Partisan Bias and the Bayesian Ideal in the Study of...

Partisan Bias and the Bayesian Ideal in theStudy of Public Opinion

John G. Bullock Yale University

Bayes’ Theorem is increasingly used as a benchmark against which to judge the quality of citizens’ thinking, butsome of its implications are not well understood. A common claim is that Bayesians must agree more as they learnand that the failure of partisans to do the same is evidence of bias in their responses to new information. Formalinspection of Bayesian learning models shows that this is a misunderstanding. Learning need not create agreementamong Bayesians. Disagreement among partisans is never clear evidence of bias. And although most partisans arenot Bayesians, their reactions to new information are surprisingly consistent with the ideal of Bayesian rationality.

In February 2004, 11 months after the Americaninvasion of Iraq, emerging evidence suggestedthat Iraq had not possessed weapons of mass

destruction at the time of the invasion. Partisans weredivided on the matter: 40% of Democrats reportedbelieving that Iraq had WMD at the time of the in-vasion, against 83% of Republicans who said the same(ABC News/The Washington Post 2004). In the fol-lowing year, the evidence became clearer. The failureof the Iraq Survey Group to locate WMD was widelypublicized, and the chief U.S. weapons inspectordeclared that WMD ‘‘were not there’’ at the time ofthe invasion (Duelfer 2004, 6). But the revelationsmade little difference: by March 2005, 35% of Dem-ocrats said that Iraq had WMD at the time of theinvasion, and 78% of Republicans said the same(ABC News/The Washington Post 2005). The gap be-tween partisans’ beliefs was as wide as ever.

Public opinion scholars will recognize this as anew instance of an old pattern. Enduring disagree-ments between Republicans and Democrats are a hall-mark of American politics, and they have long beentaken as evidence that identifying with a party biasespartisans’ subsequent political views (e.g., Berelson,Lazarsfeld, and McPhee 1954, Chap. 10). Of course,some disagreements may be due to differences in val-ues that are causally prior to party ID (Downs 1957;Fiorina 1981), but value differences cannot explainpartisan disagreements over factual questions andother matters to which values are irrelevant. Such

disagreements seem to be especially strong evidenceof partisan bias (Bartels 2002).

The most direct challenge to this view comesfrom Gerber and Green (1999), who maintain thateven these ‘‘hard cases’’ are not evidence of partisanbias. They use a Bayesian model of belief revision toargue that even when Republicans and Democratsshare the same values and are free from partisan bias,disagreements between them can endure. The argu-ment is unusual, but the model is not: Bayesianmodels like theirs are increasingly used as frameworksthrough which changes in public opinion are studied(e.g., Achen 1992; Bartels 1993; Gerber and Jackson1993; Grynaviski 2006; Husted, Kenny, and Morton1995; Lohmann 1994; Zechman 1979) and as norma-tive benchmarks against which to judge the quality ofcitizens’ thinking about politics (e.g., Bartels 2002;Gerber and Green 1999; Steenbergen 2002; Tetlock2005).

Bartels (2002, 119–23) counters that Gerber andGreen misunderstand their model: unbiased applica-tion of Bayes’ Theorem, he writes, implies not endur-ing disagreement but increasing agreement wheneverpeople revise their beliefs in response to new infor-mation. Achen (2005, 334), Grynaviski (2006, 331),and Goodin (2002) agree in a series of thoughtfularticles, with Grynaviski declaring that unbiasedBayesians ‘‘inexorably come to see the world in thesame way’’ and Goodin finding something sinister:‘‘Bayesian rationality requires minorities to cave in

The Journal of Politics, Vol. 71, No. 3, July 2009, Pp. 1109–1124 doi:10.1017/S0022381609090914

� 2009 Southern Political Science Association ISSN 0022-3816

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too completely, too quickly.’’ But none of thesearticles prove any properties of the Bayesian modelthat lies at the heart of the dispute, and as this articleshows, each of them elides vital points about theconnection between Bayesian learning and partisandisagreement. In the very long run, partisans whorevise their beliefs according to Bayes’ Theorem willagree if they receive the same evidence and interpretit in the same way. But in the shorter term—the onethat we care about when we study public opiniontrends over months, years, and election campaigns—Bayesian learning is compatible with sustained dis-agreement. And contrary to Bartels (2002, 121–23),Tetlock (2005, 128), and others, Bayesian learning iscompatible with the polarization of attitudes andbeliefs—even when people receive the same informa-tion and interpret it in the same way.

These findings show that enduring disagreementand polarization can never be ipso facto proof ofpartisan bias, because they can occur even amongunbiased Bayesians. They also matter because of theexpanding role that Bayesian models play in analysesof public opinion. Partisans are not Bayesians, butBayesian updating matters because it is increasinglythe normative standard against which partisan up-dating is judged. We have an interest in sorting outwhat it entails.

I begin by reviewing Bayes’ Theorem and con-sidering objections to its use in the study of politicalbeliefs. The next section proves properties of thedominant Bayesian model of public opinion, estab-lishing conditions under which learning will createagreement and conditions under which it will pro-mote disagreement. This model pertains only tolearning about political conditions that never change;in the following section, I generalize the model andthe results to account for learning about changingpolitical circumstances. I proceed by arguing that theconditions for ‘‘Bayesian agreement’’ established inthe previous sections are rarely met in practice. Thelast section reviews and concludes.

The Utility of Bayesian Modelsof Public Opinion

Many matters that interest political scientists—pop-ular beliefs about politics, politicians’ true preferen-ces, implications of new policies—can be thought ofas probability distributions. Like most distributions,they have means and variances, and the task that weset for ourselves is to learn about these parameters.

Messages about a politician’s ability to manage theeconomy, for example, may oscillate around a fixedbut unknown mean, which is the politician’s trueability level. Learning about politics becomes a matterof learning about probability distributions—a task towhich Bayesian statistics is especially well-suited.

The bedrock of Bayesian statistics is Bayes’ The-orem, an equation that relates conditional and mar-ginal probabilities:

pðSjEÞ5 pðEjSÞpðSÞpðEÞ ;

where S and E are events in a sample space and p(�) isa probability distribution function. Stated thus, theTheorem is merely an accounting identity, but achange in terminology draws out its significance. Thistime, let S be a statement about politics and E beevidence bearing on the statement. Before observingE, a person’s belief about S—that is, a probabilitydistribution indicating his estimate of the extent towhich S is true—is given by p(S). Because it is hisbelief about S prior to observing E, it is often calledhis prior probability of S, or simply his ‘‘prior.’’ Afterhe observes E, his estimate of the probability thatS is true is p(S|E), often called his posterior probabilityof S. And p(E|S) is the likelihood function that heassigns to the evidence; it reflects his guess about theprobability distribution from which the evidence isdrawn. Understood in this way, Bayes’ Theorem tellsus how to revise any belief after we have receivedrelevant evidence and subjectively estimated its like-lihood. It is most often applied to beliefs about futureevents (e.g., Tetlock 2005, Chap. 4), but it is funda-mentally a tool for calculating probabilities, and itapplies with equal force to attitudes about politicians,evaluations of their abilities, and all other ideas thatcan be described in probabilistic terms.1

The charge most frequently leveled against Baye-sian models of political belief revision is that peo-ple do not update their beliefs as Bayes’ Theoremdemands. The charge has merit,2 but it misses twoimportant points. One is that use of Bayesian modelsin public opinion research does not always entail an

1See Bartels (1993, 2002) and Gerber and Green (1998, 1999) forapplications of Bayesian models to political attitudes andevaluations.

2The most straightforward and best-documented violation ofBayesian updating is ‘‘cognitive conservatism,’’ the overweightingof prior beliefs when updating (e.g., Edwards 1982). Tetlock(2005), Peffley, Feldman, and Sigelman (1987), and Steenbergen(2002) argue that the effect holds generally for political beliefs.

1110 john g. bullock

assumption that people use Bayes’ Theorem to revisetheir beliefs. Often, the models instead provide a nor-mative benchmark against which to judge real beliefrevision (e.g., Bartels 2002; Steenbergen 2002; Tetlock2005). To know how far people fall from a normativeideal of belief revision, we need to specify a normativeideal, and the increasing use of Bayesian publicopinion models means that unbiased Bayesian updat-ing is that ideal.

The other important point is that Bayesianmodels of public opinion can be heuristically usefuleven if we wrongly assume that people are Bayesians,because they offer a systematic way to account for therelative influences of old beliefs and new information(Fischoff and Lichtenstein 1978, 242). Bartels’ 1993study of media effects in the 1980 presidentialcampaign is a good example. Bartels assumes Baye-sian updating among respondents in the AmericanNational Election Studies and concludes that votershad strong prior beliefs about the candidates at theoutset of the campaign.3 His data are also consistentwith the assumption that ANES subjects had weakpriors but overweighted them, i.e., that they actednot as Bayesians but as ‘‘cognitive conservatives’’(Edwards 1982). But from a political standpoint, thereis little difference between weighting strong priorsaccording to Bayes’ Theorem and overweightingweak priors. The substantive upshot is usually thesame, and so long as it is, the assumption of Bayesianupdating can be useful even when it is wrong.

Disagreement When Learningabout an Unchanging Condition

In studies of public opinion, one Bayesian learningmodel is far more common than the rest. It assumesthat prior beliefs are normally distributed and thatpeople perceive new information to be normally dis-tributed too (e.g., Achen 1992; Bartels 1993, 2002;Gerber and Green 1999; Husted, Kenny, and Morton1995; Lohmann 1994; Steenbergen 2002; Zechman1979). To see how it works, suppose that a voter istrying to learn about a politician’s level of honesty.She conceives of honesty as a continuum—there areinfinitely many degrees of honesty and dishonesty—and the politician’s level of honesty is an unknownpoint on this continuum. We call it m. Initially,the voter’s belief about m is normally distributed:

m ; Nðm̂0;s20Þ. The mean of this distribution, m̂0, is

the voter’s best guess at time 0 about the politician’shonesty. The variance of this distribution, s2

0, indi-cates the confidence that she reposes in this guess. Ifs2

0 is low, she is quite sure that the politician’s truelevel of honesty is around m̂0; if s2

0 is high, she al-lows that his true level of honesty might well be farfrom m̂0.

Later, the voter encounters a new message thatcontains information about the politician’s level ofhonesty. She interprets the message as having valuex1, where higher values suggest greater honesty on thepolitician’s part. She assumes that the message is adraw from a distribution with a mean of m.4 Thenormal distribution is usually a sensible assumption:if the message can theoretically assume any real value,and if error or ‘‘noise’’ is likely to be contributed to itby many minor causes, the central limit theoremsuggests that it is likely to be normally distributed.We write x1 ~ N(m, s2

x). The variance of thisdistribution, s2

x , captures how definitive the newinformation seems to the voter. If it comes directlyfrom a source that she trusts, the signal it sends isclear and the variance is low; but if it is a rumor, thesignal is weak and the variance is high.

By a common result (e.g., Lee 2004, 34–36), avoter with a normal prior who updates according toBayes’ Theorem in response to x1 will have posteriorbelief mjx1;Nðm̂1;s

21Þ, where

m̂1 5 m̂0

t0

t0 þ tx

� �þ x1

tx

t0 þ tx

� �; and ð1aÞ

s21 5

1

t0 þ tx; ð1bÞ

and where t0 5 1/s20 and tx 5 1/s2

x are the precisionsof the prior belief and the new message. The mean ofthe posterior belief, m̂1, is a weighted average of themean of the prior belief and the new message; theweights are determined by the precisions. Note thatalthough m̂ and x are indexed by time—we can havem̂2, x3, and so on—m is not indexed. The implicitassumption of this model is that people are learningabout a political condition that is not changing.

A useful aspect of this model is that it permitsdirect comparison of the strength of voters’ prior

3The chief moral of Bartels’ article is that media effects must alsohave been strong—stronger than previously believed—to changeprior beliefs as much as they did.

4This is tantamount to assuming that the message comes from anunbiased source. If the voter believes that the message comesfrom a biased source, she is likely to adjust for that bias beforerevising her belief. This is no obstacle to Bayesian updating (e.g.,Jackman 2005), but it is not part of the model that is typicallyconsidered in political science.

partisan bias and the bayesian ideal in the study of public opinion 1111

beliefs to the strength of the new information thatthey receive. For example, consider a Bayesian voterwhose belief about a politician’s honesty is m ~ N(2, 2).This belief can be thought of as reflecting all of therelevant information about the politician’s honestythat the voter has received throughout her life.Imagine that she then receives message x 5 4 withvariance s2

x 5 2, suggesting that she had underratedthe politician’s honesty. Because the variances of thenew message and the prior belief are equal, the newmessage seems as ‘‘strong’’ to her as all of theinformation that she had previously received. Byequations (1a) and (1b), she accords equal weightto her prior belief and the new information, and hernew belief is thus m ~ N(3, 1). Comparability ofbeliefs and information is a major virtue of Bayesianlearning models, and we will make use of it through-out this article.

Debate about the possibility of increasing dis-agreement under unbiased Bayesian updating re-volves around the properties of this model. Ofcourse, some disagreements cannot be resolved underany circumstances. They reflect value differences thatwill not be resolved by more learning.5 This is widelyunderstood in the debate over Bayesian updating ofpublic opinion. The case in which people receive thesame evidence, interpret it in the same way regardlessof value differences, and follow the same updatingrule should be the hardest case for which to prove thepossibility of increasing disagreement; it is the casethat I take up here.

When disagreement does not last—when peopledisagree before updating and agree after upda-ting—their views have converged to agreement. Beforeembarking on a series of proofs, it will help to definethis term and several others:

d Let m̂Dt be the mean of voter D’s belief about m

at time t. Let m̂Rt be the mean of voter R’s beliefabout m at time t. If m̂Dt 5 m̂Rt, D and R agree attime t. If m̂Dt 6¼ m̂Rt, they disagree at time t.

d Beliefs converge between time t and later time u ifand only if m̂Dt � m̂Rtj j. m̂Du � m̂Ruj j. They di-verge if and only if m̂Dt � m̂Rtj j, m̂Du � m̂Ruj j.

d Beliefs polarize between time t and time u if andonly if they diverge between t and u andm̂Du � m̂Dtð Þ= m̂Ru � m̂Rtð Þ, 0:

d Suppose that D and R update their beliefs inresponse to a sequence of messages x1; . . . ; xn. Ifand only if plim m̂Dn � m̂Rnð Þ5 0, their beliefs willconverge to agreement as n! ‘.

d A person learns when his belief changes inresponse to a message.

d People are unbiased if they correctly perceive thevalue and distribution of the messages that theyencounter. For example, an unbiased person willperceive message x from distribution N(m, s2

x) ashaving value x and as drawn from N(m, s2

x).A biased person encountering the same messagewill perceive it as having a value other than x or asdrawn from a distribution other than N(m, s2

x).Note that unbiased partisans necessarily interpretmessages in the same way: they agree on the‘‘locations’’ of the messages that they receive andperceive those messages to be drawn from thesame distributions.

The distinction between divergence and polarizationis crucial. Divergence implies that the gap betweenupdaters’ beliefs is widening between time t and timeu. Polarization implies both that this gap is wideningand that updaters’ beliefs are moving in differentdirections. Divergence can occur even as all beliefsmove in the direction of new evidence, but polar-ization requires that at least one person’s belief moveaway from the new evidence, a strange phenomenonthat Tetlock claims is not just ‘‘contra-Bayesian’’ butalso ‘‘incompatible with all normative theories ofbelief adjustment’’ (2005, 128). As we shall see, theseclaims are too strong.

Figure 1 illustrates the distinction between diver-gence and polarization, which is not drawn inpsychology, political science, or the popular press.In all three fields, ‘‘polarization’’ is used to describeboth phenomena; for example, when political scien-tists maintain that Bayesian updating is incompatiblewith ‘‘polarization,’’ they generally mean that itcannot even accommodate divergence (e.g., Gerberand Green 1999, 199). Maintaining the distinction isimportant to understanding Bayesian models ofpublic opinion: such models accommodate diver-gence more easily than polarization.

With these definitions in hand, we can establishconditions under which updating will produce con-vergence, divergence, and polarization of beliefs. Inall cases, we consider two unbiased Bayesians, Dand R, who have prior beliefs m;N m̂D0;s

2D0

� �and m;N m̂R0;s

2R0

� �, respectively. We assume 0 ,

s2D0, s2

R0: neither D nor R is absolutely certain of

5A related point is made in the economics literature on ‘‘agreeingto disagree’’ (e.g., Aumann 1976), which assumes people whoshare the same prior belief and whose posterior beliefs arecommon knowledge. Neither assumption is appropriate to thestudy of public opinion, which is why I do not take up the‘‘agreeing to disagree’’ literature here.


his prior belief. All proofs appear in the onlineappendix.

Proposition 1 (convergence to agreement). If D andR receive a sequence of messages x1; . . . ; x iid

neNðm;s2xÞ

where s2x is finite, their beliefs will converge to agree-

ment as n! ‘.

Proposition 1 shows that unbiased Bayesian learn-ing will indeed create agreement among partisans—eventually. The result is familiar to statisticians (seethe online appendix for discussion), but it speaks

only to belief trends as partisans receive and processan infinite amount of information. Because mostpeople update their political views infrequently andsubsist on a diet that is nearly devoid of politicalinformation—points that I elaborate later—it is moreimportant to learn what happens before they attainpolitical-information Nirvana. And in the shorterrun, Proposition 2 shows that even unbiased Baye-sians may disagree more after updating than they didbefore. Indeed, they will disagree more after updatingwhenever the condition of Proposition 2 is satisfied:

Proposition 2 (divergence). Assume that D and Rupdate in response to a sequence of messagesx1; . . . ; xt; . . . ; xn

iideNðm;s2xtÞ, where s2

xt is the vari-ance of the distribution from which message xt isdrawn. Let �x 5 +n

t 5 1xt=n, txt 5 1/s2

xt , andtx�5 +n

t 5 1txt. Divergence occurs between time 0

and time n if and only if �x� m̂R0ð ÞtR0tx�þm̂D0 � �xð ÞtD0tx� =2 min fa; bg;max fa; bg½ �, where

a 5 m̂D0� m̂R0ð Þ tD0þtx�ð Þ tR0þtx�ð Þ� tD0tR0½ � andb 5 m̂R0 � m̂D0ð Þ tD0 þ tx�ð Þ tR0 þ tx�ð Þ þ tD0tR0½ �.This condition is somewhat complicated, but it has astraightforward explanation. Just as any voter’s pos-terior belief mean is a weighted average of his priorbelief mean and the new information that he receives,the distance between any two voters’ posterior beliefmeans is a weighted average of the distance betweentheir prior belief means and the distance betweeneach voter’s prior belief mean and the newinformation:

The left-hand side of this equation, jm̂Dn � m̂Rnj, isthe distance between the voters’ posterior beliefmeans. The right-hand side is the absolute value ofa weighted average with three components. The first

component, ðm̂D0�m̂R0ÞtD0tR0

ðtD0þtx�ÞðtR0þtx�Þ, is the difference between

the voters’ prior belief means, weighted by thestrength of those prior beliefs. The second compo-

nent, ð�x�m̂R0ÞtR0tx�ðtD0þtx�ÞðtR0þtx�Þ, is the difference between R’s

prior belief mean and the new information, weightedby the strength of R’s prior belief and the new

information. The third component, ðm̂D0��xÞtD0tx�ðtD0þtx�ÞðtR0þtx�Þ,

is the difference between D’s prior belief mean andthe new information, weighted by the strength ofD’s prior belief and the new information. Proposition 2says that when the second and third components aresufficiently great, beliefs will diverge.

The simplest way for unbiased Bayesian updatingto cause divergence is best conveyed by an example.

FIGURE 1 Divergence with and without Polarization. Each panel depicts the means of two people’s beliefsat times t and u. In the left-hand panel, the beliefs are diverging but not polarizing: althoughthe gap between the means is growing, they are moving in the same direction. In the right-handpanel, the beliefs are diverging and polarizing: the gap between the means is growing and they aremoving in opposite directions. By definition (see previous page), polarization requires divergence.

belie

f mea

ns

divergence without polarizationt u t u

belie

f mea

ns

divergence with polarization

jm̂Dn � m̂Rnj5ðm̂D0 � m̂R0ÞtD0tR0 þ ð�x � m̂R0ÞtR0tx� þ ðm̂D0 � �xÞtD0tx�

ðtD0 þ tx�ÞðtR0 þ tx�Þ

��:


Consider two voters, D and R, who believe that theUnited States has the same high probability ofwinning a war m̂D0 5 m̂R0ð Þ but are not equallyconfident of this belief: D is more confident than Rof U.S. victory s2

D0 , s2R0

� �. Both then receive a mes-

sage suggesting that the chances of victory are lowerthan they believed x1 , m̂D0 5 m̂R0ð Þ. Both respond bylowering their estimates of the chance of victory—but because R has the weaker prior belief, he isswayed more by the evidence, and he revises his beliefmore than D does. Before updating, D and R agreed;after updating, they disagree: their beliefs have di-verged. Divergence can also occur when D and R donot have the same prior belief means; for example, ifR initially thinks the chances of victory slightly higherbut is much less confident of his belief, divergence isstill likely to occur.

Although the necessary-and-sufficient conditionfor divergence stated in Proposition 2 is somewhatcomplicated, two necessary conditions are muchsimpler:

Corollary to Proposition 2 (necessary conditionsfor divergence). Under the assumptions of Pro-position 2, divergence can occur between time 0and time n only if s2

D0 6¼ s2R0 and

�x =2 min m̂D0; m̂R0f g; max m̂D0; m̂R0f g½ �.Previous explorations of this model (e.g., Bartels2002) focused on the special case in which membersof different parties have equally strong prior be-liefs s2

D0 5 s2R0

� �and respond only to ‘‘moderate’’

messages that fall within the range of the priorbelief means x1; . . . ; xt; . . . ; xnð 2 minfm̂D0; m̂R0g;½maxfm̂D0; m̂R0g�Þ. The Corollary to Proposition 2shows that these are conditions under which diver-gence cannot occur. And by extension, they areconditions under which convergence must occur everytime that people revise their beliefs. If a Democratbelieves that the probability of U.S. victory in a war is10%, a Republican believes that the probability is90%, and the news always suggests 50%, beliefs mustconverge every time they are revised. Similarly, if theDemocrat puts the probability at 80% and the Repub-lican at 90%, and their beliefs are equally strong, beliefsmust converge every time that they are revised—nomatter what the news indicates about the probabilityof victory. The focus on these two conditions in pre-vious research may account for the erroneous con-viction that unbiased Bayesian learning must alwaysproduce convergence when people revise their beliefs.

The impossibility of divergence when updatershave equally strong prior beliefs deserves specialattention. This condition does not make divergence

between individuals unlikely in practice, because anytwo people are unlikely to have priors of exactly equalstrength. But is there a systematic partisan differencein belief strength—not between any given Democratand Republican, but between Democrats and Repub-licans in general? Data on this point are scarce, butfindings that liberals tend to be more ambivalent thanconservatives (Feldman and Zaller 1992; see alsoTetlock 1986) and that white men tend to holdexceptionally strong beliefs about candidates (Alvarez1998, Chap. 6) suggest that there may be a systematicpartisan difference in belief strength, too.

What of the possibility that beliefs will polarize aswell as diverge in response to new information? Itsimply cannot happen under the model considered inthis section:

Proposition 3 (polarization). Under the assumptionsof Proposition 2, polarization cannot occur.

Recall that by equation (1a), posterior beliefs areweighted averages of prior beliefs and new informa-tion. When people with different prior beliefs receivethe same information, interpret it in the same way,and revise their beliefs according to equation (1a),their beliefs must shift in the direction of the newinformation. This is all that is required to precludepolarization.

Figure 2 illustrates possible patterns of beliefrevision among unbiased Bayesians. Each paneldepicts 30 simulations in which D and R revise theirbeliefs six times in response to new information.Different lines in each panel represent differentsimulations: when the lines slope down, disagreementbetween D and R is diminishing; when the lines slopeup, disagreement is increasing. In each simulation,D and R are trying to learn about a feature of politics,m, that has a value of 2. D’s prior belief about m isalways accurate but uncertain: m ~ N(2, s2

D0), wheres2

D0, the strength of his prior belief, varies from col-umn to column. R’s prior belief is always m ~ N(1, 1).At each time t 2 1; 2; . . . ; 6, D and R update theirbeliefs in response to a message xt

iideNð2;s2xÞ, where

the precise value of xt varies from simulation to sim-ulation and where s2

x , the strength of the messages,varies from row to row.

The simulation results differ from panel to panelbecause s2

D0 and s2x differ from panel to panel, so it is

vital to consider which values of these variances areplausible. Previous research offers almost no guid-ance: measurement of prior belief strength is rare(Alvarez and Franklin 1994), and measurement of thequality of news sources is rarer still. But for ourpurposes, what is most relevant is not the individual


FIGURE 2 Bayesian Learning about an Unchanging Condition Can Increase Disagreement BetweenPartisans. Each panel depicts 30 simulations of learning by two people who update accordingto Equations 1a and 1b. In each simulation, they receive a new message at each of six times.A single line is plotted for each simulation, showing how the absolute difference between themeans of their beliefs changes as they learn from the new messages. Most of these lines trendtoward 0 in every panel, indicating that updaters’ beliefs are converging to agreement. But thereare upward-sloping line segments in every panel save those in the second column, indicatingthat disagreement between updaters can increase from one time to the next even as their beliefstrend gradually toward agreement.

In each panel, m 5 2, D’s belief at time 0 is m ~ N(2, s2D0), and R’s belief at time 0 is m ~ N(1, 1).

Lines are darker when they overlap.


variances but their ratio, which indicates the strengthof new information relative to prior beliefs. And wecan assess the plausibility of different ratios. Forexample, if s2

x 5 1 and s2D0 5 1, the 1:1 ratio implies

that each new message carries as much weight as theDemocrat’s entire lifetime of prior experience andthat six new messages collectively carry as much weightas six lifetimes of prior experience. If campaignmessages were as strong as such ratios imply, we wouldhave reason to expect colossal campaign effects, whichmanifestly do not occur (Bartels 1993; Shaw 1999).Higher ratios are thus much more plausible. Forexample, if s2

x 5 20 and s2D0 5 1, the 20:1 ratio

implies that each message is weighted 5% as heavilyas the Democrat’s lifetime of prior experience andthat six messages are collectively weighted 30% asheavily. These messages are still influential, but theydo not demolish prior beliefs as they would if s2

x 5 1.In the absence of strong evidence, the reader maydisagree about the relative plausibility of differentratios, which is why Figure 2 reports results for arange of message variances and prior belief variances.

Consider Panel 1. In this panel, the Democrat’sprior belief is twice as strong as the Republican’s(s2

D0 5 .5 versus s2R0 5 1). Each of the six messages

received by these partisans is drawn from a distribu-tion with variance s2

x 5 1, implying that the mes-sages collectively carry six times as much informationas the Republican has previously received throughouthis life, and three times as much information as theDemocrat has previously received. When new infor-mation is so much stronger than both updaters’ priorbeliefs, we should see rapid convergence. And we do:the 30 simulation lines trend sharply to 0; and at time6, they are all below 1, indicating that the partisansdisagree less at time 6 than at time 0 in all 30simulations.

The same trend toward agreement appears in allother panels. This is unsurprising: even in Panel 17,where new information is weakest relative to up-daters’ priors (s2

D0 5 .5, s2R0 5 1, s2

x 5 20), thecumulative weight of the information is 15% as greatas the weight of the Democrat’s lifetime of priorexperience, and 30% as great as the weight of theRepublican’s prior experience. Moreover, all newmessages are drawn from a distribution with m 5 2,and 2 is within the range of the updaters’ prior beliefmeans. In expectation, then, new messages will notfall outside the range of prior belief means, and anecessary condition for divergence (set forth in theCorollary to Proposition 2) will not be satisfied.

And yet, divergence from one time to the nextoccurs in 12 of the 16 panels, as indicated by the

upward-sloping line segments within each panel. (Itis absent only from the panels of the second column,where it cannot occur because D and R have equallystrong prior beliefs.) It is rarest when new informa-tion overwhelms updaters’ priors; for example, it oc-curs in expectation only 10% of the time in Panel 1.But it grows more common as information growsweaker and as the difference between the strengths ofupdaters’ priors increases. In Panel 20, where infor-mation is weakest and the gap between updaters’prior belief strengths is greatest, divergence occurs inexpectation 40% of the time that people update theirbeliefs. In all of these panels, save those in the secondcolumn, a general principle is at work: gradual con-vergence to agreement does not preclude increaseddisagreement in response to specific new messages.Indeed, increased disagreement from one time to thenext can be quite common, as it is in the panels ofthe rightmost column. And disagreement is increas-ing in Figure 2 even under conditions that sharplyfavor convergence. If m were outside the range ofupdaters’ prior belief means, if the updaters’ priorbelief strengths were less similar, or if new informa-tion were less overpowering, divergence would besharper and even more frequent.

The distinctive panels of the second column de-serve explanation. D and R have equally strong priorbeliefs in these panels, which ensures that the gapbetween their beliefs is reduced by the same amountwhenever they learn from new messages, regardless ofthe values of those messages. Within each panel of thecolumn, updating trends are therefore identical for all30 simulations. This accounts for the panels’ appear-ance: although it appears that each panel contains butone line, there are really 30 lines in each panel, eachline tracing the same path.

Disagreement When Learningabout a Changing Condition

Almost all Bayesian analyses of public opinionassume that people are trying to learn about politicalconditions that never change. The assumption is aptwhen we are trying to learn about the past (did theNew Deal prolong the Great Depression?) and per-haps when we update our beliefs over the short term(which party is better for me right now?), but it isinappropriate when the feature of politics that wewant to learn about is a moving target. The state ofthe economy changes because of demographic trendsand unforeseen events. My preferences over policieschange as new proposals are placed on the table or


taken off of it. And candidates may improve duringtheir time in office or fall under the sway of con-stituents whose views I oppose. In these cases, theunchanging-condition assumption is a poor approx-imation of reality. The assumption is embodied in themodel described in the previous section, which is thebasis for almost all Bayesian analyses of publicopinion (e.g., Achen 1992; Bartels 1993, 2002; Gerberand Green 1999; Lohmann 1994; Steenbergen 2002;Zechman 1979). But it is easy to generalize the modelso that it accommodates political change.

Suppose that a politically interesting conditionchanges according to the rule

mt 5 gmt�1þ em;t ; em;t ; Nð0; s2mÞ ð2Þ

where t is the current time and g $ 0. In thisequation, g is an autoregressive parameter thatdictates systematic change in the condition of inter-est: when g . 1, the absolute expected value of m

increases with t; when g , 1, the absolute expectedvalue of m decreases with t. For tractability, assumethat the people whose beliefs we are studying knowthe true value of g.6 em,t is a disturbance with knownand finite variance s2

m; it determines random varia-tion in m over time. When g 5 1 and s2

m 5 0, m isunchanging, just as in the model from the previoussection.

Let x1; . . . ; xn be messages containing informa-tion about m at times 1; . . . ; n. At any timet 2 1; . . . ; n, the relationship between a new messageand m is xt ~ N(mt, s2

x), where s2x is a known and

finite variance.7 The voter’s belief after updatingin response to message xt–1 is mt�1 jg;s2

m;s2x;

xt�1;Nðm̂t�1;s2t�1Þ; where xt–1 is the vector of

messages x1; . . . ; xt�1. It can be viewed either as aposterior belief (because it is the voter’s belief afterreceiving message xt–1) or as a prior belief (because itis the voter’s belief before receiving message xt).

When a political condition is changing over time,Bayesian belief revision is a two-step process. In thefirst step, the voter forecasts the value that thecondition, m, will have at time t. She does this bymultiplying the mean of her prior belief, m̂t�1, by the

rate of change in m, which is g. Her forecast is thusgm̂t�1. In the second step, the voter receives messagext and adjusts her forecast to account for it:

mt jg;s2m;s

2x; xt�1; xt;Nðm̂t ;s

2t Þ;where

m̂t 5 gm̂t�1 þ Kt xt � gm̂t�1ð Þ; ð3aÞ

s2t 5

1

1=ðg2s2t�1 þ s2

mÞ þ 1=s2x

; ð3bÞ

and Kt 5 s2t /s2

x. This second step is an ‘‘errorcorrection’’ in which the voter modifies her forecastto account for the distance between it and the newinformation that she had received ðxt � gm̂t�1Þ. Itseffect is always to draw beliefs toward the newinformation. The weight that it assumes in theupdating process is Kt , which is standardized to liebetween 0 and 1. The stronger the new information(i.e., the lower the value of s2

x), the higher the valueof Kt , and the more that beliefs will be drawn towardthe new information.

Equations (3a) and (3b) are known as the Kalmanfilter algorithm after Rudolf Kalman (1960), whoshows that they yield the expected value of mt underthe assumption of normal errors.8 For our purpose,the Kalman filter is useful because it helps to establishresults analogous to those in the previous section, butunder the assumption that people are learning aboutchanging features of politics. Again, let D and R beunbiased Bayesians whose beliefs at time t 5 0 arem0;N m̂D0;s

2D0

� �and m0;N m̂R0;s

2R0

� �, respectively.

For all finite t, assume 0 , s2Dt, s2

Rt : neither D nor Ris absolutely certain of his belief. For all times t,assume that mt is determined by Equation 2 and thatxt ~ N(mt, s2

x). Under these assumptions, the initialresults resemble those from the previous section:

Proposition 4 (convergence to agreement). If D andR receive a sequence of messages x1; . . . ; xn, their beliefswill converge to agreement as n! ‘.

Proposition 5 (divergence). Divergence occurs betweentime t and later time u if and only if KDuðxu

� gm̂DtÞ � KRuðxu � gm̂RtÞ =2 minfa; bg; maxfa; bg½ �,where a 5 ðg � 1Þðm̂Rt � m̂DtÞ and b 5 ðg þ 1Þðm̂Rt � m̂DtÞ.As with the model from the previous section, receiv-ing infinite information will cause partisans to agree,

6The same assumption is made in other analyses of learningabout changing conditions, notably Gerber and Green (1998).Divergence and polarization are less likely under this assumption.The formal results in this section are thus especially noteworthy:they show that divergence and polarization can occur even underan unfavorable assumption.

7As in Gerber and Green (1998), g, s2m, and s2

x are held constantfor simplicity of exposition. But the model described in thissection can accommodate the case in which they change overtime. See Meinhold and Singpurwalla (1983) for an example.

8For an overview of the Kalman filter, see Beck (1990). Meinholdand Singpurwalla (1983) provide a lucid introduction from aBayesian point of view.


but receiving finite amounts of information mayincrease disagreement even if both partisans interpretthe information in the same way. And as with themodel from the previous section, the condition fordivergence is somewhat complicated, but its explan-ation is straightforward. Recall that when learningabout a changing condition, the transition from priorto posterior beliefs takes place in two steps: first, oneforecasts the condition’s future value, mu ; second,one receives new information and adjusts the forecastto account for it. When g . 1, the forecastingprocess necessarily pushes beliefs apart, becausegjm̂Dt � m̂Rtj. jm̂Dt � m̂Rtj. In this case, Proposition5 says that divergence occurs when the adjustmentsfor new information also push beliefs apart, or whenthey pull beliefs together by less than the forecastingprocess pushes them apart. On the other hand, ifg , 1, the forecasting process pulls beliefs together,because gjm̂Dt � m̂Rtj, jm̂Dt � m̂Rtj. In this case,Proposition 5 says that divergence occurs when theadjustments for new information push beliefs apartmore than the forecasting process pulls them together.

To see how divergence about a changing con-dition might occur in practice, consider AdamBerinsky’s recent analysis of Congressional supportfor U.S. intervention in World War II (Berinsky 2007,987–88). In June 1940, Republican and DemocraticCongressmen sharply opposed U.S. intervention, asindicated by the antiwar bent of their statements inthe Congressional Record. But they were not equallyopposed: approximately 25% of Democratic state-ments about the war expressed support for war,against only 12.5% of Republican statements. In thenext year and a half, prowar sentiment increasedamong Congressmen of both parties. But it increasedmore quickly among Democrats, and the partisan gapin support for war therefore quadrupled: by the firstweek of December 1941, about 90% of Democraticstatements about the war were prowar, against only40% of Republican statements (Berinsky 2007, 988).

There is no sure way to draw conclusions aboutlegislators’ beliefs from their statements in the Record,but those statements suggest diverging beliefs aboutthe benefits of war, and in a way that illustrates thelogic of Proposition 5. Let t be June 1, 1940 andt þ 1; . . . ; t þ 18 be the first days of the next 18months. D and R are Congressmen trying to learnabout mt , the net benefit to Americans of a declara-tion of war. Like a politician’s honesty, this netbenefit has no natural scale; for convenience, supposethat m̂Dt 5 1:2 and m̂Rt 5 1:0. These numbers suggestthat D perceives a greater benefit of going to war thanR, just as the Record suggests that Democrats were

more supportive of war than Republicans in June1940. Let mt 5 2.5, indicating that both Congress-men underestimate the true benefit of a declarationof war. Moreover, s2

Dt 5 .5 and s2Rt 5 .2, indicating

that R’s belief is more than twice as strong as D’s.There is little random variation in the true benefit ofa declaration over time: s2

m 5 .01, suggesting that thebenefit of going to war is not, at this early stage, muchdependent on chance contingencies. But g 5 1.05,indicating that the benefit is increasing systematicallyas fascist successes mount in France, China, EastAfrica, and the Soviet Union. The news received inany particular month is relatively weak: s2

x 5 10,implying that each new month’s information mattersonly 2% as much as R’s prior learning about themerits of going to war, and 4% as much as D’s priorlearning on the same subject. But over 18 months,s2

x 5 10 implies substantial learning. The cumulativeweight of the news received in these months is 36% asgreat as the weight attached to R’s lifetime of priorexperience, and fully 90% as great as the weightattached to D’s lifetime of prior experience.

This is all that is needed to produce beliefdivergence. In June 1940, D and R will forecast thebenefit that a declaration of war will have on July 1:gm̂Dt 5 1:26, and gm̂Rt 5 1:05. The distance betweenthese forecasts (1.26 – 1.05 5 .21) will exceed thedistance between the Congressmen’s prior beliefs(1.2 – 1.0 5 .2); thus, the forecasting process willpush beliefs apart. Now suppose that both Congress-men receive new information about the benefit of adeclaration of war on July 1, xt+1 5 2.625, indicatingthat the true benefit is greater than they anticipated.(In expectation, this is the new information that theywill receive, because xt+1 is drawn from a normaldistribution that is centered in expectation at gmt 5

2.625.) They will adjust their forecasts to account forthis new information. For R, KR,t + 1 is approximately.02, reflecting the low weight that he attaches to thenew information. By equation (3a), his posteriorbelief will therefore be m̂R;tþ1 � ð1� :02Þð1:05Þð1:0Þ þ :02ð2:625Þ � 1:08. For D, KD,t + 1 isapproximately .05, reflecting his weaker prior beliefand correspondingly greater responsiveness to newinformation. His posterior belief will therefore bem̂D;tþ1�ð1�:05Þð1:05Þð1:2Þþ :05ð2:625Þ�1:33. TheCongressmen’s adjustments for new information willtherefore push their beliefs apart, just as the fore-casting process did. The combined effect of theforecasting and the adjustment for new informationwill be divergence: on June 1, the difference betweenthe Congressmen’s beliefs was 1.2 – 1.0 5 .2; on July 1,it will be 1.33 – 1.08 5 .25. And this divergence will


occur even though both Congressmen receive thesame information, interpret it in the same way, andfollow the same updating rule.

In expectation, D and R will continue to divergethrough December 1, 1941. The importance of adeclaration of war will grow as it appears more likelythat the Allies will lose without American interven-tion, and this will be reflected by change in m, whichwill grow to 6.0. A week before the attack on PearlHarbor, D’s posterior belief will be m̂D;tþ18 5 5:2, andR’s will be m̂R;tþ18 5 4:5. That is, they willboth perceive the benefits of war as greater than theydid in June 1940—but even so, the difference betweentheir beliefs will almost quadruple over the 18 months(from .2 to 5.2 – 4.5 5 .7), much as Berinsky observesa quadrupling of the gap between the proportions ofprowar statements made by Democratic and Repub-lican Congressmen over the same time span.

Partisan divergence in Congressional support forAmerican intervention ended with Pearl Harbor. Informal terms, the attack on Pearl Harbor can be seen asa new message with extremely low variance (s2

x � 0).The effect of such a message is to cause people todiscount all of the information that they had pre-viously received: in effect, posterior beliefs are deter-mined almost exclusively by the new message. In thecontext of deliberations about war, s2

x � 0 impliesthat when legislators heard the news about PearlHarbor, their beliefs about the net benefit of adeclaration of war were determined almost exclu-sively by that news, and not by the different beliefsthat they held before they heard the news. In thecontext of Bayesian disagreement, Pearl Harbor isinstructive because it shows how unusually clear newinformation can produce convergence to agreementeven among people whose beliefs had been sharplydiverging.

In this example, D and R start with beliefs thatare not equally strong, and they learn from newinformation that is ‘‘extreme’’ in the sense that it isoutside the range of their prior belief means. TheCorollary to Proposition 2 shows that these are nec-essary conditions for divergence if people are learn-ing about something that is not changing over time.But if people are learning about something that ischanging, the Corollary to Proposition 5 shows thatdivergence can occur even when they have equallystrong prior beliefs and respond to ‘‘moderate’’information:

Corollary to Proposition 5. Divergence can occurbetween t and u if s2

Dt 5 s2Rt or xu 2min m̂Dt; m̂Rtf g;½

max m̂Dt; m̂Rtf g�.

This follows almost immediately from Proposition 5.If D and R have equally strong prior beliefs(s2

Dt 5 s2Rt), they attach the same weight to the

new information that they receive (KDu 5 KRu 5 Ku).In this case, the condition for divergence boils downto (1 – Ku)g . 1. This condition can be met whetheror not new information is ‘‘extreme.’’ The Corollaryto Proposition 5 is thus a retort to the Corollary toProposition 2.

Another striking difference between learning aboutfixed and changing conditions concerns polarization:

Proposition 6 (polarization). Polarization occursbetween t and u if and only if divergence occurs betweent and u and either (a) KDuðxu � gm̂DtÞ. ð1� gÞm̂Dt

and KRuðxu � gm̂RtÞ, ð1� gÞm̂Rt or (b) KDu

ðxu � gm̂DtÞ, ð1� gÞm̂Dt and KRuðxu � gm̂RtÞ.ð1� gÞm̂Rt.

When unbiased Bayesians with normally distributedprior beliefs learn about fixed conditions fromnormally distributed messages, polarization cannotoccur. But when they learn about changing conditionsfrom normally distributed messages, polarization canoccur. To see why polarization is only possible forsuch people when conditions are changing, recall thatlearning about changing conditions takes place in twosteps, and that in the second step, people always shifttheir beliefs toward the new information that theyreceive. If this is all that happens—if beliefs do notalso change in the first step—polarization cannotoccur. And when people learn about an unchangingcondition, the second step is all that happens. Thefirst step, in which people forecast the new value of m

by accounting for its rate of change, does not occurbecause m is not changing.

Similar reasoning shows why polarization amongunbiased Bayesians will be relatively rare. Unlikedivergence, polarization requires new informationto contradict expectations: at least one person mustforecast an increase in m while information indicatesthat it has decreased or stayed the same, or forecast adecrease in m while information indicates that it hasincreased or stayed the same. For example, imaginethat M is Martin Luther King, Jr., and that H is HarrisWofford, a member of the Kennedy administrationand the main liaison between Kennedy and civilrights groups. Both men are trying to learn aboutKennedy’s support for new civil rights legislation inthe summer of 1963, just after the brutal suppressionof protesters in Birmingham, Alabama. They share animpression that Kennedy supported new legislationin June (m̂Mt; m̂Ht . 0) and that his support is increas-ing over time (g . 1). Before receiving information


about his level of support later in the summer, theyforecast this level. Then they receive new informationabout it (xu). By equation (3a), they adjust theirforecasts by moving in the direction of this informa-tion. We know that both men forecast increasedsupport from Kennedy, so if the new informationalso indicates increased support (xu . m̂Mt; m̂Ht), bothwill conclude that Kennedy’s support for new legis-lation has increased (m̂Mu . m̂Mt and m̂Hu . m̂Ht).Because their beliefs will move in the same direction,polarization will not occur.

In this example, polarization will occur onlyif the new information suggests that both Kingand Wofford were wrong about the direction ofKennedy’s support for civil rights, i.e., only if xu ,

m̂Mt; m̂Ht. And this is only a necessary condition, nota sufficient one. If the new, negative information istoo weak, King and Wofford will ignore it; both willconclude that Kennedy’s support for civil rightsincreased, and polarization will not occur. On theother hand, if the information is too strong, it willoverwhelm the forecasts of both King and Wofford;both will conclude that Kennedy’s support declined,and polarization will not occur. To polarize King andWofford, the new information must be negative andstrong enough to overwhelm one man’s optimisticforecast, but not so strong that it overwhelms bothmen’s forecasts. Formally, it must fit within a range ofvalues that is determined by the weight that each manplaces on it (KMu and KHu), the mean of each man’sprior belief about Kennedy’s position (m̂Mt and m̂Ht),and the rate at which Kennedy’s position is changing(g). This is the range that is specified in Proposition 6.

Something like this may explain what happenedin 1963. In late June of that year, King and Woffordbelieved that Kennedy’s support for civil rights wasincreasing, and they therefore expected strongersupport from Kennedy in the near future (Garrow1986, 269; Wofford 1980, Chap. 5). Both men subse-quently noticed Kennedy’s apparent wavering, whichincluded pressuring liberal Congressmen not todemand stronger legislation and an ‘‘empty show offederal response’’ to the Birmingham church bomb-ing (Branch 1998, 250; see also Garrow 1986, 296, 302;Wofford 1980, 172–74). But Wofford, who hadstronger prior beliefs about Kennedy, was largely un-moved; late in 1963, he thought Kennedy morecommitted than ever to civil rights. King, who hadlong been unsure of Kennedy’s true level of supportfor civil rights, was more affected by July’s events, andhe revised downward his view of Kennedy (Branch1998, 246–50). The result was that King and Woffordpolarized over Kennedy’s commitment to civil rights.

Figure 3 illustrates possible patterns of belief re-vision among unbiased Bayesians who are learningabout a changing condition. Each panel depicts 30simulations in which D and R revise their beliefs sixtimes in response to new information. Different linesin each panel represent different simulations: whenthe lines slope up, disagreement is increasing; whenthey slope down, it is decreasing. The strength of themessages received varies from row to row: they arestrongest in the top row (s2

x 5 1) and weakest in thebottom row (s2

x 5 20). In all panels, s2m 5 0, mean-

ing that m, the feature of politics that D and R arelearning about, does not vary stochastically overtime.9 In these respects, Figure 3 is like Figure 2.But in Figure 3, m varies systematically from one timeto the next, because g . 1 for every panel in thefigure—and this is an important difference. D and Rare now learning about a feature of politics that ischanging over time. In every panel, m starts at 2(m0 5 2) and increases as the simulation progresses;the rate of increase is determined by g, which variesfrom column to column. As in Figure 2, the priorbelief of R is m0 ~ N(1, 1) in every panel, but the priorbelief of D is now fixed at m0 ~ N(2, 1.25).10

Divergence occurs throughout Figure 3. It is rarein the top row, where information is far stronger thanprior beliefs and where disagreement therefore tendsto diminish rapidly. But in successive rows, the trendtoward agreement is less pronounced, and in thefourth and fifth rows, disagreement increases betweentime 0 and time 6 in almost every simulation.It increases even though the cumulative weight ofthe information that D and R receive is still strong. Inthe fourth row, where the variance of each message iss2

x 5 15, the six messages are collectively weighted50% as heavily as the Democrat’s prior experienceand 40% as heavily as the Republican’s. In the fifthrow, where s2

x 5 20, the six messages are collectively

9s2m is held constant to permit the effects of changes in g to

emerge clearly through comparison of the panels within Figure 3,and it is set to 0 to increase the comparability of Figure 3 toFigure 2. By equation (3a), increasing s2

m increases the weight thatD and R place on the new messages that they receive (KDt andKRt), thereby making their beliefs converge more rapidly. Thiseffect can be offset by increasing s2

x, which reduces the weightthat D and R place on the new messages.

10The Democrat’s prior belief is fixed to permit the figure to showthe effect of changes in g independent of changes in prior beliefs.I investigated belief updating patterns under a range of differentvalues for D’s prior belief. Small changes (e.g., fixing D’s prior atm0 ~ N(2, 1.5)) made only small differences to the results; largechanges (e.g., m0 ~ N(2, 10)) made larger differences, typicallyincreasing the frequency of divergence and polarization. Even inthese cases, the patterns observed across panels remained thesame.


FIGURE 3 Bayesian Learning about a Changing Condition Can Increase Disagreement BetweenPartisans. Each panel depicts 30 simulations of learning by two people who revise theirbeliefs according to Equations 3a and 3b. In each simulation, they receive a new message at eachof six times. A single line is plotted for each simulation, showing how the absolute differencebetween the means of their beliefs changes as they learn from the new messages. In the upperrows, where the cumulative effect of new information far outweighs the effect of prior beliefs, thelines trend toward 0, indicating rapid convergence to agreement. In the lower rows, whereinformation is weaker, upward-sloping line segments are common, indicating frequent beliefdivergence.

In each panel, m0 5 2, s2m 5 0; D’s belief at time 0 is m0 ; N(2, 1.25), and R’s belief at time 0 is

m0 ; N(1, 1). Lines are darker when they overlap.


weighted about 38% as heavily as the Democrat’sprior experience and 30% as heavily as theRepublican’s.

Figure 3 does not indicate when polarizationoccurs, but it would not look much different if itdid, because polarization is uncommon in the sce-narios that it depicts: it occurs twice in Panel 16, oncein Panel 19, and once in Panel 20. To see why it doesnot occur more often, consider the properties thatmessages must have to polarize D and R. Becauseg . 1, both D and R always forecast an increase in m

from one time to the next. Polarization requires thatnew information contradict these forecasts: at time 1,for example, new information must suggest that m isdecreasing. In addition, the new information must beextreme enough to overwhelm one updater’s fore-cast—e.g., extreme enough to cause m̂R1 , m̂R0—butnot so extreme that it overwhelms both updaters’forecasts, thereby causing both m̂R1 , m̂R0 andm̂D1 , m̂D0. The specific range of message values thatwill produce polarization varies from time to timeand panel to panel; as an example, consider polar-ization between time 0 and time 1 in Panel 16.Application of Proposition 6 shows that it will occurif and only if – 1.33 , x1 , – 1.08. This is a narrowrange; x1 will fall within it about six times out of athousand. Polarization under this model requiresnew messages to thread a needle, which is why itwill not happen often.

How Common Are the Conditionsfor Bayesian Agreement?

The previous sections have shown, contrary toprevious claims, that unbiased Bayesian learningcan sustain and even exacerbate disagreement. Theyhave also shown, in keeping with previous claims,that unbiased Bayesian learning can produce agree-ment between partisans even when their initialdisagreement is great. Indeed, Propositions 1 and 4show that this will happen whenever partisans receivepolitical messages so numerous and so credible thattheir prior beliefs are overwhelmed. But most parti-sans never receive so much information of such highquality about any political question. Bayesian updat-ing therefore offers no expectation of convergence toagreement: the lasting differences that we observebetween real-world partisans are just what we wouldobserve if those partisans were unbiased Bayesians.

To see that partisans are unlikely to acquireenough information to bring them to agreementeven if they update as unbiased Bayesians, consider

the wealth of information that we possess abouttheir dramatic lack of political information, nicelysummarized by Delli Carpini and Keeter (1996).Although partisans know more about politics thanpeople with no partisan leanings, strength of partyidentification bears only a weak connection to polit-ical information levels (Delli Carpini and Keeter1996, 144–49), and widespread political ignoranceamong partisans suggests that they, like nonpartisans,are not in the habit of acquiring much informationabout even those political questions that matter mostto them. Widespread inattention to politics is alsoconsistent with Achen and Bartels (2006), who arguethat events induce political thought for most peopleonly ‘‘once or twice in a lifetime,’’ and with Shaw(1999), who finds that few presidential campaignevents affect support for the candidates.

Of course, it does not take much information tooverwhelm weak prior beliefs. But partisans’ priorbeliefs—at least about political candidates, and thusabout choices in most elections—are relativelystrong, and much stronger than the cumulativeweight of media messages received over the courseof a campaign (Bartels 1993). When beliefs are thissubstantial, it is difficult to overwhelm them withnew information.

In any ordinary human span of time, then, thereis no reason to expect that partisans who disagree onpolitical matters will acquire information sufficient tomake them agree with each other. And this is so evenif they are exposed to the same information, interpretit in the same way, and revise their beliefs accordingto Bayes’ Theorem. Grynaviski’s (2006, 331) charac-terization—that mortal Bayesian learners exposed tothe same information will ‘‘inexorably come to seethe world in the same way’’—is too strong.

Conclusion

Writing about political belief revision, Bartels (2002,123) tells us that ‘‘it is failure to converge thatrequires explanation within the Bayesian framework’’(2002, 123). The chief contribution of this article hasbeen just such an explanation—an explanation thatdoes not hinge, as previous explanations have, onpartisan bias or selective exposure to congenialsources of information. Even when partisans receivethe same information and interpret it in the sameway, Bayesian updating will lead them to agreementonly if the information is of extraordinary quantity orquality. Contrary to Goodin (2002), enduring dis-agreement among Bayesians is no ‘‘paradox’’: it is the


normal state of affairs. This result is consistent withthe argument in Gerber and Green (1999) but con-trary to much else that has been written on the sub-ject, including Goodin (2002), Bartels (2002, 121–23),and Grynaviski (2006, 331).

Of course, partisans who never completely agreemay yet disagree less as they learn more about thesubject of their disagreement. But this article showsthat Bayesian learning cannot always justify eventhe weaker expectation of diminished disagreement.Indeed, it may increase disagreement instead of dimin-ishing it. Disagreement can increase through diver-gence, whereby all beliefs move in the same direction,but at different rates. Or it can increase throughpolarization, whereby different partisans’ beliefs movein different directions. Contrary to Tetlock (2005, 128),Bartels (2002, 121–23), and others, both divergence andpolarization are consistent with unbiased Bayesianlearning.

This may seem to be good news. UnbiasedBayesian updating is increasingly put forth as idealinformation processing, and the results here suggestthat partisans are closer to that ideal than we tend tosuppose, because they show that the ideal accom-modates real patterns of public opinion changeamong partisans. But there is another way to viewthe results. Rather than casting citizens’ thinkingabout politics in a rosy light, they may suggest thatBayesian updating is inadequate as a normativestandard by which to judge reactions to politicalinformation. Because it permits lasting disagreement,divergence, and polarization even among people whoreceive the same information and interpret it in thesame way, it may be insufficient for rationality in thesense of ‘‘plain reasonableness’’ (Bartels 2002, 125–26;Fischoff and Lichtenstein 1978).

Readers must judge for themselves whether‘‘plain reasonableness’’ precludes lasting disagree-ment, divergence, and polarization. What this articledemonstrates is that those patterns of public opiniondo not suffice to show that people are biased or tohelp us understand the extent of their biases. Whatwill suffice is better information about partisans’beliefs and perceptions. For more than a decade,some political scientists have clamored for moreresearch on the strength and shape—rather than justthe ‘‘location’’—of partisans’ beliefs about policiesand candidates and their perceptions of new infor-mation (Alvarez and Franklin 1994; Bartels 1986;Gerber and Green 1999; Gill and Walker 2005;Steenbergen 2002). For the study of bias in politicalinformation processing, this is exactly the rightstance. To determine how well partisan belief revision

matches up against a normative baseline, we need toknow not just the location of partisans’ beliefs but thedegree of confidence with which those beliefs areheld. Coupling measures of belief location withmeasures of belief strength will do much to help usunderstand the nature and extent of political biases,and thus to help us understand just how far partisansfall from a normative ideal.

We study Bayesian updating not because wethink it is what partisans do but because we thinkit is what they should do: in a large and increasingnumber of public opinion analyses, it is the ideal towhich real belief updating is compared. That said, thefindings reported here suggest that the gap betweenreal and Bayesian political belief revision is not aswide as many have suggested. Real partisans do notalways disagree less as they learn more. Sometimestheir disagreement is undiminished by exposure tonew information; sometimes, it is even increased.And all of these aspects of real-world opinion changeare consistent with the Bayesian ideal.

Acknowledgments

I thank Neal Beck, Jonathan Bendor, Adam Berinsky,Nathan Collins, Mo Fiorina, Alan Gerber, SarahHammond, Simon Jackman, Robert Luskin, AndrewRutten, Paul Sniderman, and Jonathan Wand forhelpful discussions. Larry Bartels, Will Bullock, FredCutler, Don Green, Stephen Jessee, Christopher Kam,Matt Levendusky, Nora Ng, Ben Nyblade, AngelaO’Mahony, Paul Quirk, and three anonymous re-viewers offered valuable comments on earlier drafts.

I wrote most of this article while I was a KillamPostdoctoral Fellow in the Department of PoliticalScience at the University of British Columbia. I thankthe Trustees of the Killam Trusts for their support.

Proofs of all formal propositions in this article aregiven at http://bullock.research.yale.edu/disagreement.All figures, simulations, and analyses can be repro-duced with files available at the same URL.

Manuscript submitted 13 March 2008Manuscript accepted for publication 5 October 2008

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John G. Bullock is assistant professor of politicalscience, Yale University, New Haven, CT 06520.


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