta({v: correspondingQUD});
};
var inferQUD = dp.cache(function(type, question) {
return (type === 'pragmatic' ? QUDPosteriorFromInference(question) :
type === 'explicit' ? QUDPosteriorFromMeaning(question) :
type === 'literal' ? Delta({v: 'identity'}) :
console.error('type not recognized'));
})
// -------
// | Models |
// -------
var getProjectedWorldPrior = dp.cache(function(qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){return qud(worldPrior())});
});
var rationality = 50;
var costWeight = 5;
var questioner = dp.cache(function(type, qudName) {
return Infer({method: 'enumerate'}, function() {
var question = questionPrior();
var possibleAnswer = Infer({method: 'enumerate'}, function(){
var trueWorld = worldPrior();
return sample(answerer(type, question, trueWorld));
});
var infGain = function(answer){
var prior = getProjectedWorldPrior(qudName);
var posterior = Infer({method: 'enumerate'}, function(){
var world = worldPrior();
observe(answerer(type, question, world), answer);
return nameToQUD(qudName)(world);
});
return KL(posterior, prior);
};
factor(expectation(possibleAnswer, infGain) * rationality);
return question;
});
});
var answerer = dp.cache(function(type, question, trueWorld) {
var truthfulAnswerPrior = makeTruthfulAnswerPrior(question, trueWorld);
var qudPosterior = inferQUD(type, question)
return Infer({method: 'enumerate'}, function(){
var qud = nameToQUD(sample(qudPosterior));
var answer = sample(truthfulAnswerPrior);
var utility = expectation(interpreter(question, answer), function(possibleWorld) {
return qud(possibleWorld) === qud(trueWorld);
});
factor(rationality * Math.log(utility) - costWeight * cost(answer));
return answer;
});
})
var runModel = function(question) {
return Infer({method: 'enumerate', model: function(){
var trueWorld = worldPrior();
var ans = answerer('pragmatic', question, trueWorld);
return _.includes(['yes', 'no'], sample(ans)) ? 'yes/no' : 'exhuastive';
}});
};
// Examine questioner behavior
viz(questioner('explicit', "QUDAmericanExpress"))
viz(questioner('explicit', "QUDAmericanExpress,Visa,MasterCard,Diners,CarteBlanche"))
// // Examine goal inference
viz(marginalize(inferQUD('pragmatic', first(questions)), function(x) {return x.slice(3).split(",").length}))
viz(marginalize(inferQUD('pragmatic', last(questions)), function(x) {return x.slice(3).split(",").length}))
viz(Infer({method: 'enumerate'}, function() {return qudPrior().slice(3).split(",").length}))
// Examine answerer behavior
viz.table(runModel(questions[1]))
viz.table(runModel(creditCardsQuestion))
We set up the mention-some disambiguation problem below, using the same structure as the previous model. This example uses a more structured world space and answer prior, but uses the same mechanism where context shifts the QUD prior.
var filterWorld = function(world) {
return _.fromPairs(reduce(function(value, memo) {
var hasNewspaper = world[value]['hasNewspaper'];
return hasNewspaper ? memo.concat([[value, world[value]]]) : memo;
}, [], _.keys(world)));
}
var pickAllNewspaperCafes = function(world) {
var filt = filterWorld(world);
return _.isEmpty(filt) ? ['none'] : _.keys(filt);
}
var pickClosestNewspaperCafe = function(world) {
var filt = filterWorld(world);
if(_.isEmpty(filt)) {
return ['none'];
} else {
var minDist = _.min(map(function(v){return v['distance']}, _.values(filt)));
return sort(filter(function(v) {return filt[v]['distance'] == minDist},
_.keys(filt)));
}
};
var KL = function(erpTrue, erpApprox){
var values = erpTrue.support([]);
return sum(map(
function(value){
var scoreP = erpTrue.score(value);
var scoreQ = erpApprox.score(value);
var probP = Math.exp(scoreP);
return probP == 0.0 ? 0.0 : probP * (scoreP - scoreQ);
}, values));
};
///
// ---------------
// | World knowledge |
// ---------------
var distances = [1,3]
var cafes = ['cafe1', 'cafe2', 'cafe3', 'cafe4']
var touristContext = "I'm new in town.";
var businesspersonContext = "I'm trying to set up a newspaper distribution business.";
var isCafeList = function(l){
var predicate = function(s) {return _.includes(cafes, s);};
return all(predicate, l);
};
var worldPrior = function(){
return {
'cafe1' : {'distance' : uniformDraw(distances), 'hasNewspaper' : flip(.5)},
'cafe2' : {'distance' : uniformDraw(distances), 'hasNewspaper' : flip(.5)},
'cafe3' : {'distance' : uniformDraw(distances), 'hasNewspaper' : flip(.5)},
'cafe4' : {'distance' : uniformDraw(distances), 'hasNewspaper' : flip(.5)}
}
}
// -------------------
// | Question knowledge |
// -------------------
var newspaperQuestion = "Where can one buy an Italian newspaper?";
// projects from a world to the relevant properties for the desired answer
var newspaperQuestionMeaning = function(world){return pickAllNewspaperCafes(world);};
var questions = [newspaperQuestion]
var questionPrior = function(){
return uniformDraw(questions);
};
// -----------------
// | Answer knowledge |
// -----------------
// (truncated) geometric distribution
var answerPrior = function(){
return flip(.8) ? [] : sort([uniformDraw(cafes)].concat(answerPrior()));
};
var cafeAnswerMeaning = function(cafeList){
return function(world){
var predicate = function(cafe) {return world[cafe]['hasNewspaper'];};
return all(predicate, cafeList);
};
};
var noneMeaning = function() {
return function(world){
var predicate = function(cafe) {return !world[cafe]['hasNewspaper'];};
return all(predicate, cafes);
};
};
// -----------
// | Interpreter |
// -----------
var meaning = function(utterance){
return (isCafeList(utterance) ? cafeAnswerMeaning(utterance) :
(_.isEqual(utterance, [])) ? noneMeaning() :
(utterance === newspaperQuestion) ? newspaperQuestionMeaning :
console.error('unknown utterance!', utterance));
};
var interpreter = cache(function(answer){
return Infer({method: 'enumerate'}, function(){
var world = worldPrior();
var answerMeaning = meaning(answer);
condition(answerMeaning(world));
return world;
});
});
// ------
// | QUDs |
// ------
var qudAll = function(world){return pickAllNewspaperCafes(world);};
var qudClosest = function(world){return pickClosestNewspaperCafe(world);};
var qudPrior = function(context){
var p = ((context === businesspersonContext) ? 1 :
(context === touristContext) ? 0 :
console.error('unknown context'));
return (flip(p) ? "qudAll" : "qudClosest");
};
var nameToQUD = function(qudName){
return (qudName === "qudClosest" ? qudClosest :
qudName === "qudAll" ? qudAll :
qudName === newspaperQuestion ? newspaperQuestionMeaning :
console.error('unknown qud name', qudName));
};
// -------
// | Models |
// -------
var getConsistentWorlds = function(answer, qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
var inferredWorld = sample(interpreter(answer));
return qud(inferredWorld);
});
};
var getProjectedWorldPrior = function(qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
return qud(worldPrior());
});
};
var explicitAnswerer = cache(function(question, trueWorld, rationality) {
var qud = nameToQUD(question);
return Infer({method: 'enumerate', maxExecutions: 1}, function(){
var answer = answerPrior();
var consistentWorldDist = getConsistentWorlds(answer, question);
factor(consistentWorldDist.score(qud(trueWorld)) * rationality);
return answer;
});
});
var explicitQuestioner = cache(function(qudName, rationality) {
return Infer({method: 'enumerate'}, function(){
var question = questionPrior();
var prior = getProjectedWorldPrior(qudName);
var informationGainDist = Infer({method: 'enumerate'}, function(){
var trueWorld = worldPrior();
var possibleAnswer = sample(explicitAnswerer(question, trueWorld, rationality));
var posterior = getConsistentWorlds(possibleAnswer, qudName);
return KL(posterior, prior);
});
factor(expectation(informationGainDist) * rationality);
return question;
});
});
var inferQUD = cache(function(context, question, rationality){
return Infer({method: 'enumerate'}, function() {
var qudName = qudPrior(context);
var q_erp = explicitQuestioner(qudName, rationality);
observe(q_erp, question);
return qudName;
});
});
var pragmaticAnswerer = cache(function(context, question, trueWorld, rationality){
var qudPosterior = inferQUD(context, question, rationality);
return Infer({method: 'enumerate', maxExecutions: 500}, function(){
var qudName = sample(qudPosterior);
var answer = answerPrior();
var consistentWorldDist = getConsistentWorlds(answer, qudName);
factor(consistentWorldDist.score(nameToQUD(qudName)(trueWorld)) * rationality);
return answer;
});
});
var world = {'cafe1' : {'distance' : 3, 'hasNewspaper' : false},
'cafe2' : {'distance' : 1, 'hasNewspaper' :true},
'cafe3' : {'distance' : 3, 'hasNewspaper' :true},
'cafe4' : {'distance' : 3, 'hasNewspaper' :true}}
console.log(businesspersonContext, newspaperQuestion);
console.log(pragmaticAnswerer(businesspersonContext, newspaperQuestion, world,10));
console.log(touristContext, newspaperQuestion);
console.log(pragmaticAnswerer(touristContext, newspaperQuestion, world, 10));
Here, we use a more sophisticated family of QUDs, and inference takes place directly over these alternatives, relative to the true world, instead of using an explicit context.
var KL = function(erpTrue, erpApprox){
var values = erpTrue.support([]);
return sum(map(
function(value){
var scoreP = erpTrue.score(value);
var scoreQ = erpApprox.score(value);
var probP = Math.exp(scoreP);
return probP == 0.0 ? 0.0 : probP * (scoreP - scoreQ);
}, values));
};
var flatten = function(xs){
if (xs.length == 0) {
return [];
} else {
return xs[0].concat(flatten(xs.slice(1)));
}
};
var timeDiff = function(timeString1, timeString2) {
var hr1 = timeString1.split(":")[0]
var hr2 = timeString2.split(":")[0]
var min1 = 60*hr1 + 1*timeString1.split(":")[1]
var min2 = 60*hr2 + 1*timeString2.split(":")[1]
return Math.abs(min2 - min1)
}
var getEveryFifthTime = function(list) {
return filter(function(num) {
return num[3] % 5 == 0;
}, list);
};
var getNonFifthTimes = function(list) {
return filter(function(num) {
return num[3] % 5 != 0;
}, list);
};
var roundToNearest = function(time) {
var hour = time.split(":")[0]
var minutes = time.split(":")[1]
var roundedMinutes = 5 * Math.round(minutes / 5)
return hour + ":" + roundedMinutes
}
///
// ---------------
// | World knowledge |
// ---------------
var times = map(function(n) {return "3:" + n;}, _.range(30, 60));
// World state is just the true current time
var worldPrior = function(){
return uniformDraw(times)
};
// -------------------
// | Question knowledge |
// -------------------
var timeQuestion = "What time is it?";
// projects from a world to the relevant properties for the desired answer
var timeQuestionMeaning = function(world){
return world
};
var questions = [timeQuestion]
var questionPrior = function(){
return uniformDraw(questions);
};
// -----------------
// | Answer knowledge |
// -----------------
// saying rounded times is easier (b/c analog watches)
var answerPrior = function(){
var ans = (flip(.25) ?
uniformDraw(getNonFifthTimes(times)) :
uniformDraw(getEveryFifthTime(times)));
return ans;
};
var timeAnswerMeaning = function(currTime){
return function(world){
return timeDiff(currTime, world) == 0;
};
};
// -----------
// | Interpreter |
// -----------
var meaning = function(utterance){
return (_.includes(times, utterance) ? timeAnswerMeaning(utterance) :
(utterance === timeQuestion) ? timeQuestionMeaning :
console.error('unknown utterance!', utterance));
};
var interpreter = cache(function(answer){
return Infer({method: 'enumerate'}, function(){
var world = worldPrior();
var answerMeaning = meaning(answer);
condition(answerMeaning(world));
return world;
});
});
// ------
// | QUDs |
// ------
var qudFactory = function(threshold) {
return function(world){
if(world < threshold) {
return roundToNearest(world);
} else {
return world;
}
};
};
// Family of quds parameterized by threshold at which "running late"
// Thresholds closer to the appointment time (provided by context) are more likely
var qudPrior = function(context){
var timeDiffs = map(function(time) {return timeDiff(time, context)},
times);
var maxDiff = Math.max.apply(null, timeDiffs);
var timeProbs = map(function(timeDiff) {return maxDiff - timeDiff;},
timeDiffs);
var qudWord = (flip(0.25) ?
"Exact" :
categorical(timeProbs, times));
return "qud" + qudWord;
};
var nameToQUD = function(qudName){
if (qudName == timeQuestion) {
return timeQuestionMeaning;
} else if (qudName === "qudExact") {
return (function(world) {return world;});
} else {
var threshold = qudName.slice(3);
return qudFactory(threshold);
}
};
// -------
// | Models |
// -------
var getConsistentWorlds = function(answer, qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
var inferredWorld = sample(interpreter(answer));
return qud(inferredWorld);
});
};
var getProjectedWorldPrior = function(qudName) {
var qud = nameToQUD(qudName);
return Infer({method: 'enumerate'}, function(){
return qud(worldPrior());
});
};
var explicitAnswerer = cache(function(question, trueWorld, rationality) {
var qud = nameToQUD(question);
return Infer({method: 'enumerate'}, function(){
var answer = answerPrior();
var consistentWorldDist = getConsistentWorlds(answer, question);
factor(consistentWorldDist.score(qud(trueWorld)) * rationality);
return answer;
});
});
var explicitQuestioner = cache(function(qudName, rationality) {
return Infer({method: 'enumerate'}, function(){
var question = questionPrior();
var prior = getProjectedWorldPrior(qudName);
var informationGainDist = Infer({method: 'enumerate'}, function(){
var trueWorld = worldPrior();
var possibleAnswer = sample(explicitAnswerer(question, trueWorld, rationality));
var posterior = getConsistentWorlds(possibleAnswer, qudName);
return KL(posterior, prior);
});
factor(expectation(informationGainDist) * rationality);
return question;
});
});
var inferQUD = cache(function(context, question, rationality){
return Infer({method: 'enumerate'}, function() {
var qudName = qudPrior(context);
var q_erp = explicitQuestioner(qudName, rationality);
observe(q_erp, question);
return qudName;
});
});
var pragmaticAnswerer = cache(function(context, question, trueWorld, rationality){
var qudPosterior = inferQUD(context, question, rationality);
return Infer({method: 'enumerate'}, function(){
var qudName = sample(qudPosterior);
var answer = answerPrior();
var consistentWorldDist = getConsistentWorlds(answer, qudName);
factor(consistentWorldDist.score(nameToQUD(qudName)(trueWorld)) * rationality);
return answer;
});
});
var appointmentContext = "4:00";
var runModel = function(group) {
return expectation(Infer({method: 'enumerate', model: function(){
var trueWorld = worldPrior();
var ansERP = pragmaticAnswerer(appointmentContext, timeQuestion, trueWorld, 1);
condition(group === "early" ?
trueWorld.slice(2) < 45 :
trueWorld.slice(2) > 45);
return Math.exp(ansERP.score(roundToNearest(trueWorld)));
}}));
};
viz(Infer({method: 'enumerate', model: function(){return qudPrior(appointmentContext)}}))
print("early rounds p = " + runModel("early"));
print("late rounds p = " + runModel("late"));