DIVISIBLE UPDATING MARTIN W. CRIPPS

Transcription

1 DIVISIBLE UPDATING MARTIN W CRIPPS Abtract A characteriation i provided of the belief updating procee that are independent of how an individual chooe to divide up/partition the tatitical information they ue in their updating Thee diviible updating procee are in general not Bayeian, but can be interpreted a a re-parameteriation of Bayeian updating Thi cla of rule incorporate over- and under-reaction to new information in the updating and other biae We alo how that a martingale property i, then, ufficient the updating proce to be Bayeian Belief Updating In thi paper we conider arbitrary procee for updating belief in the light of tatitical evidence We treat thee updating procee a determinitic map from the prior belief and a ignal tructure tatitical experiment to a profile of updated belief one for each poible ignal We place axiom on thee updating procee and how that a generaliation of Bayeian updating can be derived a a conequence of thee axiom Conider an individual who receive two ignal/piece of information/new There are everal way uch an individual can ue thee two piece of information to update their belief about the world One i to conider the joint ditribution of thee piece of information and to do jut one update An alternative which i natural when information arrive equentially or over time i to eparate the two piece of information and to update belief twice That i, to update belief once uing the firt piece of information and it ditribution And then to update thee intermediate belief a econd time uing the econd piece of information and it conditional ditribution given the firt piece of information If thee two different procedure generate the ame ultimate profile of updated belief we will ay that the updating i diviible We will characterie all the updating procee that have thi diviibility property and how that they can be interpreted a natural generaliation of Bayeian updating Furthermore we will how that diviibility plu an unbiaedne/martingale property for the updating uniquely characterie Bayeian updating The main reult of thi paper how that if updating atifie four propertie, then it i characteried by a map F from the pace of belief into itelf The four propertie or axiom are: Firt, that uninformative experiment do not reult in change in belief Date: March 5, 208 Department of Economic, Univerity College London; mcripp@uclacuk My thank are due to Ailinn Bohren, Toru Kitagawa, and Ran Speigler for their uggetion and comment

2 2 MARTIN W CRIPPS Second, that the name of the ignal do not matter jut their probability content The third enure that all updated belief are poible The fourth i the diviibility property decribed above Any updating procedure, that atifie thee four propertie, follow the tep that are illutrated in the figure below The initial belief are mapped to a Shadow Prior uing a bijection F Then, thee hadow prior are updated in the tandard Bayeian fahion uing the tatitical information that i oberved to create a Shadow Poterior Finally, the hadow poterior i mapped back to the pace of original belief uing the invere map F Belief F Shadow Prior Updated Belief F Baye Updating Shadow Poterior Figure Updating that Satifie Diviibility Our reult i that if the belief reviion protocol atifie the diviibility property, then it mut have thi tructure Where F i a homeomorphim from the pace of belief into itelf It i clear that Bayeian updating i a ubet of thi cla, F i the identity Specific propertie of F enure different propertie of the belief reviion Biae in the updating will be generated by the convexity or concavity of a function of the map F For overreaction and under-reaction to ignal we will argue that F need to be an expanive or a contraction mapping In region where F i a contraction the belief reviion will exhibit over-reaction in general and in region where F i a contraction the belief reviion will exhibit underreaction Many of the ueful propertie of Bayeian updating alo carry over to thi cla of reviion protocol For example, although the actual belief are not a martingale the hadow belief are Thu, in dynamic etting, updating that atifie diviibility will be a determinitic function of a martingale A a reult, conitency will hold for thi larger cla of updating procee, provided F map the extreme point of the belief implex to themelve That i, when thee updating procee are repeatedly applied to a equence of data, they will generate limiting belief that attach probability one to the truth Thi property of diviible updating contrat with other model of non-bayeian updating, for example Rabin and Schrag 999 and Eptein, Noor, and Sandroni 200 A quetion we alo addre i what additional axiom are neceary to characterie Bayeian updating We how in the final ection of thi paper that only one further property i ufficient for thi, that i, belief reviion i unbiaed or follow a martingale Thu any relaxation of Bayeian updating ha to either violate the martingale property, the diviibility property or one of the other axiom we impoe Why do we focu on the diviibility of updating rather than ome other property it may have? There are everal motivation for looking at diviibility Firt, the information individual oberve i typically not a unitary piece of information but a bundle of ignal What diviibility enure i that the exact way the updating i performed uing thi There are everal name ued for thi notion in different context: unbiaed, Baye plauibility have alo been ued

3 DIVISIBLE UPDATING 3 bundle of ignal i unimportant in determining the final belief And, if the individual eparate out variou feature of the bundle and ue thee individually to update their belief, then the order that they do thi ha no ultimate effect their eventual belief If updating doe not atify the diviibility property, then the updated belief will depend on how and in what order the individual ha applied the updating ee Section below for an example of thi Moreover, if diviibility i not atified, the individual ha multiple poible updated belief and an additional aumption i required to pin them down To predict the individual updated belief it will then alo be neceary to have a theory of how he choe to apply the updating rule to package of ignal Furthermore, he may alo benefit by recalling more of the hitory than her current tate of belief if the order of the updating matter In dynamic etting thi iue i mot clear If the updating atifie diviibility, then it i only neceary for the individual to keep track of their current belief and update them any time new information appear Her current tate of belief are ufficient to ummarie all pat ignal in a parimoniou way nothing ele about the pat hitory mut be remembered There i a dynamic conitency in the updating o the individual who are not required to act can imply collect information and ue it to update belief when it i neceary If the updating i not diviible, then how often the updating rule ha been applied in the pat may affect her current belief and thu may be omething he need to keep track of In ummary, if the belief updating i not diviible, then a theory of how and when updating occur i required Such a theory need to addre the trade-off between the memory cot of toring accumulated ignal and the proceing cot of when and how to update An Example of Non-Diviible Updating In thi ection we conider the model of non-bayeian learning decribed Eptein, Noor, and Sandroni 200 or conervative Bayeian updating of Hagmann and Loewentein 207 and apply it to an example 2 In thee model of learning, the updated belief are a linear combination of the prior and the poterior We will how, uing an example, that thi model of updating doe not atify the diviibility property and eek to explain why the updated belief vary for different protocol Conider an individual who i waiting for a bu and i learning about the arrival proce of bue There are two tate for the arrival proce: In the good tate a bu will arrive in period t = 0,, with probability αα t α 0, while in the bad tate a bu arrive in period t with probability ββ t where α < β She ha initial belief µ 0, that the tate i good and a bu will arrive at ome point If no bu arrive in period t = 0, then a Bayeian would revie thee belief downward in the light of thi bad new to αµ µβ+µα However, in thi model of non-bayeian learning 2 Both of thee paper ue thi model of updating a a part of more general tudy of dynamic choice when there i learning or information deign We do not addre choice iue here, focuing olely on belief reviion

4 4 MARTIN W CRIPPS he revie her belief to a weighted average of the prior and Bayeian poterior: αµ µ = λµ + λ µβ + µα, λ 0 Thi i a particularly elegant model a it preerve the martingale property of the Bayeian poterior The value λ i a parameter of the updating procedure, which can in general depend on the current hitory, but in thi example it i treated a a fixed property of the updating Thi model of updating generalie Bayeian updating λ =, where λ adjut the effect the bad new of no arrival ha the belief Chooing λ < allow the individual to be under-confident or conervative in their updating of their belief Converely, λ > allow the individual to be overconfident about the new information they have received Thu, λ allow a range of update from extreme dogmatim λ = 0 or jumping to certainty λ µ α Now conider what the individual belief could be after t period waiting for a bu without an arrival One poible application of thi updating procedure i to iterate the updating proce decribed in for each of the t period the individual ha been waiting, to get αµ τ µ τ = λµ τ + λ, τ = 0,,, t µ τ β + µα Thi updated belief in period t, µ t, clearly ha required coniderable mental agility on the part of the individual But it require le memory, a the individual doe not need to keep track of all pat event the current value of the belief µ i all that he need to know An alternative application of the updating procedure would be to uppoe that the individual took all their current information at time t and performed one update of their prior Thi i the update for example that i done by omeone who only get to the bu top in period t and learn a bu ha not yet arrived Such a peron would do one α Bayeian update and arrive at the updated belief t µ µβ t +µα 3 Hence the non-bayeian t updating protocol would in thi cae have an updated belief: α t µ µ t := λµ + λ µβ t + µα t The updated belief µ t ha required a different kind of mental agility requiring the individual to keep track of the pat hitory of event and their original prior The feature of the updating procedure we eek to remedy below i that the iterated and one-hot update reult in different belief: µ t µ t The two different way of uing the ame information ha generated different updated belief The only time µ t = µ t i when λ = and the updating i fully Bayeian There i no imple comparative tatic that decribe when one of the updating procedure decribed above make the individual more optimitic than the other If we take the initial belief µ = 2 for implicity, then µ 2 = λ 2 + λ α2 and the iterated β 2 +α 2 updating give: µ = λ 2 +λ α β+α, and µ αµ 2 = λµ +λ µ β+µ α A calculation 3 α t i the probability that no bu arrive in the firt t period in the good tate

5 DIVISIBLE UPDATING 5 ee Appendix how that µ 2 µ 2 λγ γ = µ + γ µ + γ + γ 2 + γ 3, γ := α β, µ 0, /2 Thi implie that µ 2 > µ 2 if and only if λ < The intuition for thi comparative tatic i a follow Varying λ tend to emphaie either the data or the prior The iteration of the updating mitigate thi effect To ee how thi work in practice firt uppoe that λ < and the prior i given increaed weight In the one-hot update, µ 2, there i only one opportunity for the bad new two period without bue to drive the prior down But, when updating i iterated, it weaken the effect of the increaed weight on the original prior It, intead, place ome weight on the intermediate prior, µ Hence, µ 2 will tend to be maller than µ 2 when the information that arrive i bad new, becaue µ 2 place le weight on the original prior than µ 2 Converely, when λ > the data i given increaed weight in the updating The onehot update, µ 2, give thee two period of bad new exceive weight Wherea the iterated update, µ 2, decreae the weight given to the firt period of the bad new by averaging it with the prior So, the individual i le optimitic when they do a one-hot up date becaue it maximie the weight given to the data that ha been collected The iteration now dilute the effect of the over-emphaized data 4 Of coure, the two way of determining period t belief, µ t and µ t, are not the only way the belief in period t could be arrived at An individual could update everal time over the period = 0,, t how many time thi occurred would depend on the agent cot of cogitation Thu there i a whole family of potential updated belief at time t and a time pae thi family grow 2 Related Literature There i a growing Economic literature, both experimental and theoretic, invetigating the conequence of a non-bayeian updating of belief, ee for example: Rabin and Schrag 999, Ortoleva 202, Angriani, Guarino, Jehiel, and Kitagawa 207, Levy and Razin 207, Brunnermeier 2009, Bohren and Hauer 207 among many other Much of thi literature combine iue of updating and deciion taking Thi i not what the current paper doe it focue olely on the reviion of belief and the propertie one might want to place on thi reviion One theme of thi literature ha been to invetigate the propertie of a particular aumption about how updating may occur The aim here i omewhat different, that i to try to undertand what updating procedure are conitent with a given property One exception to the focu on deciion taking i Eptein, Noor, and Sandroni 200 who provide a model of updating that capture the under and overreaction to new information Their model of updating i ditinct from the cla we conider in everal repect and ha already been conidered at length in the example above 4 When t varie continuouly it i poible to get a cleaner expreion for µt and a imilar comparative tatic on µ t and µ t can be obtained Thi i alo done in the Appendix

6 6 MARTIN W CRIPPS In Gilboa and Schmeidler 993 the notion of diviibility termed there commutativity i introduced and i argued to be an important feature of belief updating particularly in the context of updating ambiguou belief Hanany and Klibanoff 2009, ha perhap the cloet connection with thi one Here it i hown that there i a unique reweighted Bayeian update that generate a given et of dynamically conitent preference They moreover how that thi rule atifie commutativity, a property equivalent to diviibility Thee reweighted Bayeian update are a ubet of the cla of updating rule that decribed here the rule that update a in Figure Here we decribed all continuou updating rule that atify commutativity/diviibility In Zhao 206 a et of weaker axiom are hown to characterie an updating rule that doe not atify diviibility, but doe atify an order independence property imilar to the one dicued in the appendix, however, thi property i required to hold only for independent event There are link between the notion of diviibility and the model dynamic choice under uncertainty In particular the literature on recurive preference in dynamic etting, or dynamically conitent preference update rule Here agent are required to act conitently in ituation where information arrive over many period and thu implicitly behave a if they update diviibly; ee for example Eptein and Zin 989 or Eptein and Schneider 2003 In the tatitic literature the notion of prequentiality, Dawid 984, emphaie the idea that that forecating hould be an iterative procedure and there may be difference between iteratively revied forecat and other kind of forecating procedure 2 A Model of Belief Updating In thi ection we decribe our model of belief reviion and the mot important axiom we will impoe on the updating The approach taken i inpired by the axiomatic interpretation of entropie: ee for example Shannon and Weaver 949,Tverberg 958 or Aczél and Dhombre 989 p66 We will not adopt the terminology of prior and poterior, reerving thee term for the Bayeian updating only Intead the agent i aumed to be equipped with belief that are revied when information arrive to form updated belief There are K poible value for an unknown parameter {, 2,, K} := Θ and the agent ha the initial belief, µ = µ,, µ K Θ, about the value of thi parameter There i a tatitical experiment E that the agent can conduct which provide further information on 5 The experiment comprie a finite et of ignal {, 2,, n} = S and parameter-dependent probabilitie for the ignal p = p,, p n We will only conider experiment with trictly poitive probabilitie for all ignal Hence, we aume p > 0 for all S and Θ, or that p o S 6 We will alo want to conider the probabilitie of a given ignal S, hence we will define p := p,, p K 0, K to be thee probabilitie Thu p are the row of the matrix E and p are it column In 5 See Torgeren 99, for example, for the general propertie of tatitical experiment 6 We ue o S to denote the interior of S

7 DIVISIBLE UPDATING 7 ummary, the agent ha prior µ Θ and acce to the experiment E := p Θ o S K An updating proce take the experiment, E, and the belief, µ, and map them to the updated belief for each poible ignal outcome, {, 2,, n} The outcome of the updating proce i a profile of n poible updated belief {µ Θ : S} one for each poible ignal realiation S We define the updating function U n to be the map from the belief and the experiment to the profile of updated belief, that i U n : Θ o S K Θ n, for n = 2, 3, We will alo write µ,, µ n = U n µ, p Θ = U n µ, E 7 We will impoe ome condition that the updating mut atify The firt condition i that if the ignal are identical then there i no updating That i if p = p for all, Θ then the updated belief i the ame a the original belief Axiom Uninformativene U n µ, E = µ,, µ, if p = p for all, Θ The econd condition on the updating i that the name of the ignal are unimportant for how the belief are revied, it i only the probabilitie in the experiment that matter Thu permuting the order of the ignal jut permute the order of the updated probabilitie Axiom 2 Symmetry For any n, any permutation ω : {, 2,, n} {, 2,, n}, any µ, and any E = p Θ o S K, U n µ, ωp Θ = Un ω µ, E,, Un ωn µ, E, where ωp := p ω,, p ωn and U n = U n,, U n n The final Axiom in thi ection require that the updating i non-dogmatic and continuou Continuity i atified by many model of belief reviion in the literature, but may not hold if there are fixed cot of contemplation and belief reviion; ee Ortoleva 202 for an example of dicontinuou updating Non-dogmatic reviion, looely tated, allow an agent to have arbitrary updated belief after a ignal if they oberve ufficiently peruaive evidence To be more precie, it ay that if there are only two poible ignal, then for any initial belief that attache poitive probability to every parameter value and any given updated belief after ignal =, there exit a unique experiment or profile of probabilitie for the ignal = uch that the updated belief after ignal = ha thi value Thi doe not require that all poible profile of belief can be generated from a uitable tatitical experiment It only that the range of updating function for a given ignal i the entire et o Θ The additional requirement that thi map i a bijection i required for complete olution of the functional equation we later olve Without uniquene only local olution for U n exit 8 The uniquene 7 In thi model the updated belief are a determinitic function of the ignal and experiment Thi i not conitent with all model of updating For example in Rabin and Schrag 999 the updated belief are randomly determined by a bia that i realied after the ignal i oberved To capture thi model of updating it would be neceary for the function U n to take value in Θ n 8 See, for example, Berg 993

8 8 MARTIN W CRIPPS property would again be violated by model updating that have fixed cot of contemplation In uch model there may be et of experiment for which it i imply not worth reviing belief, hence there would be many experiment with the update equal to the prior Axiom 3 Non-Dogmatic The function U 2 : Θ o S K Θ 2 i continuou For any µ, µ o Θ there exit a unique Ẽ o S K uch that U 2 µ, Ẽ = µ 2 Binary Experiment We end thi ection by introducing the function u that decribe the updating in the cae where there are only two ignal a binary experiment Thi will play a key role in what follow For a binary experiment with only two ignal p = p, p 2 and E = p, p 2 Θ we can write U 2 µ, E U 2 µ, p, p 2, U 2 2 µ, p, p 2, where p + p 2 = and p = p,, p K are the probabilitie of the ignal {, 2} for each of the parameter Becaue of the axiom of ymmetry the two function on the right above are identical when the argument are tranpoed, that i, U 2 µ, p, p 2 U 2 2 µ, p 2, p So, we define u : Θ 0, K Θ a 2 uµ, p := U 2 µ, p, p That i, uµ, p decribe the updated belief after a binary experiment where the ignal = occurred and p were the parameter-dependent probabilitie of the ignal = and p were the probabilitie for = 2 Thi allow u to write the full profile of updated belief for binary experiment only in term of one function u: and U 2 µ, E uµ, p, uµ, p 2 We will ue diviibility to extend thi decompoition o that it hold for experiment with arbitrary number of ignal 4 Axiom alo implie that the function u atifie 3 uµ, p = µ, µ Θ, p 0, 3 Axiom on Iterated Updating: Diviibility and Order Independence We now conider two type of condition one might place on the updating of belief The firt condition on the updating we conider, Axiom 4, i termed diviibility It i deigned to capture the fact that the update i independent of how information i proceed whether it i proceed in an iterative way or a one-off proce The econd condition we conider, Axiom 6 given in the Appendix, i that the updating of belief i independent of the order that information arrive That i, revering the order that two piece of information arrive ha no effect on the ultimate profile of belief Although

9 DIVISIBLE UPDATING 9 thee axiom appear different, we how in the Appendix that they are actually equivalent and o we will only ue the diviibility axiom Conider two poible way of learning the ignal The firt i a one-tep proce where a ignal i generated according to the experiment E = p Θ and then revealed to the individual The econd i a two-tep proce where, firt, with probabilitie p, p Θ the ignal = or the ignal i revealed to the agent in a impler experiment Then, in the cae where the outcome wa obtained in the firt experiment, a ignal from the et {2,, n} i generated from a econd experiment with the probabilitie p p Θ where p i the vector p with the th element omitted In what follow we will ue E := p p Θ to denote the experiment that occur conditional on 9 The Axiom 4 ay that thee two different procee of learning have no effect on the individual ultimate profile of belief Thi aertion ha two ditinct element Firt it ay that learning the ignal i = when there are n other ignal ha the ame effect on the updated belief a learning the ignal i = when there i a binary experiment and the other ignal occur with probabilitie p Formally thi require that Unµ, E uµ, p The econd element of the Axiom 4 ay that the updated belief an individual ha when they ee the ignal > in a one-off experiment, that i Un µ, E, are the ame a the belief they would have in the two tep-proce That i, if they were firt told the ignal wa not = and updated their belief to uµ, p and then were told the ignal wa {2,, n} from the experiment E and engaged in the further update to U n uµ, p, E 0 The form of thi function require a little explanation The updated belief after the ignal i the th component of the vector U n when the firt ignal i not preent Hence the change in the upercript on U n Axiom 4 Diviibility For all E = p Θ o S K, µ Θ, and n 3 U n µ, E = [uµ, p, U n uµ, p, E ] If thi axiom i combined with ymmetry and iteratively applied, it implie that any multi-tep procedure that i baed upon the experiment E will reult in the ame updated profile of belief In the appendix we conider a econd property for belief updating that might be deirable that the order that information arrive ha no effect on the ultimate profile of belief That i, if ignal arrive in an order and thi order i re-arranged, then the ultimate profile of belief i unaffected by thi alteration We decribe an axiom on the updating that generate thi property and then how that thi Axiom i equivalent to the diviibility axiom 9 Recall that p o Θ and o > p 0 We ue to denote the vector,,, of appropriate length

10 0 MARTIN W CRIPPS 4 Characteriation of Diviible Updating In thi ection a propoition i proved that give a full characteriation of any updating procedure U n that atifie the Axiom, 2, 3, and 4 We will how that any uch updating i characteried by a homeomorphim F that map belief to hadow prior Then, thi hadow prior i updated in a fully Bayeian manner to create a hadow poterior Finally the hadow poterior i mapped back by F to form the individual updated belief Thi i illutrated in the figure below belief F hadow prior U updated belief F Baye Updating hadow poterior Figure 2 Updating Function U that Satify Axiom, 2, 3, and 4 The firt tep in the argument i to how that uµ, p, the function that determine the updated belief after ignal, mut be homogeneou degree zero in p if it atifie Axiom, 2, and 4 The intuition for thi reult i quite imple Suppoe that there are three poible ignal S = {, 2, 3} and that one ignal, ay = 2, i equally likely under all tate We now apply the Axiom 4 to the two cae: one where i determined in a one-off experiment and the econd where there i firt an experiment with binary outcome = 2 and 2 and then {, 3} i elected with the appropriate conditional probabilitie A oberving = 2 i uninformative, Axiom implie the firt tage of the two-tep proce lead to no updating of the prior However the econd tage experiment, when a ignal 2 i elected, ha increaed the relative probabilitie p of the ignal {, 3} Thu the equality in Axiom 4 implie that the belief after the one-tep experiment with low relative probabilitie p of the ignal {, 3} are equal to the belief after the two-tep experiment with higher relative probabilitie of the ignal Thu caling up the probabilitie had no effect on the updating of belief Thi i the definition of homogeneity degree zero Lemma Suppoe updating atifie Axiom, 2, and 4, then the function uµ, p defined in 2 i homogeneou degree zero in p And 4 U n µ, E uµ, p,, uµ, p n Proof Suppoe that n = 3 Conider two way the agent can proce the ignal = : a She could be told the outcome of an = or > experiment, which would reult in the updated belief 5 b The agent could firt be told 2 in an = 2, 2 experiment and then the outcome of a final experiment where {, 3}, which would reult in the two tep econd updating uµ, p u u µ, p 2, p p Θ 2

11 DIVISIBLE UPDATING The combination of Axiom 2 and 4 applied to 5 and 6 implie uµ, p u u µ, p 2, p 7 p Θ p 2 [0, p ] 2 If p 2 = /λ for all, then 3 implie u µ, p 2 µ So the right of 7 become uµ, λp Θ and we get the condition uµ, p u µ, λp, λ [, min p ] Hence the function u i homogeneou degree zero in p if Axiom 4 hold The equation 4 follow by combining Axiom 4 with ymmetry We now move on to etablihing the main reult which give the form of the function uµ, p defined in 2 The intuition for thi characteriation of the function uµ, p come from the fact that there many different intermediate experiment that will generate the ame final updated belief Thi i ummaried in the functional equation, given below, that wa derived in Lemma uµ, p u u µ, p, p p Θ p Thi ay that the update for the ignal, uµ, p, i alo equal to the update for all equivalent intermediate experiment when the ignal did not occur, then the individual had to do two update They would firt revie their belief to u µ, p when didn t occur Then, they would do a econd update upon oberving One could turn thi equation into a family of PDE and thereby determine u However, there are technique to olve thi particular functional equation without the aumption of differentiability continuity i enough The firt tep i to turn it into a linear functional equation by taking logarithm Giving the equation uµ, p u u µ, p, p p One imple olution to thi equation i to add the argument together: uµ, p = µ + p Suitably adapted to the fact that µ, p, and u are vector of probabilitie and the homogeneity reult above, thi imple olution give Baye updating For a more general olution one can firt notice that the functional equation tell u about the contour of the function u Thi i becaue a p varie the argument on right, that i u µ, p and p p vary in a way that doe not change the value of u It i, then, relatively imple to ee that each uch contour of the function u i a tranlation of the other Thu once the form of one contour ha been determined all other contour are jut tranlation of it Hence chooing an arbitrary function to determine the hape of one contour and another to determine the value taken by each contour i ufficient to determine the entire function Thi i how the family of olution to the functional equation given in the Propoition are generated The role of Axiom 3 here i to enure that we are looking for a continuou olution to the functional equation, and that it hold globally not jut locally If continuity fail the et of olution to even the implet functional equation become enormou The role of uniquene i to enure a global olution exit, F i a homeomorphim, otherwie there may be many local olution to the functional equation and only local olution See Aczél and Dhombre 989 Chapter for an example of thi

12 2 MARTIN W CRIPPS exit Thee local olution alo have the ame form a the global olution, ee Berg 993 Propoition If the updating U n atifie the Axiom, 2, 3, and 4, then there exit a homeomorphim F : Θ Θ uch that 8 uµ, p = F F µp Θ F µp,, F K µp K Θ F µp ; where F µ F µ, F 2 µ,, F K µ, and U n µ, E = uµ, p,, uµ, p n Proof We begin by reducing the dimenion of the variable in the function uµ, p Define w : o Θ R K ++ a follow µ wµ,, µ K :=,, µ K ; µ K µ K where R ++ := {x R : x > 0} The function w i a bijection, from o Θ to R K ++ and it ha the invere w x,, x k = x + K i= x,, i x K + K i= x, i + K i= x i We will now re-define the variable of the function u Firt we define the variable φ := wµ, which require u to retrict µ o Θ and ue continuity to define uµ, p when µ i on the boundary of thi et We will alo define π := wp The fact uµ, p i homogeneou degree zero in p 0, K, by Lemma, implie that there i no lo by thi tranformation We will alo apply the tranformation w to the function and define vφ, π wuµ, p Hence we have tranformed a function u : o Θ o Θ o Θ to a function v : R K ++ RK ++ RK ++ If we re-write 7 with thi new notation we get vφ, π vvφ, ρ, π/ρ, where ρ := w p 2 and π/ρ := w p p Θ 2 v : R K ++ RK ++ RK ++ ; Now we take logarithm to do a final tranformation of thi function Let u define φ := ln φ, π := ln π, ρ = ln ρ, and ṽ φ, π ln vφ, π Then, a final re-writing of 7 with thi new notation give ṽ φ, π ṽṽ φ, ρ, π ρ, ṽ : R K R K R K If we define y = π ρ and z = ρ thi then become the functional equation ṽ φ, y + z ṽ ṽ φ, z, y, φ, y, z R K Thi i called the tranlation equation and wa originally olved in it multivariate form by Aczél and Hozú 956 Given the uniquene property in the regularity aumption, Axiom 3, the reult decribed in Mozner 995 page 2 apply Thu any continuou olution to thi equation ha the property that here exit a continuou bijection g : R K R K uch that 9 ṽ φ, π = g [g φ + π]

13 DIVISIBLE UPDATING 3 Now we will revere the tranformation of thi problem Firt we will remove the logarithm in 9 to get Then we will invert the g function ln vφ, π = g [gln φ + ln π] g ln vφ, π = gln φ + ln π Now we will introduce the function hx = gln x = h x,, h K x to implify thi expreion h vφ, π = hφ + ln π h vφ, π = ln e hφ + ln π h vφ, π = ln e hφ π,, e hk φ π K e h vφ,π = e hφ π,, e hk φ π K Finally, we define fx e hx and thi become fvφ, π = f φπ,, f K φπ K Now ubtitute the w tranformation to get 0 f w uµ, p = f wµ p p K,, f K wµ pk p K We will now define the function F : o Θ o Θ o that the following diagram commute, that i, f w w F Thi i poible a w i invertible F o Θ o Θ w R K f ++ R K ++ F i a bijection and continuou hence it i a homeomorphim We will extend thi definition of F where neceary to the boundary of Θ uing continuity Uing thi we can re-write 0 a w F uµ, p = w F µ p F µp = F K µp K p K w,, w K F µ pk,, F K µp K F K µp K Now applying w to both ide thi give F u µ, p F µp Θ F µp,, F K µp K Θ F µp Applying F to both ide of thi give equation 8 in the propoition The other diplayed equation in the propoition follow from a ubtitution of 8 into 4 p K, A an example of thi updating rule in action, conider again the model of the arrival of bue in Section Recall that µ wa the probability the individual attached to α the good tate and that a Bayeian updater would have a poterior t µ α t µ+β t µ if there had been t period without a bu arriving

14 4 MARTIN W CRIPPS An individual who update diviibly i decribed by a homeomorphim F : µ, µ F µ, F 2 µ and it invere F : µ, µ F µ, F2 µ If uch an individual arrived at the bu top and learnt that there had been t period without a bu arriving, then they would update their belief about the good tate to µ t = F F µα t F µα t + F 2 µβ t One more period without a bu would then lead the individual to do the further update µ t+ = F F µ t α F µ t α + F 2 µ t β However, the updating rule for µ t given here implie that F µ t = F µα t F µα t + F 2 µβ t, and F 2 µ t = F µ t If thee are ubtituted into the expreion for µ t+ we get µ t+ = F F µα t+ F µα t+ + F 2 µβ t+ Thu the belief µ t+ of our individual are exactly thoe of a new arrival at the bu top We now give two general clae of homeomorphim F that generate particular clae of diviible updating rule 4 Geometric Weighting The firt example of a homeomorphim F : Θ Θ we conider reult in an updating function that ha already been ued to model overreaction or under-reaction to new information F µ = µ α µα,, µ α K K µα An explicit form for F only exit when α = α for all F x = µ /α µ/α,, µ /α K µ/α ;, for α = α, We can now imply ubtitute thi functional form into 8 to generate a belief updating rule that atifie our Axiom In general thi give the updating rule u µ, p Θ F µ α p µ α K,, K p K p p When α = α for all u µ, p Θ Θ µα Θ µα µ p /α Θ µ p /α,, µ K pk /α Θ µ p /α

15 DIVISIBLE UPDATING 5 From thi expreion it i clear that thi updating rule exaggerate or alter the probabilitie that enter into the normal Bayeian formula Thi can be interpreted a overreaction or under-reaction to new information But, in Section 52, we will alo how that thi generate a bia in the updating When α α, there i no explicit olution for the full updated ditribution However, it i poible to proceed in a lightly different way to undertand how the relative probabilitie are updated Firt, write F u µ, p Θ µ α p Θ µα and then define µ,, µ K = u µ, p Θ to get µ α,, µ K αk µ α µ α,, p µ α p Θ µα,, p µ α K K p K Then dividing the entry in thi vector by the entry we get u µ, p α Θ u µ, p Θ α = µα µ α p ; p Θ µα p u µ, p Θ u µ, p Θ = µ µ µ α K K p K Θ µα p p p /α } {{ } if α=α =α In the cae where the parameter α differ it i till poible to give an explicit expreion for the relative ize of the updated belief in and Thi updating rule i generalie the overconfidence/under-confidence protocol decribed in Angriani, Guarino, Jehiel, and Kitagawa 207 and Bohren and Hauer 207, for example If /α > then the individual i overreact to new information on parameter value and i overconfident by placing too much weight on their obervation Thi i achieved by exaggerating the difference in the ignal and i more clearly een in the ratio of the updated belief given above Converely, if /α < then the individual i under-react to new information about their learning on parameter value they place too much weight on their prior and do not adjut their belief a much a a Bayeian would In thee two paper, the under and over-reaction i uniform acro all parameter But, the functional form permit the agent to be over-react for ome parameter and under-react for other Thu there i the poibility of elective over and under-reaction where the agent more readily change belief about ome parameter but not about other 42 Exponential Weighting and Other Homeomorphim For all Θ, let f : [0, ] R + be a trictly increaing and continuou function that atifie f 0 = 0 Then let u define F, a below f µ F µ = f µ,, f K µ K f µ

16 6 MARTIN W CRIPPS Without an explicit form for F we can ue the relation F u µ, p Θ f µ p Θ f µ p,, f K µ K p K Θ f µ p Again if we write the updated belief a u µ, p Θ = µ,, µ K, then the ratio of any two entrie give f µ f µ = f µ p f µ p A an example of uch a function we could chooe f µ = e β/µ, which reult in a tranformation of belief that i imilar to a multinomial logit e β /µ e β K/µ K F µ =,, for β > 0 e β /µ e β /µ Thi function F doe not net Bayeian updating there are no value of the parameter β for which thi function i the identity Suppoe that β = β for all Then F doe map point cloer to µ when β i mall and move point away from µ when β i large A β 0, o F µ converge to µ for all interior µ And a β, o F µ converge to a ditribution that put all weight on the larget element of µ Thu the extreme of thi function are imilar to thoe in the previou example And, uing the intuition from our previou example, we would expect large value of β to be aociated with under-reaction and mall value of β to be aociated with over-reaction to new information An explicit form for the invere function F i not given here However, by taking the ratio of any two entrie ay ˆ and in thee vector we get Alternatively e βˆ/µ ˆ e β /µ = e βˆ/µˆpˆ e β /µ p β µ βˆ µ ˆ = β βˆ pˆ + ln µ µˆ p Thu the updating of belief reult in a linear hift in the invere probabilitie of each parameter value If βˆ = β = β it i eay to ee that large value of β decreae the dependence of thi hift on on the ratio of the probabilitie Thu there i the conjectured under-reaction in thi cae 5 Propertie of Diviible Updating In thi ection we decribe ome of the general propertie of the updating rule that atify the Axiom, 2, 3, and 4 In particular we will how that thee updating rule obey conitency and decribe condition on the function F that enure updating i biaed in particular way We are going to need to impoe the condition that the updating repect certainty That i, if the individual ha a prior that attache probability one to the parameter and oberve ignal that only occur with poitive probability under thi prior, then they do not revie their belief We let e Θ denote the prior that attache

17 DIVISIBLE UPDATING 7 probability one to the parameter, that i, it i a vector with unity in the entry and zero elewhere U n e, E = e,, e, for all E o S K 5 The Conitency of Diviible Updating Here we conider the limiting propertie of the updating rule 8 a increaing amount of information on the parameter Θ i collected For a given value of the parameter we will decribe the limiting updated belief a the experiment E i repeatedly ampled We will how that the updating converge almot urely and that, if the experiment i informative, then the updated belief converge with probability one to the ditribution e certainty that the parameter i Propoition 2 2 Thi reult on conitency for diviible updating contrat with example of non-bayeian updating in the literature that do not atify conitency: Rabin and Schrag 999 and Eptein, Noor, and Sandroni In general model of learning there are two propertie that might be deirable: the individual attache probability one to their belief converging, 2 the belief do converge with probability one In general there are learning protocol where neither, all, or one of thee propertie hold The convergence reult below i conditional on the parameter value, thu, convergence independent of the initial belief the individual attache to each poible parameter Thu both of the above propertie are atified by diviible learning We will begin with a decription of the model where the individual repeatedly ample from the ame experiment for a given parameter value Fix a parameter value and let the tochatic proce { t } t=0 S be independently ampled from the ditribution p o S We will ue P to denote the probability meaure on S of thi proce We will alo inductively define a belief tochatic proce {µ t } t=0 Θ o that µ t+ i the updated value of µ t when the ignal t i oberved The formal definition i: µ 0 Θ and 2 µ t+ := u µ t, p t, t = 0,, Propoition 2 how that for all the proce {µ t } t=0 converge P almot urely provided the updating atifie Axiom, 2, 3, 4 Furthermore, if the updating atifie condition and the ignal are informative, then µ t converge to e, P almot urely That i the updating atifie conitency Propoition 2 If the updating U n atifie the Axiom, 2, 3, 4, then for all there exit µ Θ uch that µ t µ, P almot urely If the updating alo repect certainty, p p for all, and µ 0 o Θ, then, µ = e with P probability one 2 Thi property of Bayeian updating i uually termed conitency, ee Diaconi and Freedman 986 for example 3 In Lehrer and Teper 205 the notion of conitency i ued a an axiom to characterie Bayeian updating

18 8 MARTIN W CRIPPS The proof of thi reult i given in the appendix The proof i quite trivial; it jut applie the uual proof of the conitency of Bayeian updating ee for example De- Groot 970 to the hadow-belief reviion proce It then ue the property that belief reviion repect certainty to enure that when the hadow belief approach certainty they are mapped back by a continuou invere function to belief that alo approach certainty 52 Bia in Diviible Updating In thi ection we decribe clae of diviible updating rule that exhibit biae for, or againt, the truth That i, we will give ufficient condition on the function F defined in 8 to generate uch biae To make the expreion of thee condition imple we will conider a model with only two parameter Θ = {, } where the individual form the updated belief u µ, p when they ee the ignal We will ay the updating i locally conitent if µ i expected to increae when i true Thi i the inequality µ E u µ, p Here E i an expectation take relative to the objective ditribution of the ignal Thu, thi inequality ay that an outide Bayeian oberver expect the individual to have an increaed belief in when it it i true and they oberve the outcome of the experiment E Local conitency hold for Bayeian updating which i globally conitent and other model of updating that are approximately Bayeian 4 We will alo ay updating i locally inconitent if µ [repectively, µ ] i expected to decreae [repectively, increae] when i true Thee are the inequalitie µ E u µ, p, µ E u µ, p Thi kind of bia doe not arie here, becaue of individual are ignoring or miinterpreting their ignal a in Rabin and Schrag 999, for example It arie becaue when i true the individual i low to move their belief in upward in repone to poitive evidence but quick to move belief down when evidence in favour of i oberved Thee two effect on average give a downward movement of belief Thu, the bia could be interpreted more a reluctance to move to extreme belief rather than a univerally poitive bia in favour of parameter Thi negative movement in belief alo appear to be at odd with the reult in Propoition 2, that belief ultimately converge to the truth for all diviible updating procee Thi conflict i reolved a we will how that an individual cannot be globally inconitent, it i only poible for the individual to be locally inconitent on a ubet of the tate In thi two-parameter cae the homeomorphim F : µ, µ µ, µ can be decribed by it effect on it firt element We will ue the trictly increaing function f : [0, ] [0, ] to decribe how F operate on it firt element F µ fµ, fµ Before tating the reult we will give a definition of locally convex and locally concave function Thi allow u to think about the local biae in the updating rather than only 4 Thi i a feature of ome of the argument in Bohren and Hauer 207 for example

19 DIVISIBLE UPDATING 9 global propertie We introduce the et Rµ to decribe all poible updated belief that the experiment E could generate given the initial belief µ and the updating function u More preciely we let Rµ [0, ] be the mallet interval that contain all the poible value for the updated belief when the initial belief i µ { Rµ := µ : min f fµ p µ max f } fµ p fµ p + fµ p fµ p + fµ p The definition of local concavity/convexity of a function g imply require g to be concave/convex on the range of updated belief that could occur when the original belief i µ, that i, the interval Rµ If the experiment i particularly informative or if f increae rapidly thi require g to be concave/convex on a large interval but it alo may be a relatively mall interval for uninformative experiment or unreponive function f Definition The function g : [0, ] R i locally convex at µ, if it i a convex function on the et Rµ The function g[0, ] R i locally concave at µ, if it i a concave function on the et Rµ Equipped with thi definition we can provide ufficient condition for the biae decribed above The proof of thi propoition i given in the appendix Propoition 3 If the function f i increaing: i And /f i locally convex at µ, then µ E u µ, p and updating i locally conitent ii And /f i locally concave at µ, then µ E u µ, p and updating i locally inconitent If f0 = 0 then /f i not locally concave for all µ For Bayeian updating fµ i the identity and in thi cae /fµ = µ i convex In thi cae Propoition 3 i the uual reult that a Bayeian ha belief that are a ubmartingale and on average are revied upward Now we will reviit the example in Section 4 where F ha the geometric weighting form and α = α = α When there are only two parameter thi can be implified µ to fµ = α µ α + µ thi function i illutrated for α < in Figure 52 below Firt α oberve that /fµ = + µ µ α i convex when α So, by cae i of Propoition 3, updating i alway locally conitent when α However, when α < there exit an interval of value µ 2 + α, ] where /fµ i concave In thi region the updating i locally inconitent cae ii of the Propoition Thu a the individual get cloer to certain that i the true parameter, the expected movement of the belief i away from certainty One could interpret thi a a ort of mean reverion in the updating The convergence to certainty i lower a a reult It i clear from the figure below that when α < f tend to move belief to more central poition A the experimenter become more convinced that the parameter i, µ 2 +α, the updating on average move the updated belief away from thee extreme value reviing belief downward in repone to bad new on much fater than they are increaed in repone to good new Although, the bad new i le likely than the good new, the curvature of f

20 20 MARTIN W CRIPPS i ufficient to enure that the expectation of thee two effect i le than the current belief in fµ f 0 0 µ Figure 3 fµ = µ α µ α + µ α, for α < 53 Sufficient Condition for Under and Overreaction to Information In thi ection we provide a ufficient condition for the updating rule from Propoition to overreact or under-react to new information We will argue that one caue of overreaction i that the homeomorphim F contract pace or move point cloer together Once the hadow prior ha been updated to a hadow poterior, the ditance between the hadow prior and hadow poterior i a meaure of how much Bayeian learning ha occurred If the invere function F expand pace or F contract pace, then when the hadow poterior and prior are mapped back to the belief pace they are even further apart The repone to the ignal ha become more exaggerated and overreaction i preent Similarly, if the function F expand pace and move point further apart Then the learning that occurred in the hadow Bayeian world get reduced when it i mapped back to the belief pace by F, becaue the invere will move point cloer together A a reult the Bayeian learning in the hadow pace i undertated and there i under-reaction to new information Thi i not the complete explanation of the caue of under-reaction, becaue the amount of Bayeian learning that occur i alo influenced by the prior that i being updated For example if the prior i cloe to certainty, then it will not be revied much in repone to a given experiment Thu under-reaction to an experiment can be generated if µ i mapped to an F µ that i cloe to certainty, becaue in thi cae the Bayeian update at F µ will be mall Thu there are two effect that influence under and over-reaction to information: how F expand or contract pace and the extent to which F map µ to the extreme value There are homeomorphim in which both thee effect tend to work together Thi i the cae if F mapped all interior belief cloer to the centre of Θ Then, the Bayeian update of the reultant hadow prior would be larger becaue the hadow

21 DIVISIBLE UPDATING 2 prior i more uncertain and le extreme And the invere map F would expand pace a F contract pace and move thee Bayeian update further apart 5 A in the previou ection we are able to give ufficient condition for over- and underreaction for the cae where there are two parameter, Θ = {, } In thi cae we can treat the homeomorphim F µ fµ, fµ a being determined by a one-dimenional function f : [0, ] [0, ] 6 To define over-reaction or under-reaction it i neceary to do three thing: firt, elect a meaure of the belief, econd chooe a meaure of variability of belief; and third pecify a benchmark model of updating to make comparion with The reult have a impler form if we meaure belief uing the odd ratio µ µ = µ µ The variance will be our meaure of variability of belief and the Bayeian update a the benchmark for comparion Thu our benchmark meaure of variability i the variance of the odd ratio of the Bayeian update If we ue µ to denote the Bayeian update, then the odd ratio ha the variance [ µ ] Var µ [ µ = Var µ p p ] = µ µ 2 Var [ p The final variance i zero if and only if the ignal in the experiment are uninformative p = p It i clear thi benchmark i affect both by the information content of the experiment and by the initial prior Now we can define under-reaction to new information to be the ituation where the variance of the odd ratio of the update i le than the variance of the Bayeian update and over-reaction to hold when it i greater Definition 2 The updating exhibit under-reaction at µ if [ ] u µ, p 2 [ µ p 3 Var < Var u µ, p µ p and it exhibit overreaction at µ if [ ] u µ, p 2 [ µ p 4 Var > Var u µ, p µ p We now are able to tate a reult on when diviible updating exhibit over and underreaction in the two-parameter cae Propoition 4 Suppoe that the updating i decribed by the homeomorphim F µ fµ, fµ, that f i differentiable, and that there exit a contant β > uch that β µ fµ β, β µ fµ β ; for all µ [0, ] Then: i if max µ Rµ f µ β 4, then the updating exhibit overreaction at µ i if min µ Rµ f µ β 4, then the updating exhibit underreaction at µ 5 An example of uch a homeomorphim i given in Figure 52 above 6 The principal reaon the general cae of thi reult for more than two parameter i out of reach i that, apart from the linear cae, there i no imple relationhip between Var[µ] and Var[Fµ] when µ i multidimenional ], ] p ]