# How To Grade a Test Without Knowing the Answers — A Bayesian ...

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How To Grade a Test Without Knowing the Answers A BayesianGraphical Model for Adaptive Crowdsourcing and Aptitude Testing

Yoram Bachrach yobach@microsoft.com

Microsoft Research, Cambridge, UK

Tom Minka minka@microsoft.com

Microsoft Research, Cambridge, UK

John Guiver joguiver@microsoft.com

Microsoft Research, Cambridge, UK

Thore Graepel thoreg@microsoft.com

Microsoft Research, Cambridge, UK

Abstract

We propose a new probabilistic graphicalmodel that jointly models the difficulties ofquestions, the abilities of participants and thecorrect answers to questions in aptitude test-ing and crowdsourcing settings. We devisean active learning/adaptive testing schemebased on a greedy minimization of expectedmodel entropy, which allows a more efficientresource allocation by dynamically choosingthe next question to be asked based on theprevious responses. We present experimentalresults that confirm the ability of our modelto infer the required parameters and demon-strate that the adaptive testing scheme re-quires fewer questions to obtain the same ac-curacy as a static test scenario.

1. Introduction

Collective decision making is a well-studied topic insocial choice, voting and artificial intelligence. It haslong been known that decisions based on aggregatingthe opinions of several agents can be of higher qualitythan those based on the opinions of single individuals.The Condorcet Jury Theorem (de Caritat et al., 1785),dating back to the 18th century, is concerned with agroup of individuals attempting to reach a binary de-cision by majority vote; it is assumed that one of the

Appearing in Proceedings of the 29 th International Confer-ence on Machine Learning, Edinburgh, Scotland, UK, 2012.Copyright 2012 by the author(s)/owner(s).

two outcomes of the vote is correct, and that each in-dividual independently chooses the correct responsewith probability p. The theorem states that if p > 12 ,then adding more agents increases the probability ofmaking the correct decision, and the probability thatthe collective decision will be correct approaches 1in the limit of infinitely many participating agents.

Recent technological advances make it easier to shareopinions and knowledge, enabling us to harness thecollective intelligence of crowds for solving tasks.Companies can use crowdsourcing to carry out busi-ness tasks, using platforms such as Amazon Mechani-cal Turk. Such services allow us to collect the opinionsof many individuals, but leave open the question ofhow to aggregate the collected data to reach decisions.

A key technique for solving tasks using collectiveintelligence is to obtain information from multiplesources and aggregate it into a single complete solu-tion. Consider a crowd of experts who are assignedwith many similar classification tasks, such as classi-fying many news articles according to their topic (pol-itics, business, entertainment etc.). We refer toeach expert as a participant and to each classificationtask as a question. Suppose each participant expressesher opinion regarding the correct answer for each ques-tion, in the form of a response, chosen by her from thelist of possible answers for that question. Similarlyto Condorcets Jury Theorem, we make the simplify-ing assumption that for each of the questions only oneanswer is correct. We call such a domain a multipleproblem domain. Given the responses provided by theparticipants regarding the various questions in a mul-

How To Grade a Test Without Knowing the Answers

tiple problem domain, how should we best determinethe correct answer for each of the items? Which ques-tions are easy and which are hard? How can we findthe most competent participants in the crowd? Whichquestions best test the ability of a participant?

Given the correct answers to each of the items, it iseasy to find the most competent participants, or dif-ferentiate the easy and hard questions a participantwho has answered almost all the questions correctly islikely to be more skilled than a participant who hadvery few correct responses. Typically, however, thecorrect answers are not known in advance the wholepoint of crowdsourcing a classification task is to deter-mine the correct classifications for the items.

A possible solution to the above problem is to firstevaluate the skill of each expert by asking her to pro-vide responses to a set of items for which the correctanswer is known (sometimes called a gold-set). Aprominent example of this, where all correct answersare known, is intelligence testing (Anastasi et al.,1997). Psychologists have studied human intelligence,and designed IQ tests for evaluating the aptitude ofindividuals. These tests have been shown to be predic-tive of a persons performance in many domains, suchas academic attainment and job success (Anastasiet al., 1997). Such tests typically ask participants torespond to questionnaires composed of many multiple-choice questions, and allow ranking participants ac-cording to their individual skill levels after examiningthe responses. The properties of responses to IQ testshave been widely studied by psychometricians, so suchdatasets can serve as a testing ground for exploring in-ference models for multiple problem domains.

Our Contribution: We propose a new family ofgraphical models for analyzing responses in multipleproblem domains and evaluate the models on a dataset of completed questionnaires of a standard IQ test.The proposed framework enables us to jointly inferthe correct answer for each question (when these arenot known in advance), the difficulty levels of the ques-tions, and the ability of each participant. We show howthe model can: determine a probability distributionover answers for a given question by aggregating theresponses of participants based on their abilities andthe questions difficulties; test the ability levels of par-ticipants efficiently by finding the best next questionto ask in an adaptive way, depending on the previousresponses; automatically calibrate aptitude tests froma set of questions and the responses provided by theparticipants, determining the relative difficulty levelsof the questions and their ability to discriminate be-tween participants of similar but uneven skill levels.

2. Related Work

Measuring intelligence is a key topic in psychology.Psychologists showed that peoples performance onmany cognitive tasks is strongly correlated, so asingle statistical factor called general intelligenceemerges (Anastasi et al., 1997). A measure for the per-formance of groups of people in joint tasks, called col-lective intelligence, was investigated (Woolley et al.,2010). This approach focuses on explicit collabora-tion and interaction between members of the crowd.Although in our setting participants do not interactdirectly, one can view our model as a method for infer-ring correct answers to questions given the responses ofa crowd of individuals. The number of correct answersinferred can serve as a measure of the intelligence of thecrowd. In this sense, our work is somewhat similar toother approaches which also use aggregated responsesto IQ tests for measuring collective intelligence (Lyle,2008; Bachrach et al., 2012; Kosinski et al., 2012).

Psychometricians developed a body of theory calledtest theory which analyzes outcomes of psychologi-cal testing, such as the ability levels of participants orthe difficulty of questions in a test, trying to improvereliability in such tests (Anastasi et al., 1997). Oneparadigm for designing tests of mental abilities, is theitem-response theory (Hambleton et al., 1991) (IRTfor short). IRT has been used to develop high-stakesadaptive tests such as the Graduate Management Ad-mission Test (GMAT). IRT is based on the idea thatthe probability of a participant providing the correctresponse to a question is a function of both a param-eter of the question and a parameter of the item (forexample, the questions difficulty and the persons ap-titude). When applying aptitude tests, the parameterof the person is latent (cannot be directly observed),and only its manifestation, in the form of the partici-pants responses, can be directly observed. Our frame-work relies on a probabilistic graphical model (Koller& Friedman, 2009), using themes similar to IRT.

Many papers deal with merging opinions, ranging frominformation aggregation in the semantic web (Kasneciet al., 2010) to prediction markets (Pennock & Sami,2007). Frameworks such as Probabilistic RelationalModels (Getoor et al., 2007) combine a logical repre-sentation with probabilistic semantics, and allow in-ference to aggregate information and opinions. Onebasic method for collective decision making is voting.Voting was studied in social choice theory (Sen, 1986),which focuses on how participants can manipulate bylying about their preferences. We assume that theexperts responses are their true opinion and focus onthe inference problem. One application of our model is

How To Grade a Test Without Knowing the Answers

aggregating crowdsourced opinions. A machine learn-ing approach for doing so which does not model taskdifficulty was proposed in (Raykar et al., 2010) and atechnique that models task difficulty but uses an EMapproach was proposed in (Whitehill et al., 2009). An-other method based on graphical models is (Welinderet al., 2010). In that model questions are endowedwith features, which could represent concepts or top-ics and participants have different areas of expertisematching these topics. Our model focuses on a gen-eral domain in the spirit of test theory and IRT, anddoes not rely on specific features. An active learningapproach for labeling data was proposed in (Yan et al.,2011) and is similar to our adaptive IQ testing tech-nique. Another approach akin to ours is the TrueSkillsystem (Herbrich et al., 2007), which uses a graphi-cal model to estimate the relative skill levels of peoplebased on past contests.

3. Joint Probabilistic Model ofDifficulty, Ability, and Response

We present a probabilistic model for analyzingmultiple problem domains, which we refer to asthe Difficulty-Ability-REsponse estimation model, orDARE for short. The inputs to the model are re-sponses that participants give to multiple choice ques-tions. Additional inputs may be ground truth informa-tion for some or all of the questions. The model fallsinto the framework of probabilistic graphical models.Such models allow structurally describing the genera-tive process assumed to underlie the observed data interms of latent and observed random variables. In thedomain of interest, information such as the correct re-sponse to a question, the ability of a participant, andthe difficulty of a question are modeled as unobservedvariables whereas the given response to a question bya user is viewed as an observed variable. The struc-ture of the model is determined by the conditional in-dependence assumptions made about the variables inthe model. Pearl (Pearl, 1988) introduced BayesianNetworks (directed graphical models), which encodeassumptions of conditional independence as a graphwhose vertices represent variables and whose edgesrepresent dependencies between variables. We use themore general notion of a factor graph, see e.g. (Koller& Friedman, 2009), to describe the factorial structureof the assumed joint probability distribution amongthe variables. After defining the structure of the modelas a factor graph and setting the observed variables totheir observed values, approximate message passing al-gorithms (Koller & Friedman, 2009) can infer marginalprobability distributions of unknown variables of in-terest such as the correct response to a question, the

ability of a participant, or the difficulty of a question.

3.1. The DARE Model

We model a situation in which a set P of participantsis available to answer a set Q of multiple choice ques-tions. We assume that for each question q Q thereare Rq possible answers, only one of which, yq Rq, iscorrect. We model the process by which participantsp P produce responses rpq Rq to questions q Q.We assume that: a) Every participant has an under-lying ability ap R which determines her ability todetermine the correct answer to questions q Q. b)Each question q has an inherent difficulty dq R whichdetermines how likely it is that participants p P willknow the correct answer to question q.

We propose a joint probabilistic model whose factorgraph is given in Figure 1: The model has two parts,one modeling the probability of participant p knowingthe correct answer to question q (left of cpq in Fig-ure 1), and one relating the true answer yq to questionq to the response rpq given by participant p dependingon them knowing the correct answer as represented bycpq of the answer (right of cpq in Figure 1). Knowledgeof the correct answer, cpq {T, F}, is modeled as aninteraction of the ability ap R of participant p, andthe difficulty dq R of question q. Specifically, it isassumed to depend on the difference tpq := apdq via:

P (cpq = T |tpq, q) :=

(q(x tpq))(x) dx

= (qtpq

). (1)

Here denotes the standard Gaussian density (x) :=21

exp(x2/2) and denotes the (sigmoidal) cu-mulative Gaussian distribution (t) :=

t (x) dx;

() denotes the step function, and the precision qdetermines how discriminative question q is. The in-tegral representation emphasizes that the probabilitycan be viewed as emerging from a binary process re-sulting from evaluating the step function on variablet with added Gaussian noise of variance 1.

The response rpq is modeled as a mixture of two distri-butions. If participant p knows the correct answer toquestion q, cpq = T , we constrain the response rpq tomatch the correct answer, rpq = yq, otherwise we as-sume that rpq is sampled uniformly at random from theavailable answers, rpq DiscreteUniform(Rq). Notehow this mixture is expressed as a gate (dashed pair ofboxes in Figure 1), which switches the factor connect-ing to rpq depending on the state of the variable cpq.Gates were introduced in (Minka & Winn, 2008) asa powerful and flexible notation that simplifies factor-graph representations of mixture models. They can be

How To Grade a Test Without Knowing the Answers

used to represent context-dependent conditional inde-pendence relations, and are suited for implementationsof approximate message passing inference.

In order to do inference on the model, we need todefine prior distributions for the variables of inter-est. We assume factorizing Gaussian priors for theabilities ap Normal(p, 2p) and difficulties dq Normal(q,

2q ). We choose a Gaussian prior as it lets

us specify a range of plausible values based on twoparameters (mean and variance) per variable, and ad-mits a relatively simple approximate inference. Thefactorization assumption reflects the belief that a pri-ori knowing the difficulty of one question would notbe informative about the difficulty of another ques-tion, and similarly for the abilities of participants. Wealso assume factorizing discrete uniform priors for thetrue answers yq DiscreteUniform(Rq) and for theresponses rpq DiscreteUniform(Rq) for participant-question pairs. Finally, we define factorizing Gammapriors for the precision parameters q Gamma(k, ).The Gamma prior is conveniently parameterized by ashape parameter k and a scale parameter , and is theconjugate prior for the precision parameter := 2

of the normal distribution if the mean is known.This choice simplifies inference by approximate mes-sage passing because the posterior also takes the func-tional form of the Gamma distribution.

Based on the above specification we defined a jointprobability distribution p(ap, dq, tpq, q, cpq, rpq, yq) forspecific pairs of question q and participant p. Assum-ing exchangeability of questions q and participants pwe get a model with two plates as depicted in Fig-ure 1, where one plate runs over participants p Pand the other over questions q Q (plates denote areplication of the fraction of the graphical model theycontain). From a generative point of view, this modelsa table with |P | rows (one for each participant p) and|Q| columns (one for each question q), where entriesare the responses rpq of participants to questions.

3.2. Probabilistic Inference

We show how the model can infer quantities of interest.Generally, the data is given in the form of two incom-plete sets: A set of m participant-question-responsetriples R := {rp1q1 , . . . , rpmqm} and a set of n ground-truth question-answer pairs y := {yq1 , . . . , yqn}. Onespecial case is when all the ground-truth question-answer pairs are known. This is the traditional testscenario as used in aptitude tests including schooltests, GMAT, IQ tests, etc. Another special case iscrowdsourcing domains where we may not have anyground-truth available, but can obtain participant-

Figure 1. Factor graph for the joint difficulty-ability-response estimation (DARE) model.

question-response triples depending on budget andtime constraints. As shown in Section 4, provid-ing some ground-truth question-answer pairs (a gold-set), can improve the accuracy of the inferred answersyq because it can be used to assess the abilities ap ofthe participants p more accurately, leading to moreaccurate inference on the answers yq. Generally, forevery observed response rpq, we set the discrete priordistribution p(rpq) to a single point distribution con-centrated on the observed response rpq, and similarlyfor every known ground-truth question-answer pair yq .

Given the data R and y, we wish to infer sev-eral approximate marginal (posterior) distributions:the discrete distribution p(yq|R,y) over correct an-swers yq, which assign a probability qr [0, 1] toeach of the possible responses r Rq; the Gaussiandensity p(ap|R,y) over abilities ap of participants pwith means p and variances

2p; the Gaussian den-

sity p(dq|R,y) over difficulties dq of questions q withmeans q and variances

2q ; the Bernoulli distribution

p(cpq|R,y) over correctness cpq of participant ps re-sponse to question q given by probabilities pq; thediscrete distribution p(rpq|R,y) over responses rpq ofparticipant p to question q, which assign a proba-bility pqr [0, 1] to each of the possible responsesr Rq; the Gamma distribution p(q|R,y) over theprecision/discrimination parameter q, with scale pa-rameters q and shape parameters kq.

Inference in the model is done using approximatemessage passing (see (Koller & Friedman, 2009)).We used Infer.NET (Minka et al., 2010), a pack-age for probabilistic inference. Specifically, we usedthe expectation-propagation (EP) algorithm presentedin (Minka, 2001). EP allows us to calculate marginaldistributions of interest on a given factor graph by it-eratively calculating messages along edges that propa-gate information across the factor graph. In our case,EP provides only an approximation to the exact solu-

How To Grade a Test Without Knowing the Answers

tion because a) the underlying factor graph is loopy ,and b) the messages at the junction between cpq, tpq,and q are approximations, and so are the messages go-ing in and out of the gate connected to cpq. Thus EPis run iteratively until convergence, so its running timeis linear in the input size (variables and observations).

3.3. Active Learning and Adaptive Testing

Having a joint probabilistic model of the data has anumber of advantages, including the ability to querydifferent distributions of interest and to handle missingdata in a principled way. In addition, maintaining in-formation about the uncertainty present in the modelallows us to reason about the impact of future obser-vations on the model uncertainty. This idea forms thebasis of active learning, a variant of which is known inthe psychometrics literature as adaptive testing.

Often there is a considerable cost associated with ob-taining additional data points, so one can use themodel of the data to determine which measurements totake next so as to improve the inferred knowledge ac-cording to a pre-determined criterion. In the absenceof problem specific information, a reasonable goal is re-ducing uncertainty in estimates of model parametersas measured by the entropy of the posterior distribu-tions, an idea put forward in (MacKay, 1992).

Suppose we have determined a set of model parametersof interest, denoted here as a vector w. Our goal is tofind a criterion by which to decide which response rpqto elicit in order to maximally reduce the posteriorentropy S(w) of those parameters defined as:

S(w) =

p(w) log

m(w)

p(w)dw ,

where m(w) is an arbitrary base measure whichdoes not influence the outcome (see (MacKay, 1992)).We consider two posterior distributions, pm(w) :=p(w|R,y) before inclusion of the new data point,and pm+1(w) := p(w|R,y, rpq) after inclusion ofdata point rpq. We then aim at maximizing the en-tropy reduction S(rpq) := S(pm(w)) S(pm+1(w))over the choice of response rpq to elicit. Sincethe actual response rpq is unknown, this choice canonly be guided by the expected entropy reductionEpm(rpq|R,y)[S(rpq)] , where the expectation is takenover the predictions of the model before inclusion ofthe new data point, i.e., based on the predictive dis-tribution pm(rpq|R,y) obtained by message passing.

In its full generality, this active learning scheme canguide the full observation/measurement process in-cluding all possible responses rpq and ground truthanswers yq. However, here we focus on the case of

adaptive testing, where all the ground-truth answersyq are available, and where the goal is to determinethe ability of a participant p as accurately as possi-ble, using as few questions as possible. In this spe-cial case, the parameter vector w only includes theability ap of participant p. The posterior distributionpm(ap) before inclusion of the new observation is Nor-mal, pm(ap) := Normal(ap;p.m,

2p,m), and so is the

posterior distribution after inclusion, pm+1(ap|rpq) :=Normal(ap;p,m+1(rpq),

2p,m+1(rpq)). The entropy of

a univariate Gaussian with parameters and 2 is12 ln(2e

2), so the entropy reduction S(rpq) is:

S(rpq) =1

2ln(2p,m/

2p,m+1(rpq))

Thus the response minimizing posterior variance ispreferred. Given participant p, for each possible ques-tion q the expectation Epm(rpq|R,y)[S(rpq)] is calcu-lated by examining the following quantities for all pos-sible responses rpq Rq: a) their probabilities pq, andb) the resulting posterior variances 2p,m+1(rpq) in theupdated model. From these we compute the expectedentropy reduction for each question q:

1

2

rpqR

pq ln(2p,m/

2p,m+1(rpq))

We then pick the question q that reduces the expectedentropy the most.

4. Empirical Analysis

We empirically tested the DARE model discussed inSection 3.1 using a dataset of responses to a standardintelligence test, called Ravens Standard ProgressiveMatrices (SPM) (Raven), which falls within the cat-egory of multiple choice domains. It consists of sixtyquestions, each of which consists of a matrix of shapeswith one element missing and eight possible answers.Each answer is a possible shape that completes the ma-trix, but only one answer is correct. A sample item,similar1 to those in SPM is shown in Figure 2. SPMis one of the most popular intelligence tests, and wasused both for research and clinical purposes.

The sample consisted of 120 individuals who filledSPM for its standardization in the British market in2006 (Raven). The mean number of correct responses,called raw score, was 99.57 (STD=14.16).

4.1. Unobserved Correct Answers

First, we investigate the DARE models ability to han-dle missing correct answers yq. In this case the model

1The SPM test is copyright protected, so we only pro-vide an example question similar to those in the real test.

How To Grade a Test Without Knowing the Answers

Figure 2. Item similar to those in the SPM test

allows us to compute the probability p(yq|R,y) that agiven answer yq is correct. To minimize the probabil-ity of error we select the mode of that distribution asthe models answer for that question. When providedwith the responses of all 120 participants the DAREmodel correctly infers the correct responses for 46 ofthe questions. Note, that the number of errors is nottoo surprising because some items in the test were verydifficult and few participants answered them correctly.The highest raw score was fifty so even the top scoringparticipant answered ten items incorrectly.

We can calculate a participants raw IQ score with re-spect to the true correct answers yq or with respect tothe predicted correct answers yq, and we refer to thelatter score as the model raw score. In crowdsourcingsituations when the correct answer for each question isunknown, one can use the model raw scores as an esti-mate of participants abilities. Figure 3 shows a scatterplot, in which each point represents a participant; theposition on the x-axis represents the participants rawIQ score, and the position along the y-axis their modelraw score. As Figure 3 indicates, there is a very strongcorrelation between the true raw IQ scores and modelraw scores (R2 = 0.7243), and the difference betweenthe two scores is rather small across all participants.

One can think of DARE as an aggregator that re-ceives the responses of a crowd of participants, andoutputs the inferred answer for each question. Ex-isting work (Lyle, 2008; Bachrach et al., 2012) testssimpler aggregators using IQ test data. The formeruses majority voting, and the latter does consider theability levels of participants but assumes all items tobe of equal difficulty. Another possible simplifying as-sumption, not examined in this earlier work, is that allthe participants have equal ability. In contrast, in theDARE model, the probability of a participant to knowthe correct answer depends both on the difficulty dqof the question and the ability ap of the participant,

Figure 3. Estimates of skill levels for missing informationregarding the correct answers to the questions.

with the above scenarios as special cases.

We refer to the model with different question difficul-ties as the question model, and the model with differentparticipant abilities as the participant model. We ex-amine how such simplifications affect the models abil-ity to infer correct answers as a function of the amountof available data. Figure 4 shows how well the ques-tion, participant and DARE models perform in thisregard. For any given crowd size, shown on the x-axis,we randomly selected 10, 000 subsets of participantsof that size. For each such crowd we inferred the cor-rect answer yq to each question using the model, andused the number of questions for which the inferredanswer yq was equal to the true correct answer y

q as a

measure of the models performance. The y-axis is thequality of each model, averaged over the 10, 000 sam-pled crowds. Figure 4 shows the ability of all modelsto infer correct answers increases with the amount ofdata. It also shows that DARE outperforms the sim-pler models. Interestingly, only modeling participantsability of is better than only modeling question diffi-culty (which is equivalent to majority vote).

Figure 4. Effect of crowd size on correct responses inferred.

How To Grade a Test Without Knowing the Answers

Crowdsourcing Data: To examine the appli-cability of our model to crowdsourcing, we testedour model on the TREC 2011 Crowdsourcing Trackdataset (Lease & Kazai, 2011), generated by crowd-sourced workers which classified search engine re-sponses for queries (relevant / irrelevant). Each query-document pair is a question as workers must deter-mine if the document is relevant for the query. Thisdataset is sparse, as most workers only examined fewquestions, and all questions have at most 10 answersin total, and includes ground-truth judgements. Weisolated the 369 questions with the most answers (8per question), and the 84 workers who answered themost questions (at least 30 answers each). Our anal-ysis shows that DARE slightly outperforms majorityvoting on this dataset. Majority voting gets 206 ques-tions correct, while DARE gets 210 correct. We alsotested how well DARE estimate participants skills,similarly to Figure 3. Although for this crowdsourcingdataset the resulting scatter plot is quite noisy (r2 of0.79), it is similar to the one in Figure 3.

4.2. Partial Information on Correct Answers

We now examine situations where participants are firsttested on a gold-set of questions for which the cor-rect answer is known. Consider choosing i questionsand making the correct answer to these questions ob-servable to the model. This does not reveal the cor-rect answer to the remaining |Q| i questions, butit does allow the model to better estimate the abil-ity levels of the participants, which in turn allows themodel to better infer the correct answer to these re-maining items. Figure 5 shows this effect in DARE.The x-axis represents the number of revealed ques-tions and the y-axis represents the proportion of theremaining questions for which the model inferred theright answer. For each number i of revealed items,we sampled 100, 000 crowds of 20 participants and irevealed questions (uniformly at random), and the lo-cation on the y-axis is the average proportion of theremaining questions for which the model inferred theright answer over this sample. As the figure shows,having a larger gold-set increases the models abilityto infer the correct response for the remaining items.

4.3. Adaptive Skill Testing

We now show how DARE can be used for adaptiveskill testing. Given a budget of b questions to ask,our goal is to infer the participants ability levels. Weuse DARE to estimate a participants raw IQ scoreafter only observing this participants responses to aset of asked questions (revealed responses). In astatic approach, for each budget b we choose a specific

Figure 5. Effect of partial information on correct answers.

question set used for all the participants, and measurethe RMSE in estimated raw IQ across all participants.To choose the best set of questions of size b for thestatic approach, one must enumerate over all possible(|Q|

b

)question sets of size b, and use the one minimiz-

ing the error. This is intractable when |Q| is large, sowe heuristically choose the question set. We selectedstatic question set for a given budget b by choosingquestions that equally partition the participant pop-ulation in terms of the fraction of participants whosolved the question2 For example, with a budget b = 2we selected a question that roughly two thirds of theparticipants solved correctly and one that roughly onethird of the participants solved incorrectly.

We also implemented the adaptive testing scheme ofSection 3.3 and compared it to the baseline static ap-proach. Under the adaptive approach, the next ques-tion to ask depends on the participants response toearlier questions, so we reveal the participants re-sponses one at a time. To measure the RMSE for agiven budget b, we simulated the adaptive process foreach of the participants and averaged the errors acrossall participants. Figure 6 shows RMSEs for the staticand adaptive approaches for different budget levels. Itshows the adaptive approach has a smaller error in itsinferred ability levels for any given budget. 3

5. Conclusions and Limitations

We presented the DARE model for inferring the cor-rect answers, difficulty levels of questions and abil-

2The static approach essentially has access to informa-tion regarding the difficulty of the questions which is notnormally be available. As our analysis shows, our activeapproach beats the static approach even when the staticapproach can use such information.

3The standard deviation for the RMSEs is 1.07 for theadaptive scheme and 0.99 for the static scheme.

How To Grade a Test Without Knowing the Answers

Figure 6. Static and adaptive skill testing.

ity levels of participants in multiple problem domains.Our evaluation of the model shows that joint infer-ence of these quantities is possible to a high level ofaccuracy and that it is indeed possible to grade a testwithout knowing the answers. We showed that in oursetting modeling participants ability levels is more im-portant than questions difficulty levels, that includinga gold-set helps, and that active learning leads tomore efficient testing.

Our approach is subject to several limitations. Ourevaluation used an IQ dataset, whereas crowdsourc-ing tasks may exhibit different properties, such as agreater homogeneity in task difficulty levels. Also, weassume that participants answer to the best of theirability, but participants may be selfish agents withvarying motives. For a game theoretic treatment ofsuch issues see (DiPalantino & Vojnovic, 2009; Gaoet al., 2012).

Many questions are open for future research. Arethere better models for aggregating responses, or mod-els better tailored to other domains? How can onetractably compute the optimal non-adaptive test for agiven population? Can we use similar models to inferthe ability levels of individuals when only their perfor-mance within the context of a group is known?

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