A statistical model for aggregating judgments by incorporating peer predictions

03/14/2017 ∙ by John McCoy, et al. ∙ MIT 0

We propose a probabilistic model to aggregate the answers of respondents answering multiple-choice questions. The model does not assume that everyone has access to the same information, and so does not assume that the consensus answer is correct. Instead, it infers the most probable world state, even if only a minority vote for it. Each respondent is modeled as receiving a signal contingent on the actual world state, and as using this signal to both determine their own answer and predict the answers given by others. By incorporating respondent's predictions of others' answers, the model infers latent parameters corresponding to the prior over world states and the probability of different signals being received in all possible world states, including counterfactual ones. Unlike other probabilistic models for aggregation, our model applies to both single and multiple questions, in which case it estimates each respondent's expertise. The model shows good performance, compared to a number of other probabilistic models, on data from seven studies covering different types of expertise.

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Introduction

It is a truism that the knowledge of groups of people, particularly experts, outperforms that of individuals [43] and there is increasing call to use the dispersed judgments of the crowd in policy making [42]. There is a large literature spanning multiple disciplines on methods for aggregating beliefs (for reviews see [9, 6, 7]), and previous applications have included political and economic forecasting [3, 27], evaluating nuclear safety [10] and public policy [28], and assessing the quality of chemical probes [31]. However, previous approaches to aggregating beliefs have implicitly assumed ‘kind’ (as opposed to ‘wicked’) environments [16]. In a previous paper, [35] we proposed an algorithm for aggregating beliefs using not only respondent’s answers but also their prediction of the answer distribution, and proved that for an infinite number of non-noisy Bayesian respondents, it would always determine the correct answer if sufficient evidence was available in the world.

Here, we build on this approach but treat the aggregation problem as one of statistical inference. We propose a model of how people formulate their own judgments and predict the distribution of the judgments of others, and use this model to infer the most probable world state giving rise to the observed data from people. The model can be applied at the level of a single question but also across multiple questions, to infer the domain expertise of respondents. The model is thus broader in scope than other machine learning models for aggregation in that it accepts unique questions, but can also be compared to their performance across multiple questions. We do not assume that the aggregation model has access to correct answers or to historical data about the performance of respondents on similar questions. By using a simple model of how people make such judgments, we are able to increase the accuracy of the group’s aggregate answer in domains ranging from estimating art prices to diagnosing skin lesions.

Possible worlds and peer predictions

Condorcet’s Jury Theorem [8], of 1785, considers a group of people making a binary choice with one option better for all members of the group. All individuals in the group are assumed to vote for the better option with probability . This is also known as the assumption of voter competence. The theorem states that the probability that the majority vote for the better alternative exceeds , and approaches as the group size increases to infinity.111See [42] for a review of modern extensions to Condorcet’s Jury Theorem.. Following this theorem, much of the belief aggregation and social choice theory literature assumes voter competence and thus focusses on methods which use the group’s consensus answer, for example by computing the modal answer for categorical questions or the mean or median answer for aggregating continuous quantities. 222The work on aggregating continuous quantities using the mean or median also has an interesting history, dating back to Galton estimating the weight of an ox using the crowd’s judgments [14].

To build intution both how our model relates to, but also departs from, previous work on aggregating individual judgments, we begin with a model of Austen-Smith and Banks [1]

who argue that previous work following the Concordet Jury Theorem starts the analysis ‘in the middle’ with members of the group voting according to the posterior probabilities that they assign to the various options, but without the inputs to these posteriors specified. They describe a model with two possible world states

and two possible options to select and assume that everyone attaches a utility of 1 to the option that is the same as the actual world state, and a utility of 0 to the other option. Throughout, we conceptualise the possible world states as corresponding to the different possible answers to the question under consideration, with the correct answer, based on current evidence, corresponding to the actual or true state of the world.333In this paper, we ignore utilities and assume that people vote for the world state that they believe is most probable. The inputs we elicit from respondents are suffficient to implement mechanisms that make answering in this way incentive compatible, for example by using the Bayesian Truth Serum [34][19]. The actual world state is unknown to all individuals, but there is a common prior that the world is in state A, and each individual receives a private signal .444Throughout the paper, we describe other models using a consistent notation, rather than necessarily that used by the original authors. Crucially, Austen-Smith and Banks assume that signal is strictly more likely than signal in world and signal is strictly more likely than signal in world . That is, they assume that the distribution on signals is such that signal has probability greater than in world and signal has probability greater than 0.5 in world . After receiving her signal, each individual updates her prior belief and votes for the alternative she believes provides the higher utility.

In the Austen-Smith and Banks model, the majority verdict favors the most widely available signal, but the assumption that the most widely available signal is correct may be false for actual questions. For example, shallow information is often widely available but more specialized information which points in a different (correct) direction may be available only to a few experts. The model thus does not apply to situations where the majority is incorrect.

We previously proposed a model that does not assume that the most widely available information is always correct [35]. Using the same setup as Austen-Smith and Banks, we require only that signal is more probable in world than in the other world, This i does not imply that signal is necessarily the most likely signal in world . If the world is in state then we refer to signal as the correct signal since a Bayesian respondent receiving signal places more probability on the correct world than a respondent receiving any other signal. Our model does not assume that in any given world state the most likely signal is also the correct signal. For example, under our model, signal may have probability 0.8 in state and probability 0.7 in state and thus be more likely than signal in both world states. Under this signal distribution, if the actual state was then the majority would receive signal , and, assuming a uniform common prior, would vote for state and so would be incorrect.555Assuming this signal distribution and a uniform common prior, consider a respondent who received signal . Then, their estimate of the probability that the world is in state is and since this quantity is higher than 0.5, respondents receiving signal vote that the world is most likely in state . But if the actual world is state , respondents have a .7 probability of receiving signal and hence the majority of respondents will vote incorrectly.

We call the model proposed in this paper the ‘possible worlds model’ (PWM) since to determine the actual world state it is not sufficient to consider only how people actually vote, but, instead, one needs to also consider the distribution of votes in all possible worlds. That is, to determine the correct answer, one requires not simply what fraction of individuals voted for a particular answer under the actual world state, but also what fraction would have voted for that answer in all counterfactual world states. A useful intuition is that an answer which is likely in all possible worlds should be discounted relative to an answer which is likely only in one world state: simply because 60% of respondents vote for option does not guarantee that the world is actually in state since perhaps 80% of respondents would have voted for if this was actually the case. If an aggregation algorithm had access to the world prior and signal distributions for every possible world, then recovering the correct answer would be as simple as determining which world had an expected distribution of votes matching the observed frequency of votes.666The algorithm would also require access to the world prior to determine the expected vote frequencies, which may not match the expected signal frequencies. The new challenge presented by our possible worlds model is how to tap respondent’s meta-knowledge of such counterfactual worlds, and our solution is to do this by eliciting respondent’s predictions of how other respondents vote. We model such predictions as depending on both a respondent’s received signal and their knowledge of the signal distribution and world prior, which allows us to infer the necesary information about counterfactual worlds.777In a previous paper [35] we showed that given our weaker assumptions on the signal distribution, eliciting only votes or even full posteriors over worlds from ideal Bayesian respondents is not sufficient to identify the actual world for multiple choice questions with any number of options. On the other hand, we provide formal conditions under which eliciting votes and predictions about others votes will always uncover the correct answer. For binary questions, the answer which is more popular than predicted can be shown to be the best answer under such conditions. The probabilistic generative model proposed in this paper assumes that such predictions are generated by respondents as noisy approximations to an exact Bayesian computation.

Our assumptions about people’s predictions of others are supported by work in psychology that has robustly demonstrated that people’s predictions of others answers relate to their own answer, and, in particular, are consistent with respondents implicitly conditioning on their own answer as an ‘informative sample of one” [11, 12, 17, 23, 36]. People show a so-called false consensus effect whereby people who endorse an answer believe that others are also more likely to endorse it. For example, in one study [39], about 50% of surveyed undergraduates said that they themselves would wear a sign saying “Repent” around campus for a psychology experiment and predicted that 61% of other students would wear such a sign, whereas students who would not wear such a sign predicted that 30% would. As Dawes pointed out in a seminal paper [11]

there is nothing necessarily false about such an effect, it is rational to use your own belief as a sample of the population and base your prediction of others on your own beliefs. In our model, people’s beliefs do not directly affect their predictions, but rather both their own beliefs and predictions are both conditional on the private signal which they received. In this paper, we use the model discussed above to develop statistical models of aggregation. We thus turn now to previous attempts that tackle the aggregation problem as one of Bayesian inference, attempting to infer the correct answer given data from multiple individuals.

Aggregation as Bayesian inference

One approach to the problem of aggregating the knowledge or answers of a group of people is to treat it as a Bayesian inference problem, and model how the data elicited from respondents is generated conditional on the actual world state. The observed data is then used to infer a posterior distribution over possible world states. Unlike our model, other such aggregation models either require data from respondents answering multiple questions or require the user to specify a prior distribution over the answers.

Bayesian approaches to aggregating opinions are not new [46, 29]. Using such an approach, the aggregator specifies a prior distribution over the variable of interest and updates this prior with respect to a likelihood function associated with information from respondents about this variable [9]. For example, Bayesian models with different priors and likelihood functions have been compared for the problem of determining the value of an indicator variable where each expert gives the probability that the indicator variable is turned on [5].

We will compare the PWM to two other hierarchical Bayesian models for aggregation: a Bayesian Cultural Consensus Theory model [21, 33] and a Bayesian cognitive hierarchy model [24]. We give formal definitions of these two models in section Comparison models, and explain here how they relate to other models for aggregating opinions.

Cultural consensus models [38, 2, 37, 45] are a class of models used to uncover the shared beliefs of a group. Respondents are asked true or false questions, and these answers are used to infer each respondent’s cultural competence and the culturally correct consensus answers. The standard cultural consensus model, called the General Concordet Model [38], assumes that, depending on question difficulty and respondent competence, people either know the correct answer or make a guess. The Bayesian cultural consensus model [21, 33]

is a hierarchical Bayesian model that adds hyperpriors and a noise model to the General Concordet Model.

Outside of cultural consensus theory, a number of hierarchical Bayesian models for aggregation have been developed that attempt to model how people produce their answers given some latent knowledge, and then aggregate information at the level of this latent knowledge. For example, when playing “The Price is Right” game show, people’s bids for a product may not correspond to their knowledge of how much the product is worth (because of the competive nature of the show), and so infering their latent knowledge and aggregating at this level will give more accurate estimates about the product than aggregating at the level of their bids [26]. Such hierarchical Bayesian models also include models for aggregating over multidimension stimuli, for example combinatorial problems [48] and travelling salesman problems [47]. Modeling the cognitive processes behind someone’s answer also allows individual hetrogeneity to be estimated, for example the differing knowledge that more or less expert respondents have in ranking tasks [25] or using a cognitive hierarchy model to account for people’s differing levels of noise and calibration when respondent’s are answering binary questions using probabilties.[24, 44] .

The models discussed above reflect important advances for solving the problem of belief aggregation, but they remain focussed on attempting to derive the consensus answer, allowing for differences in how this consensus answer is represented and reported. 888One of the clearest expressions of this is from [24], “A more concrete and perhaps more satisfying answer is that the model works by identifying agreement between the participants in the high-dimensional space defined by the 40 questions. Thinking of each person’s answer as a point in a 40-dimensional space makes it clear that, if a number of people give similar answers and so correspond to nearby points, it is unlikely to have happened by chance. Instead, these people must be reflecting a common underlying information source. It then follows that a good wisdom of the crowds answer is near this collection of answers, people who are close to the crowd answer are more expert, and people who need systematic distortion of their answers to come close to this point must be miscalibrated. Based on these intuitions, the key requirement for our model to perform well is that a significant number of people give answers that contain the signal of a common information source.”

Previously[35], we described a simple principle for aggregating answers to binary questions based on the possible worlds model: select the answer which is more popular than people predict, rather than simply the most popular answer. While this principle has the advantage of simplicity, developing a probabilistic generative model based on the possible worlds model has several benefits. First, such a generative model yields a posterior distribution over world states, rather than simply selecting an answer. Second, generative models allow us to easily incorporate other kinds of information into the model. For example, they facilitate the introduction of latent expertise variables allowing the model to be run across multiple questions. Third, the generative model allows one to explicitly model the noise in respondent’s beliefs, rather than assuming that respondents are perfect Bayesians. Fourth, as discussed later, the generative model lends itself to a number of possible extensions.

A generative possible worlds model

A model for single questions

In this section we discuss applying the model to single questions and in the next section we discuss inferring respondent level parameters by applying the model across multiple questions. We present the core model for binary questions here and return in the discussion section to our modeling assumptions, and consider alternatives and extensions, for example to non-binary multiple choice questions. The graphical model [20, 22] for respondents voting on a single questions is shown in Figure 1. Suppose that respondents each give their personal answer to a multiple choice question with two possible answers and predict the distribution of votes given by other respondents. We assume that each of the possible answers to the question corresponds to a different underlying world state and denote the actual state of the world as . For example, if respondents are asked, “Is Philadelphia the capital of Pennsylvania?” the two possible worlds states are true and false with false being the actual world state, since the capital of Pennyslvania is actually Harrisburgh. A respondent indicates their vote for the answer that they believe is most likely, with superscripts indexing a particular respondent throughout. Respondents also give their prediction (‘’ for meta-prediction) of the fraction of respondent’s voting for each answer. For binary questions, is completely specified by the fraction of people predicted to vote for option . Respondents dmetimes additionally asked to indicate the confidence that they have in their answer being correct, and in section Extensions of the possible worlds model we develop a model that uses such confidence judgments.

Figure 1: The single question possible worlds model (PWM) which is used to infer the underlying world state based on a group’s votes and predictions of the votes of others. In keeping with standard graphical model plate notation [20, 22]

, nodes are random variables, shaded nodes are observed, an arrow from node

to node denotes that is conditionally dependent on , a rectangle around variables indicates that the variables are repeated as many times as indicated in the lower right corner of the rectangle.

The prior over worlds represents information that is common knowledge among all respondents. It is given by a binomial distribution with parameter

and the hyperprior over

is a uniform Beta distribution. Each respondent is assumed to receive a private signal that represents additional information above the common knowledge information captured by the prior, with respondent

receiving signal . We assume that there are the same number of possible signals as there are possible world states with for binary questions. The signal that a respondent receives is determined by the actual world and a signal distribution. The signal distribution is represented in the binary case as a

left stochastic matrix (i.e. the columns sum to 1) where the

-th entry corresponds to the probability of an arbitrary respondent receiving signal when (assuming a fixed ordering of world states and signals). In other words, each respondent’s signal is sampled from a categorical distribution with the -th column of the signal distribution matrix giving the probabilities of the different signals. Note that the model for single questions assumes that respondents are identical except for the signal that they receive, and that for any given signal all respondents have the same probability of receiving it.

The prior we specify over the signal distribution does not impose the substantive assumption that if then signal is necessarily the most probable signal. Instead, we allow the possibility that respondents receiving an incorrect signal may be the majority. Specifically, we assume that the signal distribution is sampled uniformly from the set of left stochastic matrices with the constraint that for all . This prior provides a constraint on the world in which a given signal is more likely, not a constraint on which signal is more likely for a given world state, and does not imply that the majority of respondents receive a signal corresponding to the correct answer. That is, this prior over signal distributions guarantees that signal is more likely to be received in state than in state , but allows signal to be more likely than signal in state . The constraint that we do place on the signal distribution allows one to index signals and world states and make the model identifiable, analogous to imposing an identifiability constraint to alieviate the label switching problem when doing inference on mixture models [13, 18].

Respondents are modeled as Bayesians who share common knowledge [40, 30] of the signal distribution and of the prior over world states. Respondents know their received signal, but not the actual world state, as denoted by the lack of an edge in Figure 1 between the world state node and the respondent vote node. A respondent receiving signal has all the information necessary to compute a posterior distribution over the world states. A respondent is assumed to wish to vote for the world state that is most likely under their posterior distribution over world states conditional on their received signal. For the case of two worlds and two signals this implies that respondents receiving signal will wish to vote for world , except in the case where the signal distribution and world prior is such that world is more likely irrespective of the signal received. We allow for noise in the voting data by making a respondent’s vote a softmax decision function of the posterior distribution, with the voting noise parameter giving the temperature of this function. The voting noise parameter is drawn from a distribution. The parameters of this prior distribution were fixed in advance of running the model on the datasets.

We now turn to modeling the predictions given by respondents about the votes of other people. A respondent who received signal has the information required to compute the posterior distribution over the signals received by an arbitrary respondent , since can be computed by marginalizing over possible worlds. Observe that a respondent’s received signal affects their predicted probability that an arbitrary respondent receives a particular signal since how likely an arbitrary respondent is to receive each signal depends on the world state, but a respondent’s predicted probability of the world state depends on the signal that they received. In other words, for respondent to compute the probability that respondent will wish to vote for a particular world, respondent can simply sum up the probabilities of all the signals which would cause to do this. Hence, each ideal respondent has a posterior distribution, conditioned on their received signal, over the votes given by other respondents. Actual respondent’s predictions of the fraction of respondents voting for

are assumed to be sampled from a truncated Normal distribution on the unit interval with a mean given by their posterior distribution on another respondent’s voting

and variance

(prediction noise) which is the same for all respondents and which is sampled uniformly from .

A generative possible worlds model for multiple questions

Figure 2 displays a generative possible worlds model that applies to respondents answering multiple questions. It closely follows the single question model, but incorporates respondents’ expertise across different questions. For each question, we sample a world prior, world, signal distribution, and noise parameters as with single questions, imposing no relationship across questions. The essential difference relative to previous work [24], is that we capture differences in respondent expertise in terms of how likely they are to receive the correct signal, rather than in absence of error in reporting answers. We call this ‘information expertise’.

The information expertise parameter for respondent , denoted , has support and affects how likely a respondent is to receive the correct signal. If the probability of receiving signal in world according to the signal distribution is (i.e. ) then the probability of a respondent with information expertise receiving signal is increased by and the probability of receiving a different signal is decreased by the same amount. That is, the probability of receiving signal in world increases linearly with the information expertise from the probability given by the signal matrix to 1. For example, suppose the actual world was and there was a 0.4 probability of receiving signal in this world. Then, if the probability of respondent receiving signal is 0.4, if then the probability of respondent receiving signal is , and if the probability of respondent receiving signal is 1.The information expertise parameter does not determine how accurately a respondent will answer questions in absolute terms, but rather relative to the difficulty of the question.For example, for an easy question where the signal distribution gives an 80% chance of receiving the correct signal, even if someone has the lowest possible expertise of 0 they will still be very likely to give the correct answer to the question.

This model of information expertise does not allow respondents that are less likely to receive the correct signal than the probability given by the signal matrix. Of all the respondents, the answers of those with expertise 0 are the most uninformative to the model.If we additionally allowed negative expertise values between 0 and -1, which linearly decreased the probability of receiving the signal corresponding to the actual world, then someone with expertise -1 would always receive the signal opposite to the actual world and so provides the same informational content as someone with an information expertise of 1. We assume a uniform prior distribution on .999Suppose we allowed information expertise values between -1 and 1 with a uniform prior, and our data consists only of questions where the majority is incorrect for every question and a small subset of respondents are correct for all the questions. Then, using the model we may infer that the small subset of respondents have expertise values near (and so are in a minority only because they received the wrong signal) and that the anwer to select is the one given by the incorrect majority. If we only allow expertise values between and then we cannot simply infer that the minority received the incorrect signal and may infer that the correct answer is the one endorsed by the minority, depending on the predictions given by respondents.

Respondents are modeled as formulating their personal votes and predictions without taking information expertise into account. In the next section, we develop a model which allows that respondents know their own information expertise.

Figure 2: The multiple question Possible World Model which is applied across questions for respondents answering questions. It is similar to the single question model, but it includes, for each respondent, an information expertise variable which determines how likely an individual is to receive the correct signal compared to the baseline given by the signal matrix.

Extensions of the possible worlds model

We test two extensions to the generative possible worlds model described above. First, for both the single question and multiple questions version of the model, we discuss incorporating information about the confidence that respondents have in their answers. As for the other elicited responses, a respondent’s confidence is modeled as being generated from the Bayesian posterior probability, conditional on their received signal, of the answer they voted for, plus noise. That is, if respondent received signal and voted for option , their confidence is a noisy report of . A respondent’s confidence is assumed to be sampled from a Normal distribution (truncated between 0 and 1) which has a mean given by the Bayesian posterior on the answer which they voted for, and a variance which corresponds to the noise governing the confidence of all respondents.101010For binary questions, respondents should give a confidence from 50% to 100%, since if someone had less than 50% confidence in an option they should have voted for the alternative option. We assume a truncated Normal distribution with support from , rather than from , to allow for respondent voting error. The noise

is sampled from a uniform distribution on

.

The second extension to the model we discuss assumes that respondents know their own information expertise, rather than simply assuming it is 0. In the version of the model above, a respondent’s expertise only affects their probability of receiving a particular signal, but since everyone assumes their own expertise is 0, all respondents who receive signal have the same posterior beliefs over worlds and other respondents’ signals. Suppose a respondent has accurate knowledge of their own information expertise. This implies that where so that a respondent’s distribution over worlds takes into account their own information expertise. This is in contrast to the basic version of the model where without an information expertise term. Assuming that respondents know their own expertise in this way has the effect that given two respondents receiving the same signal, the one who knows that she has high information expertise will put higher probability on the answer they endorse than the one who believes that he has low information expertise. The respondents will also put different probabilities on the signals received by other respondents, since they have differing beliefs about the likelihood of different possible worlds. A large disadvantage to assuming that respondents know their own information expertise is the increase in computation required for inference with this model.111111We discuss inference via Metropolis Chain Monte Carlo in a later section. If we assume that respondents know their own information expertise then the posterior distributions over worlds and signals (conditional on the received signal) have to be computed separately for every respondent rather than only once, and this occurs every inference step.As we will see in the results section, these extensions turn out not to improve performance on the datasets from the seven studies in this paper.

Comparison models

Our model can be applied to individual questions, but other generative models for aggregation require multiple questions. To compare the results of our model to other methods we run it on both the different questions individually, without learning anything about individual respondents, and also across questions to learn something about respondents answering multiple questions. We compare our model to majority voting, selecting the surprisingly popular answer [35], and the linear and logarithmic pools all of which also only require individual questions. We also compare our model’s performance to other hierarchical Bayesian models that require multiple questions. Specifically, we compare our model to the Lee and Danileiko cognitive hierarchy model [24] and the Bayesian Cultural Consensus model [33].Bayesian Cultural Consensus

Cultural Consensus Theory [38, 2, 45] is a prominent set of techniques and models that are used to uncover shared cultural knowledge, given answers from a group of people. The theory deals with respondents answering a set of binary questions that all relate to the same topic. Respondent’s answers are used to determine the extent to which each individual knows the culturally correct answers (their ‘cultural competence’) and the cultural consensus is then determined by aggregating responses across individuals with the answers of culturally competent people weighted more heavily. Hence, unlike our model, cultural consensus models cannot be applied to single questions and nor can they be applied to questions with continuous answers. As in the Lee and Danileioko model above, the overall assumption is thus that the consensus answer is the correct one. The formal models of cultural consensus we consider build on the “General Concordet Model” [38] and can be thought of as factoring an agreement matrix, where each element gives the extent to which every two individuals agree (corrected in a particular way for guessing).

In keeping with our focus on aggregation as inference, we use the Bayesian Cultural Consensus model [21, 33, 32], shown in Figure 3, which is a generative model based on the General Concordet Model. The model is applied to data from respondents answering dichotomous questions. Respondents are indexed with and questions with and for each question , a respondent votes for either true or false, denoted by . The model assumes that for each question there is a culturally correct answer . For question , a respondent knows and reports with probability and otherwise guesses true with probability corresponding to a respondent specific guessing-bias. The competence of respondent at answering question is given by the Rasch measurement model, , which is here a function of the respondent’s ability and the question difficulty . The competence of a respondent for a question increases with the respondent’s ability, and decreases with the question’s difficulty. When ability matches difficulty the probability of the respondent knowing the answer for the question is 0.5. Uniform priors are assumed for all parameters of the model. The complete set of answers given by respondents can thus be expressed as a probabilistic function of the culturally correct answer for all questions as well as the difficulty of each question, and the ability and guessing bias parameters of each respondent. The model is sometimes constrained in various ways (for example, by assuming homogeneous question difficulty, homogeneous ability across respondents, or neutral guessing bias,) but we do not impose these constraints and allow all parameters to vary across respondents and questions. The posterior distribution over the culturally correct answer for each question provides the aggregate answer, and the inferred values of respondent ability and guessing-bias give information about a respondent’s performance.

Figure 3: Graphical model representation of the Bayesian Cultural Consensus Model [33].

A cognitive hierarchy model for combining estimates

We measure the performance of the model developed by Lee and Danileiko [24] which was discussed in the introduction and is depicted in Figure 4. The model requires multiple questions answered by the same set of respondents so that individual level parameters can be learnt from the data.

Figure 4: Graphical model representation of the cognitive hierarchy model [24].

The cognitive hieararchy model assumes that respondents each answer the same set of questions. Respondents are indexed by , questions are indexed by , and the answer of respondent to question is denoted . Respondents answers are their subjective probabilities that the world is in a particular state (e.g. their estimated probability that the answer to the question is true), and so . A latent true probability is assumed to be associated with each question and two individual-level parameters govern how respondents report this true probability. First, based on psychological results about how people perceive probabilities, each respondent’s perception of the true probability varies depending on how well calibrated the respondent is. In the model, a respondent with calibration parameter , perceives a probability

which assumes a linear-in-log-odds calibration function. The model further incorporates a respondent-level parameter

which the authors term ‘expertise’ or ‘level of knowledge’ by assuming that the reported probability

is sampled from a Gaussian distribution centred around the perceived probabiltiy

with variance given by the reciprocal of . That is, the larger the , the more likely respondent is to report a probablity closer to their perceived probability. A uniform prior distribution on the unit interval is assumed for both and , and the calibration parameter for each respondent has a prior.

Evaluating the models

Data

We evaluate the models on data from running seven studies with human participants. The seven studies were first described in [35], and additional details about the experimental protocols for all of these studies can be found in the supplementary information published with the paper.121212All studies were performed after approval by the MIT Committee on the Use of Humans as Experiment Subjects, and respondents gave informed consenst using procedures approved by the committee. The first three studies consist of data from respondents answering questions of the form “Is city X the capital of state Y?” for every U.S. state where city X was always the most populous city in the state. The first of these studies () was done in a classroom at MIT (“MIT class states study”), the second () was done in a laboratory at Princeton (“Princeton states study”) and the third () was done in a laboratory at MIT (“MIT lab states study”). Respondents indicated whether they thought the answer to each question was true or false and predicted the fraction of respondents answering true. For example, the first statement was “Birmingham is the capital of Alabama” to which the correct response is false because although Birmingham is Alabama’s most populous city, the capital of Alabama is Montgomery. For the third study, respondents additionally gave their confidence of being correct on a scale from 50 percent to 100 percent and predicted the average confidence given by others. The prediction of the average confidence given by others is not used in the graphical model, but we return to this kind of prediction in the discussion section.

Two other studies asked respondents to judge the market price of ninety pieces of Twentieth Century art. Two groups of respondents ( for both groups) - either art professionals, mostly gallery owners (“Art professionals study”), or M.I.T. graduate students who had not taken an art or art history course (“Art laypeople study”) - worked through a paper based survey where each page contained a color reproduction of a Twentieth Century artwork, along with information about it dimensions and medium. Respondents indicated which of four bins they believed a piece’s price fell into: under $1000, $1000 to $30 000, $30 000 to $1 000 000, or over $1 000 000.131313All prices in this paper refer to American dollars. We can collapse these answers into low and high prices with $30 000 as the cut-off. Respondents also predicted the percentage of art professionals and the percentage of MIT students that they believed would predict a price over $30 000. We only use their predictions for people in the same group as themselves, i.e. we do not consider predictions made by M.I.T. students about art professionals, and vice versa.141414Respondents also answered questions about subjectively liking a piece (both their own and predicting others), but we do not analyze this data here. The actual asking price for each piece was known to us, with 30 of the 90 pieces having a price over $30 000.

A sixth study had qualified dermatologists examine images of lesions (“Lesions study”) and judge whether the pictured lesions were benign or malignant. Respondents also predicted the distribution of judgments on an eleven point scale which we converted to a judged percentage, and gave their confidences on a six point Likert scale which was likewise converted to a percentage. All the lesions had been biopsied, and so whether each lesion was actually malignant is known to us. For our analysis, we collapse two groups of dermatologists one of which () saw a set of 40 benign lesions and 20 malignant and the other () saw a set of 20 benign lesions and 40 malignant. Thus, there were 80 images in total, half of which were benign.

Our last study had respondents (), recruited from Amazon’s Mechanical Turk, answer an online survey with 80 statements which they were asked to evaluate as either true or false (“Trivia study”). The statements were from the domains of history, science, geography, and language. They included, for example, “Japan has the world’s highest life expectancy”, “The chemical symbol for Tin is Sn”, and “Jupiter was first discovered by Galileo Galilei”. 151515The correct answers are false (Monaco is higher), true, and false (Galileo was the first to discover its four moons, but not the planet itself. One of the earliest recorded observations is in the Indian astronomical text, the Surya Siddhanta, from the fifth century).For half of the questions, the correct answer was false. For each questions, respondents indicated which answer they thought was correct, predicted the probability that their answer was correct with a six point Likert scale, and predicted the percentage of people answering the survey who would answer true.

Applying the possible worlds model

Markov Chain Monte Carlo (MCMC) inference was performed using using the Metropolis-Hastings algorithm. The signals were marginalized out when doing inference. We represent the world prior using the probability of the world in state , and the signal distribution matrix as the probability of receiving signal in each state. We use truncated Normal proposal distributions (centred on the current state with different fixed variances for each parameter) for the world prior, noise, expertise and signal probabilities (maintaining the constraint that the probability of signal in state is higher than in state ), and we propose the oppsite world state at each metropolis step. When doing inference across multiple questions, the parameters for a particular question (the signal matrix, prior over worlds, and world) are conditionally independent of those for another question given the individual level expertise. We thus run MCMC chains for each question in parallel with the individual level parameters fixed, interspersed with an MCMC chain only on the individual level parameters. For running the model on questions separately, we use 50 000 Metropolis Hastings steps, 5000 of which are burn-in. For running the model across multiple questions, we run 100 overall loops, the first 10 of which were burnin, where each loop contains 2000 steps for the question parameters and 150 steps for the respondent-level parameters. We assess convergence using the Gelman-Rubin statistic, the Geweke convergence diagnostic, and by comparing parameter estimates across multiple chains.

Applying the Bayesian cultural consensus model

The Bayesian Cultural Consensus Model uses only the votes of respondents. Cultural consensus models assume that there is a unidimensional answer key to the questions which the group is asked. A heuristic check that the data is indeed unidimensional is to compute the ratio of the first to the second eigenvalue of the agreement matrix with a ratio of 3:1 or higher indicating sufficient unidemnsionality

[33, 45]. For the datasets considered in this paper the ratio of the first to second eigenvalues were 2.76 for the MIT states-capitals class dataset, 2.62 for the Princeton states-capitals dataset, 3.32 for the MIT lab states-capitals dataset, 2.7 for the MIT art data, 8.92 for the Newbury art data, 2.81 for the trivia data, 10.26 for the respondents who saw the lesions data with the split 20 malignant and 40 benign lesions, and 6.73 for the respondents who saw the lesions data with the split 40 malignant and 20 benign lesions. Most of the datasets are higher than the traditional standard for unidimensionality, and the model performed well on datasets not meeting this standard. The model learns a respondent-level guess-bias towards true. The states-capitals and trivia questions explicitly deal with true and false answers, in the art studies we coded true as refering to the high price option (over $30 000), and in the lesions study we coded true as referring to the malignant option.161616It is an open empirical question whether one would see any differences in the responses to the question “True or false, is this question malignant?” versus “Is this lesion benign or malignant?”

The Bayesian Cultural Consensus Toolbox [33] specicifies the model using the JAGS model specification. This was again altered to allow for unbalanced observational data. Gibbs sampling was run for 1000 steps of burn-in, followed by 10 000 iterations, using 6 independent chains and a step-size of two for thinning. As is standard when applying MCMC to these models [33, 21], convergence was assessed using the Gelman-Rubin statistic and comparing the parameter estimates across chains. Three of the inferred parameters are of interest: the cultural consensus answer for each question, the guessing bias of each respondent, and the ability of each respondent.

Applying the cognitive hierarchy model

The cognitive hierarchy model requires subjective probabilities from respondents and so we apply the model only to data where this information is available, specifically the MIT lab states study, the lesions study, and the trivia study. We used the JAGS (Just another Gibbs sampler) model specification provided by Lee and Danileiko with their paper, but altered it to allow for unbalanced observational data since not every respondent answered every question: in the MIT lab states dataset and the trivia dataset occassionally a respondent simply missed a question and in the lesions dataset only about half the respondents answered some of the questions due to the experimental design. Gibbs sampling was run for 2000 steps of burn-in, followed by another 10000 iterations, using 8 independent MCMC chains and standard measures of autocorrelation and convergence were evaluated to ensure that the samples approximated the posterior. We infer for each question the latent true probability, and for each respondent their calibration and expertise parameters.

Results

The Bayesian cultural consensus model and cognitive hierarchy model cannot be applied to individual questions, but our generative possible worlds model is applied both to individual questions in a study and also applied across questions in a study to learn respondent-level expertise. The cognitive hierarchy model is only applied to studies where confidences were elicited, whereas the other two models are applied to the voting data from all seven studies. We compare the models both with respect to their ability to infer the correct answer to the questions and their ability to infer the expertise of respondents. The possible worlds model allows us to infer a prior over world states and we compare the inferred world prior to a proxy for common knowledge about the likelihood of a city being a state capital, specifically the frequency of mentions of city-state pairs on the Internet.

Inferring correct answers to the questions

Of the three probabilistic models discussed, only the generative possible worlds model can be applied to individual questions. There are other aggregation methods, however, that can be applied to individual questions. For all studies we compute for comparison the answer endorsed by the majority (counting ties as putting equal probability on each answer) and for studies where confidences were elicited we also compute a linear opinion pool given by the mean of respondent’s personal probabilities, and a logarithmic opinion pool given by the normalized geometric means of the probabilities that respondent’s assign to each answer

[9]. We also show the result of selecting the surprisingly popular answer [35], which for binary questions is simply the answer whose actual vote frequency exceeds the average vote frequency predicted by the sample of respondents. We discuss the results of applying the possible worlds model both for questions separately and to all the questions in a study together.

Figure 5 shows the results of each method in terms of Cohen’s kappa coefficient, where a higher coefficient indicates a higher degree of agreement with the actual answer. Cohen’s kappa is a standard measure of categorical correlation, and we display it rather than the percentage of questions correct since for some of the studies the relative frequencies of the different correct answers are unbalanced, which means that a method can have high percentage agreement if it does not discriminate well and is instead biased towards the more frequent answer. 171717Cohen’s kappa is computed as where is the relative osberved agreement between the method and the actual answer, and

is the agreement expected due to chance, given the frequencies of the different answers output by the method. There is not a standard technique to accomodate ties when computing the kappa coefficient for binary questions. In the case of ties, we construct a new set of answers with double the number of original answers. Answers which were not ties simply appear twice in the new set, and answers which were originally ties appear once with one answer and once with the other answer. The kappa coefficient is invariant to doubling the number of answers, and the standard error is a multiple of the original number of questions and so the doubling of the number of answers can be easily accounted for when computing standard errors.

A disadvantage of the kappa coefficient is that it does not use the probabilities reported by a method, but only the answer which the method determines is more likely. Figure 6 shows the result of each method in terms of its Brier score, where a lower score indicates that a method tends to put high probability on the actual answer. There are a number of similar formulations of the Brier score which we compute here as the average squared error between the probabilistic answer given by the method and the actual answer. For methods which output an answer rather than a probability we take the probabilities to be zero, one, or 0.5 in the case of ties.

Figure 5: Performance of the aggregation methods for each dataset shown with respect to the kappa coefficient, with error bars indicating standard errors. The lighter colored bars show methods that require data from multiple questions.
Figure 6: Performance of the aggregation methods for each dataset shown with respect to the Brier score, with error bars indicating bootstrapped standard errors. The lighter colored bars show methods that require data from multiple questions.

We first consider the methods that act on questions individually, which are shown with lighter bars in the two figures showing model performance. The linear and logarithmic pools give similar answers (the minimum kappa coefficient comparing the logarithmic and linear pools is ), and so we do not show the logarithmic pool results separately. Compared to the other methods that operate on questions individually, the generative possible worlds model outperforms majority voting, and the linear and logarithmic pool across studies if we consider the accuracy of the answer selected by each method. This is displayed with respect to Cohen’s kappa in Figure 5, but we can also compare the number of errors more directly. Across all 490 items, PWM applied to questions separately improved on the errors made by majority voting by 27% (, all p-values shown are from a two-sided matched pairs sign test on correctness unless otherwise indicated). Across the 210 questions on which confidences were elicited, PWM applied to separate questions improved on the errors made by majority vote by 30% () and over the linear pool errors by 20% (). As can be seen, the possible worlds single question model is able to uncover the correct answer even when run on individual questions where the majority is incorrect. In terms of selecting one of two binary answers, the performance of the PWM on separate questions and the surprisingly popular answer is similar ( across the 490 questions, and the answers are not significantly different by a two-sided matched pairs sign test, ), which is to be expected since they build on the same set of ideas to use people’s predictions of the distribution of answers given by the sample. However, for all studies PWM separate question voting has a lower Brier score than the surprisingly popular answer, since it produces graded judgments rather than simply selecting a single answer.

We also show the performance of models that require multiple questions (lighter colored bars): the Bayesian cultural consensus model, the cognitive hierarchy model, and PWM applied across questions. The PWM across questions improves over the single question PWM (), although in terms of the kappa coefficient this improvement is small except for two of the states-capitals studies. The cultural consensus model also shows good performance across datasets. It is similar to PWM across question voting in terms of the kappa coefficient, except for the MIT states-capitals study where its performance is not as good. This is also reflected in the difference between the correctness of the two methods in terms of absolute numbers of questions correct which favors the PWM applied across questions (. The cognitive hierarchy model selects similar answers to the linear pool () resulting in similar accuracy at selecting the correct answer as measured by Cohen’s kappa, but better performance with respect to the the probability it assigns to the correct answer (as measured by Brier score). The cognitive hierarchy model shows similar performance to the cultural consensus and possible worlds models on the trivia and lesions studies, but impaired performance on the MIT lab states-capitals study.

We earlier discussed two possible extensions to the possible worlds model: incorporating confidences and assuming that respondents are aware of their own expertise. On the questions where confidences were elicited, we applied the PWM with confidences incorporated both for separate questions and across questions. Applied to questions separately, the answers given by the PWM with and without confidences were similar ( on the selected answers, on the returned probabilities). This was also the case when running the PWM across questions with and without confidences (, . Hence, incorporating confidence made little difference to the possible world model results. We also ran the model (both with and without confidences) assuming that people knew their own expertise. This again made little difference to the results for either the model without confidences ( for answers, for probabilities) or the model incorporating confidences ( for answers, for probabilities).

The inferred world prior and state capital mention frequency statistics

The PWM allows one to infer the value of latent question-specific parameters other than the world state, such as the complete signal distribution and the prior over worlds. The accuracy of these values is difficult to assess in general, but we analyze the inferred world prior in the state capitals studies. Previous work in cognitive science has demonstrated that in a variety of domains people have prior beliefs that are well calibrated with the actual statistics of the world [15]. We use the number of Bing search results of the city-state pair asked about in each question (specifically the search query “City, State”, for example “Birmingham, Alabama”) as a proxy for how common mentions of the city-state pair are in the world. For all three state capitals studies, the inferred world prior on the named city being the capital (using the PWM applied to individual questions) has a moderate correlation with the Bing search count results under a log transform (MIT class: , Princeton: , MIT lab: ). This suggests that the model inferences about the world prior may reflect common knowledge about how salient the named city is in relationship to the state.

Respondent-level parameters

As well as comparing results from the models to the actual answers, we can also evaluate how well the model predicts the performance of individual respondents. The PWM applied across questions as well as the cultural consensus model and cognitive hierarchy model all infer respondent-level expertise parameters. Figure 7 shows how these respondent-level expertise parameters correlate with the kappa coefficient of each respondent. For studies where confidences were elicited, the pattern of results is the same if respondent performance is measured using the Brier score.

Figure 7: Pearson correlations of inferred respondent level expertise parameters from each model against the accuracy of each respondent, measured by their kappa coefficient. Error bars show bootstrapped standard errors.

Both the PWM expertise parameter and the cultural consensus competence parameters show high accord with individual respondent accuracy. For the 220 total respondents in all the studies, the correlation of respondent-level kappa accuracy with PWM expertise is r=0.79 (all respondent-level correlations are significant at the p<0.005 level) and with cultural consensus competence is . For the 97 respondents in the studies where confidences were elicited, the correlation of respondent-level kappa accuracy with cognitive hierarchy expertise is , with PWM expertise is r=0.78 and with cultural consensus expertise is .

For every respondent, two other pieces of information that may help predict performance are the fraction of times that the respondent was in the majority, and the fraction of times that the respondent voted true. We examine the relationship between the various expertise parameters and respondent performance if we partial out these two additonal factors. Across all studies this partial correlation of respondent-level kappa acurracy with PWM expertise is and with cultural consensus competence is . Across the studies where confidence was elicited this partial correlation of respondent-level kappa accuracy with cognitive hierarchy expertise is , with PWM expertise is , and with cultural consensus competence is .

For completeness, we also report on the other respondent-level parameters inferred by the models. The cultural consensus guess bias parameter correlates highly with the fraction of questions for which a respondent says true (), but not with the kappa accuracy of a respondent (, ). The cognitive hierarchy model calibration parameter has a correlation of with the kappa accuracy of respondents for the studies where confidence was elicited.

Factors affecting model performance

In this section, we discuss factors that affect the peformance of the different generative models, and under what circumstances we expect the different models to perform well. We discuss how an inconsistent ordering across answer options affects the two models, and the role of the predictions of others answers in the possible worlds model.

A consistent coding of answers

In these studies, the question format did not vary across items in terms of which direction was designated as True and which as False. If respondents have a bias toward a particular response (True or False), this bias could be detected by some models and neutralised. However, these questionnaires could have randomized the polarity of questions, e.g., replacing some questions that ask whether X is true with questions asking whether X is false. Ideally, our inference of respondent expertise should not be affected by such inessential changes in question wording.

To see how methods might fare with question rewording, we constructed a new dataset where the first half of the questions in each study are coded in reverse, and another dataset where the second half of the questions in each study is coded in reverse. To reverse a question, each respondent’s vote is swapped (i.e. a vote for true becomes one for false and vice versa) as is the correct answer to the question. Additionally, a respondent’s prediction of the fraction of people voting true becomes their prediction of the fraction of people voting false. We applied the Bayesian cultural consensus model, the cognitive hierarchy model, and the PWM applied across questions to the half-reversed datasets. The cognitive hierarchy model and the possible worlds model were not affected by this transformation (the average of the kappa coefficients for the two half-reversed datasets were within 0.05 of the original kappas for every study). This was not the case for the cultural consensus model, which had much lower performance on some of the studies with the half-reversed datsets. In particular, for the MIT class states study, the Princeton states study, and both art studies it had a kappa of approximately for the half-reversed datasets, in comparison to good performance on the original datasets. This decrease in performance of the cultural consensus model is because it relies in part on a respondent parameter which indicates the bias towards answering true. The other models, by contrast, do not have such a parameter and so can deal with sets of questions such that there is not a consistent coding across questions. More generally, the other two models can deal with sets of questions where there is no ordering on the answer options which is the same across questions, for example a set of questions asking which of two novel designs for a product will be more successful.

The role of peer predictions

The PWM uses the predictions of other people’s votes to infer the model parameters, since personal votes alone are insufficient to determine the signal distributions in the non-actual worlds. We evaluate the effect of these predictions on the inferences from the PWM by lesioning the PWM to not use predictions. This lesioning of the model results in inferences that are very similar to that given by majority voting. Across the seven datasets, the median spearman correlation between the inferred probability of the world being in state true by the lesioned model and the fraction of people voting for true is .

More generally, even in situations where peer predictions are available, these can be more or less useful depending on how accurately respondents can give these predictions and how much variation there is across questions - if everyone simply always predicts 50% of people will answer true for every question the predictions will not be useful for improving the accuracy of the model inferences.

Discussion

The generative PWM presented in this paper is a step towards developing statistical methods for inferring the true world state which rely not on the crowd’s consensus answer, but rather allow for individuals to have differential access to information. The PWM depends on assumptions about: (1) the common knowledge that respondents share, (2) the signals that respondents receive, and (3) the computations that respondents make (and how they communicate them). We discuss each of these in turn, and the possible extensions that they suggest.

Common knowledge

One set of modeling assumptions concerns the knowledge shared by respondents. Specifically, the PWM assumes that respondents share common knowledge of the world prior and signal distribution. However, neither of these assumptions are entirely correct: people neither exactly know these quantites, and nor are beliefs about these quantities identical across people. As discussed when comparing the inferred world prior in the states capitals studies to Bing search results, in some cases the world prior may reflect statistics of the environment which may be learnt by all respondents. In other cases, expert respondents may have a better sense of the world prior. For example, in diagnosing whether somebody has a particular disease based on their symptoms, knowledge of the base rate of the disease helps diagnosis but may not be known to everyone.

One could extend the model to weaken the common prior assumption in various ways, although one could not simply assume that everyone had a different belief about the prior over worlds. A model could instead assume, for example, that respondents receiving the same signal share a common prior over worlds, but that respondents receiving different signals have different prior beliefs.181818One would still need some kind of assumption about what respondents believed about the prior possessed by respondents receiving other signals. Alternatively, one could develop models where all respondents had noisy access to the actual world prior and signal distribution, and formulated beliefs about other respondents knowledge of these quantities. For example, each respondent could receive a sample from a distribution around the actual world prior and actual signal distribution. In the case of respondents answering many questions, one could attempt to learn parameters that governed the accuracy of a particular respondent’s knowledge of these distributions. One could also incorporate domain knowledge available to the aggregator into the hyperprior over world priors. For example if one has external knowledge of the base rate frequency of benign versus malignant lesions one could choose a hyperprior that would make a prior matching this base rate more probable.

Signal structure

The PWM assumes that there are the same number of signals as world states. It further assumes that the signals themselves have no structure: they are simply samples from a binomial distribution (in the case of two worlds) with signal more common in world . This treats the information or insight available to a respondent coarsely in that it does not allow for more kinds of information available to respondents than there are answers to a question, or for respondents with different pieces of information to endorse the same answer. Models with more signals than worlds could be developed with constraints on the worlds in which different signals were more probable. For example, one could imagine a hierarchical signal sampling process whereby each signal was a tuple where the first element indicated the world in which the signal was most likely and the second element gave the rank of the probability of that signal amongst other signals with the same first element. That is, there would be a constraint on the signal distribution such that signal , say, would be more likely in world than in any other worlds, and of the other signals more likely in world than in any other world it was the third most probable in world . It would then be necessary to maintain these constraints when sampling the signal distribution. More generally, models with other kinds of assumptions about the signals that respondents receive could be developed to more faithfully model the information available to respondents when dealing with complex questions - for example, respondents could receive varying numbers of signals, a mix of public and private signals, or signals that are not simply nominal variables but rather have richer internal structure.

Respondent computations

The PWM assumes that repondents can compute Bayesian posteriors over both answers and the votes given by others, and noisely communicate the results of their computations. The model could be extended by modeling respondents as more plausible cognitive agents, rather than simply as noisy Bayesians. For example, the cognitive hierarchy model recognizes, based on much work in the psychology of decision making [49], that respondents will be differentially calibrated with respect to the probabilities they perceive and a similar calibration parameter could be added to the possible worlds model for the predictions of others answers. Modeling respondent’s predictions of other people could also incorporate what is known about this process from social psychology. As one example, as well as showing a false-consensus effect, people also exhibit a false-uniqueness bias such that they do not take their own answer sufficiently into account when making their predictions (e.g. [4, 41]). This could be modeled as their predictions resulting from a mixture of their prior over signals (i.e. their knowledge of the signal distribution) and their posterior over signals with the mixture weight given by a respondent level false-uniqueness parameter.

In the proposed PWM, respondents are assumed to not take some information into account when predicting the votes given by others. We allow for some noise when modeling how respondent’s vote, but do not assume that respondents themselves take this noise into account when predicting the votes of others. In the model discussed here, respondents do not attribute information expertise to other respondents, but one could instead develop models where each respondent assumes some distribution of information expertise across other respondents as well as modeling the noise that they believe is present across the voting patterns of other people.

Given the assumption that respondents compute a posterior over worlds and over other’s signals, additional statistics relating to either of these posteriors could be elicited and modeled. For example, in one of the states capitals studies respondents were asked to predict the average confidence given by other respondents, which can be computed from these two posteriors. Such additional information could potentially help sharpen the model inferences. In practice, communicating such questions to respondents, respondent difficulty in reasoning about such questions, and respondent fatigue would all impose constraints on the amount of additional data that could be elicited in this manner.

Non-binary questions

Lastly, we discuss extending the PWM to non-binary multiple choice questions. Most of this extension is straightforward, since the conditional distributions have natural non-binary counterparts. The prior over worlds becomes a multinomial rather than binomial distribution, with the world hyperprior drawn from a Dirichlet, rather than Beta, distribution. The signal distribution for each world is likewise a multinomial distribution. Respondents can still compute a Bayesian posterior distribution over worlds and the answers of others. Respondent votes can still be modeled with a softmax decision rule from the Bayesian posterior over multiple possible worlds. Respondents can still compute a Bayesian posterior distribution over the votes of others, but one needs a method of adding noise to this posterior. One possibility is to sample from a truncated normal distribution around each element of the posterior and then normalize the resultant draws to sum to 1, another is to sample from a Dirichlet distribution with a mean or mode determined from the Bayesian posterior and an appropriate noise parameter.

Conclusion

We have presented a generative model for inference that can be applied to single questions and infer the correct answer even in cases where the majority or confidence-weighted vote is incorrect. The model shows good performance compared to models both when applied to questions separately and when applied across multiple questions. It maintains this performance when the answers for questions do not have a consistent ordering. It additionally allows one to infer respondent level expertise parameters that predict the actual accuracy of individual respondents. While the possible worlds model that we have proposed allows multiple extensions it is already a powerful method for aggregating the beliefs of groups of people.

Acknowledgments

We thank Dr. Murad Alam, Alexander Huang and Danica Mijovic-Prelec for generous help with Study 3, and Danielle Suh with Study 4. Supported by Sloan School of Management, Google, Inc., NSF SES-0519141, and Intelligence Advanced Research Projects Activity (IARPA) via the Department of Interior National Business Center contract number D11PC20058. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions expressed herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DoI/NBC, or the U.S. Government.

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