A tutorial on conformal prediction

Transcription

1 A tutorial on conformal prediction Glenn Shafer and Vladimir Vovk praktiqekie vyvody teorii vero tnote mogut bytь obonovany v kaqetve ledtvi gipotez o predelьno pri dannyh ograniqeni h loжnoti izuqaemyh vleni On-line Compreion Modelling Project (New Serie) Working Paper #3 Firt poted June 9, Lat revied June 9, Project web ite:

2 Abtract Conformal prediction ue pat experience to determine precie level of confidence in new prediction. Given an error probability ɛ, together with a method that make a prediction ŷ of a label y, it produce a et of label, typically containing ŷ, that alo contain y with probability ɛ. Conformal prediction can be applied to any method for producing ŷ: a nearet-neighbor method, a upport-vector machine, ridge regreion, etc. Conformal prediction i deigned for an on-line etting in which label are predicted ucceively, each one being revealed before the next i predicted. The mot novel and valuable feature of conformal prediction i that if the ucceive example are ampled independently from the ame ditribution, then the ucceive prediction will be right ɛ of the time, even though they are baed on an accumulating dataet rather than on independent dataet. In addition to the model under which ucceive example are ampled independently, other on-line compreion model can alo ue conformal prediction. The widely ued Gauian linear model i one of thee. Thi tutorial preent a elf-contained account of the theory of conformal prediction and work through everal numerical example. A more comprehenive treatment of the topic i provided in Algorithmic Learning in a Random World, by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).

3 Content Introduction 2 Valid prediction region 2 2. An example of valid on-line prediction Fiher prediction interval A numerical example On-line validity Confidence ay le than probability Exchangeability 8 3. Exchangeability and independence Backward-looking definition of exchangeability The betting interpretation of exchangeability A law of large number for exchangeable equence Conformal prediction under exchangeability 3 4. Nonconformity meaure Conformal prediction from old example alone Example: Predicting a number with an average Are we complicating the tory unnecearily? Conformal prediction uing a new object Example: Claifying iri flower Example: Predicting petal width from epal length Optimality Example are eldom exactly exchangeable On-line compreion model Definition Conformal prediction Example The exchangeability-within-label model The on-line Gauian linear model Acknowledgement 48 Reference 48 A Validity 50 A. A claical argument for independence A.2 A game-theoretic law of large number A.3 The independence of hit for Fiher interval

4 Introduction How good i your prediction ŷ? If you are predicting the label y of a new object, how confident are you that y = ŷ? If the label y i a number, how cloe do you think it i to ŷ? In machine learning, thee quetion are uually anwered in a fairly rough way from pat experience. We expect new prediction to fare about a well a pat prediction. Conformal prediction ue pat experience to determine precie level of confidence in prediction. Given a method for making a prediction ŷ, conformal prediction produce a 95% prediction region a et Γ 0.05 that contain y with probability at leat 95%. Typically Γ 0.05 alo contain the prediction ŷ. We call ŷ the point prediction, and we call Γ 0.05 the region prediction. In the cae of regreion, where y i a number, Γ 0.05 i typically an interval around ŷ. In the cae of claification, where y ha a limited number of poible value, Γ 0.05 may conit of a few of thee value or, in the ideal cae, jut one. Conformal prediction can be ued with any method of point prediction for claification or regreion, including upport-vector machine, deciion tree, booting, neural network, and Bayeian prediction. Starting from the method for point prediction, we contruct a nonconformity meaure, which meaure how unuual an example look relative to previou example, and the conformal algorithm turn thi nonconformity meaure into prediction region. Given a nonconformity meaure, the conformal algorithm produce a prediction region Γ ɛ for every probability of error ɛ. The region Γ ɛ i a ( ɛ)-prediction region; it contain y with probability at leat ɛ. The region for different ɛ are neted: when ɛ ɛ 2, o that ɛ i a lower level of confidence than ɛ 2, we have Γ ɛ Γ ɛ 2. If Γ ɛ contain only a ingle label (the ideal outcome in the cae of claification), we may ak how mall ɛ can be made before we mut enlarge Γ ɛ by adding a econd label; the correponding value of ɛ i the confidence we aert in the predicted label. A we explain in 4, the conformal algorithm i deigned for an on-line etting, in which we predict the label of object ucceively, eeing each label after we have predicted it and before we predict the next one. Our prediction ŷ n of the nth label y n may ue oberved feature x n of the nth object and the preceding example (x, y ),..., (x n, y n ). The ize of the prediction region Γ ɛ may alo depend on thee detail. Reader mot intereted in implementing the conformal algorithm may wih to turn directly to the elementary example in 4.2 and 4.3 and then turn back to the earlier more general material a needed. A we explain in 2, the on-line picture lead to a new concept of validity for prediction with confidence. Claically, a method for finding 95% prediction region wa conidered valid if it had a 95% probability of containing the label predicted, becaue by the law of the large number it would then be correct 95% of the time when repeatedly applied to independent dataet. But in the on-line picture, we repeatedly apply a method not to independent dataet but to an accumulating dataet. After uing (x, y ),..., (x n, y n ) and x n to predict y n, we ue (x, y ),..., (x n, y n ), (x n, y n ) and x n+ to predict y n+, and o

5 on. For a 95% on-line method to be valid, 95% of thee prediction mut be correct. Under minimal aumption, conformal prediction i valid in thi new and powerful ene. One etting where conformal prediction i valid in the new on-line ene i the one in which the example (x i, y i ) are ampled independently from a contant population i.e., from a fixed but unknown probability ditribution Q. It i alo valid under the lightly weaker aumption that the example are probabilitically exchangeable (ee 3) and under other on-line compreion model, including the widely ued Gauian linear model (ee 5). The validity of conformal prediction under thee model i demontrated in Appendix A. In addition to the validity of a method for producing 95% prediction region, we are alo intereted in it efficiency. It i efficient if the prediction region i uually relatively mall and therefore informative. In claification, we would like to ee a 95% prediction region o mall that it contain only the ingle predicted label ŷ n. In regreion, we would like to ee a very narrow interval around the predicted number ŷ n. The claim of 95% confidence for a 95% conformal prediction region i valid under exchangeability, no matter what the probability ditribution Q the example follow and no matter what nonconformity meaure i ued to contruct the conformal prediction region. But the efficiency of conformal prediction will depend on Q and the nonconformity meaure. If we think we know Q, we may chooe a nonconformity meaure that will be efficient if we are right. If we have prior probabilitie for Q, we may ue thee prior probabilitie to contruct a point predictor ŷ n and a nonconformity meaure. In the regreion cae, we might ue a ŷ n the mean of the poterior ditribution for y n given the firt n example and x n ; in the claification cae, we might ue the label with the greatet poterior probability. Thi trategy of firt guaranteeing validity under a relatively weak aumption and then eeking efficiency under tronger aumption conform to advice long given by John Tukey and other [25, 26]. Conformal prediction i tudied in detail in Algorithmic Learning in a Random World, by Vovk, Gammerman, and Shafer [28]. A recent expoition by Gammerman and Vovk [3] emphaize connection with the theory of randomne, Bayeian method, and induction. In thi article we emphaize the on-line concept of validity, the meaning of exchangeability, and the generalization to other on-line compreion model. We leave aide many important topic that are treated in Algorithmic Learning in a Random World, including extenion beyond the on-line picture. 2 Valid prediction region Our concept of validity i conitent with a tradition that can be traced back to Jerzy Neyman introduction of confidence interval for parameter in 937 [9] and even to work by Laplace and other in the late 8th century. But the hift of emphai to prediction (from etimation of parameter) and to the on-line etting (where our prediction rule i repeatedly updated) involve ome 2

6 rearrangement of the furniture. The mot important novelty in conformal prediction i that it ucceive error are probabilitically independent. Thi allow u to interpret being right 95% of the time in an unuually direct way. In 2., we illutrate thi point with a well-worn example, normally ditributed random variable. In 2.2, we contrat confidence with full-fledged conditional probability. Thi contrat ha been the topic of endle debate between thoe who find confidence method informative (claical tatitician) and thoe who init that full-fledged probabilitie baed on all one information are alway preferable, even if the only available probabilitie are very ubjective (Bayeian). Becaue the debate uually focue on etimating parameter rather than predicting future obervation, and becaue ome reader may be unaware of the debate, we take the time to explain that we find the concept of confidence ueful for prediction in pite of it limitation. 2. An example of valid on-line prediction A 95% prediction region i valid if it contain the truth 95% of the time. To make thi more precie, we mut pecify the et of repetition enviioned. In the on-line picture, thee are ucceive prediction baed on accumulating information. We make one prediction after another, alway knowing the outcome of the preceding prediction. To make clear what validity mean and how it can be obtained in thi on-line picture, we conider prediction under an aumption often made in a firt coure in tatitic: Random variable z, z 2,... are independently drawn from a normal ditribution with unknown mean and variance. Prediction under thi aumption wa dicued in 935 by R. A. Fiher, who explained how to give a 95% prediction interval for z n baed on z,..., z n that i valid in our ene. We will tate Fiher prediction rule, illutrate it application to data, and explain why it i valid in the on-line etting. A we will ee, the prediction given by Fiher rule are too weak to be intereting from a modern machine-learning perpective. Thi i not urpriing, becaue we are predicting z n baed on old example z,..., z n alone. In general, more precie prediction i poible only in the more favorable but more complicated et-up where we know ome feature x n of the new example and can ue both x n and the old example to predict ome other feature y n. But the implicity of the et-up where we predict z n from z,..., z n alone will help u make the logic of valid prediction clear. 2.. Fiher prediction interval Suppoe we oberve the z i in equence. After oberving z and z 2, we tart predicting; for n = 3, 4,..., we predict z n after having een z,..., z n. The 3

7 natural point predictor for z n i the average o far: z n := n z i, n but we want to give an interval that will contain z n 95% of the time. How can we do thi? Here i Fiher anwer [0]: i=. In addition to calculating the average z n, calculate 2 n := n (z i z n ) 2, n 2 i= which i ometime called the ample variance. We can uually aume that it i non-zero. 2. In a table of percentile for t-ditribution, find t n 2, the point that the t-ditribution with n 2 degree of freedom exceed exactly 2.5% of the time. 3. Predict that z n will be in the interval Fiher baed thi procedure on the fact that z n z n n n n z n ± t n 2 n n n. () ha the t-ditribution with n 2 degree of freedom, which i ymmetric about 0. Thi implie that () will contain z n with probability 95% regardle of the value of µ and σ A numerical example We can illutrate () uing ome number generated in 900 by the tudent of Emanuel Czuber (85 925). Thee number are integer, but they theoretically have a binomial ditribution and are therefore approximately normally ditributed. Here are Czuber firt 9 number, z,..., z 9 : 7, 20, 0, 7, 2, 5, 9, 22, 7, 9, 4, 22, 8, 7, 3, 2, 8, 5, 7. (3) Czuber tudent randomly drew ball from an urn containing ix ball, numbered to 6. Each time they drew a ball, they noted it label and put it back in the urn. After each 00 draw, they recorded the number of time that the ball labeled with a wa drawn ([5], pp ). Thi hould have a binomial ditribution with parameter 00 and /6, and it i therefore approximately normal with mean 00/6 = 6.67 and tandard deviation 500/36 = (2) 4

8 From them, we calculate z 9 = 6.53, 9 = The upper 2.5% point for the t-ditribution with 8 degree of freedom, t , i 2.0. So the prediction interval () for z 20 come out to [9.55, 23.5]. Taking into account our knowledge that z 20 will be an integer, we can ay that the 95% prediction i that z 20 will be an integer between 0 and 23, incluive. Thi prediction i correct; z 20 i On-line validity Fiher did not have the on-line picture in mind. He probably had in mind a picture where the formula () i ued repeatedly but in entirely eparate problem. For example, we might conduct many eparate experiment that each conit of drawing 00 random number from a normal ditribution and then predicting a 0t draw uing (). Each experiment might involve a different normal ditribution (a different mean and variance), but provided the experiment are independent from each other, the law of large number will apply. Each time the probability i 95% that z 0 will be in the interval, and o thi event will happen approximately 95% of the time. The on-line tory may eem more complicated, becaue the experiment involved in predicting z 0 from z,..., z 00 i not entirely independent of the experiment involved in predicting, ay, z 05 from z,..., z 04. The 0 random number involved in the firt experiment are all alo involved in the econd. But a a mater of the analytical geometry of the normal ditribution [8, 9], Fiher would have noticed, had he thought about it, that thi overlap doe not actually matter. A we how in Appendix A.3, the event z n t n 2 n n n z n z n + t n 2 n n n for ucceive n are probabilitically independent in pite of the overlap. Becaue of thi independence, the law of large number again applie. Knowing each event ha probability 95%, we can conclude that approximately 95% of them will happen. We call the event (4) hit. The prediction interval () generalize to linear regreion with normally ditributed error, and on-line hit remain independent in thi general etting. Even though formula for thee linear-regreion prediction interval appear in textbook, the independence of their on-line hit wa not noted prior to our work [28]. Like Fiher, the textbook author did not have the on-line etting in mind. They imagined jut one prediction being made in each cae where data i accumulated. We will return to the generalization to linear regreion in There we will derive the textbook interval a conformal prediction region within the on-line Gauian linear model, an on-line compreion model that ue lightly weaker aumption than the claical aumption of independent and normally ditributed error. (4) 5

9 2.2 Confidence ay le than probability. Neyman notion of confidence look at a procedure before obervation are made. Before any of the z i are oberved, the event (4) involve multiple uncertaintie: z n, n, and z n are all uncertain. The probability that thee three quantitie will turn out o that (4) hold i 95%. We might ak for more than thi. It i after we oberve the firt n example that we calculate z n and n and then calculate the interval (), and we would like to be able to ay at thi point that there i till a 95% probability that z n will be in (). But thi, it eem, i aking for too much. The aumption we have made are inufficient to enable u to find a numerical probability for (4) that will be valid at thi late date. In theory there i a conditional probability for (4) given z,..., z n, but it involve the unknown mean and variance of the normal ditribution. Perhap the matter i bet undertood from the game-theoretic point of view. A probability can be thought of a an offer to bet. A 95% probability, for example, i an offer to take either ide of a bet at 9 to odd. The probability i valid if the offer doe not put the peron making it at a diadvantage, inamuch a a long equence of equally reaonable offer will not allow an opponent to multiply the capital he or he rik by a large factor [24]. When we aume a probability model (uch a the normal model we jut ued or the on-line compreion model we will tudy later), we are auming that the model probabilitie are valid in thi ene before any example are oberved. Matter may be different afterward. In general, a 95% conformal predictor i a rule for uing the preceding example (x, y ),..., (x n, y n ) and a new object x n to give a et, ay Γ 0.05 ((x, y ),..., (x n, y n ), x n ), (5) that we predict will contain y n. If the predictor i valid, the prediction y n Γ 0.05 ((x, y ),..., (x n, y n ), x n ) will have a 95% probability before any of the example are oberved, and it will be afe, at that point, to offer 9 to odd on it. But after we oberve (x, y ),..., (x n, y n ) and x n and calculate the et (5), we may want to withdraw the offer. Particularly triking intance of thi phenomenon can arie in the cae of claification, where there are only finitely many poible label. We will ee one uch intance in 4.3., where we conider a claification problem in which there are only two poible label, and v. In thi cae, there are only four poibilitie for the prediction region:. Γ 0.05 ((x, y ),..., (x n, y n ), x n ) contain only. 2. Γ 0.05 ((x, y ),..., (x n, y n ), x n ) contain only v. 3. Γ 0.05 ((x, y ),..., (x n, y n ), x n ) contain both and v. 6

10 William S. Goett Ronald A. Fiher Jerzy Neyman Figure : Three influential tatitician. Goett, who worked a a tatitician for the Guinne brewery in Dublin, introduced the t-ditribution to Englih-peaking tatitician in 908 [4]. Fiher, whoe applied and theoretical work invigorated mathematical tatitic in the 920 and 930, refined, promoted, and extended Goett work. Neyman wa one of the mot influential leader in the ubequent movement to ue advanced probability theory to give tatitic a firmer foundation and further extend it application. 4. Γ 0.05 ((x, y ),..., (x n, y n ), x n ) i empty. The third and fourth cae can occur even though Γ 0.05 i valid. When the third cae happen, the prediction, though uninformative, i certain to be correct. When the fourth cae happen, the prediction i clearly wrong. Thee cae are conitent with the prediction being right 95% of the time. But when we ee them arie, we know whether the particular value of n i one of the 95% where we are right or the one of the 5% where we are wrong, and o the 95% will not remain valid a a probability defining betting odd. In the cae of normally ditributed example, Fiher called the 95% probability for z n being in the interval () a fiducial probability, and he eem to have believed that it would not be uceptible to a gambling opponent who know the firt n example (ee pp of [2]). But thi turned out not to be the cae [20]. For thi and related reaon, mot cientit who ue Fiher method have adopted the interpretation offered by Neyman, who wrote about confidence rather than fiducial probability and emphaized that a confidence level i a full-fledged probability only before we acquire data. It i the procedure or method, not the interval or region it produce when applied to particular data, that ha a 95% probability of being correct. Neyman concept of confidence ha endured in pite of it hortcoming. It i widely taught and ued in almot every branch of cience. Perhap it i epecially ueful in the on-line etting. It i ueful to know that 95% of our prediction are correct even if we cannot aert a full-fledged 95% probability for each prediction when we make it. 7

11 3 Exchangeability Conider variable z,..., z N. Suppoe that for any collection of N value, the N! different ordering are equally likely. Then we ay that z,..., z N are exchangeable. Exchangeability i cloely related to the idea that example are drawn independently from a probability ditribution. A we explain in the next ection, 4, it i the baic model for conformal prediction. In thi ection we look at the relationhip between exchangeability and independence and then give a backward-looking definition of exchangeability that can be undertood game-theoretically. We conclude with a law of large number for exchangeable equence, which will provide the bai for our confidence that our 95% prediction region are right 95% of the time. 3. Exchangeability and independence Although the definition of exchangeability we jut gave may be clear enough at an intuitive level, it ha two technical problem that make it inadequate a a formal mathematical definition: () in the cae of continuou ditribution, any pecific value for z,..., z N will have probability zero, and (2) in the cae of dicrete ditribution, two or more of the z i might take the ame value, and o a lit of poible value a,..., a N might contain fewer than n ditinct value. One way of avoiding thee technicalitie i to ue the concept of a permutation, a follow: Definition of exchangeability uing permutation. The variable z,..., z N are exchangeable if for every permutation τ of the integer,..., N, the variable w,..., w N, where w i = z τ(i), have the ame joint probability ditribution a z,..., z N. We can extend thi to a definition of exchangeability for an infinite equence of variable: z, z 2,... are exchangeable if z,..., z N are exchangeable for every N. Thi definition make it eay to ee that independent and identically ditributed random variable are exchangeable. Suppoe z,..., z N all take value from the ame example pace Z, all have the ame probability ditribution Q, and are independent. Then their joint ditribution atifie Pr (z A &... & z N A N ) = Q(A ) Q(A N ) (6) for any 2 ubet A,..., A N of Z, where Q(A) i the probability Q aign to an example being in A. Becaue permuting the factor Q(A n ) doe not change their product, and becaue a joint probability ditribution for z,..., z N i determined by the probabilitie it aign to event of the form {z A &... & z N A N }, thi make it clear that z,..., z N are exchangeable. 2 We leave aide technicalitie involving meaurability. 8

12 Pr(z = H & z 2 = H) Pr(z = H & z 2 = T) Pr(z = T & z 2 = H) Pr(z = T & z 2 = T) Table : Example of exchangeability. We conider variable z and z 2, each of which come out H or T. Exchangeability require only that Pr(z = H & z 2 = T) = Pr(z = T & z 2 = H). Three example of ditribution for z and z 2 with thi property are hown. On the left, z and z 2 are independent and identically ditributed; both come out H with probability 0.9. The middle example i obtained by averaging thi ditribution with the ditribution in which the two variable are again independent and identically ditributed but T probability i 0.9. The ditribution on the right, in contrat, cannot be obtained by averaging ditribution under which the variable are independent and identically ditributed. Example of thi lat type diappear a we ak for a larger and larger number of variable to be exchangeable. Exchangeability implie that variable have the ame ditribution. On the other hand, exchangeable variable need not be independent. Indeed, when we average two or more ditinct joint probability ditribution under which variable are independent, we uually get a joint probability ditribution under which they are exchangeable (averaging preerve exchangeability) but not independent (averaging uually doe not preerve independence). According to a famou theorem by de Finetti, an exchangeable joint ditribution for an infinite equence of ditinct variable i exchangeable only if it i a mixture of joint ditribution under which the variable are independent [5]. A Table how, the picture i more complicated in the finite cae. 3.2 Backward-looking definition of exchangeability Another way of defining exchangeability look backward from a ituation where we know the unordered value of z,..., z N. Suppoe Joe ha oberved z,..., z N. He write each value on a tile reembling thoe ued in Scrabble c, put the N tile in a bag, hake the bag, and give it to Bill to inpect. Bill ee the N value (ome poibly equal to each other) without knowing their original order. Bill alo know the joint probability ditribution for z,..., z N. So he obtain probabilitie for the ordering of the tile by conditioning thi joint ditribution on hi knowledge of the bag. The joint ditribution i exchangeable if and only if thee conditional probabilitie are the ame a the probabilitie for the reult of ordering the tile by ucceively drawing them at random from the bag without replacement. 9

13 Figure 2: Ordering the tile. Joe give Bill a bag containing five tile, and Bill arrange them to form the lit Bill can calculate conditional probabilitie for which z i had which of the five value. Hi conditional probability for z 5 = 4, for example, i 2/5. There are (5!)/(2!)(2!) = 30 way of aigning the five value to the five variable; (z, z 2, z 3, z 4, z 5 ) = (4, 3, 4, 7, 7) i one of thee, and they all have the ame probability, /30. To make thi into a definition of exchangeability, we formalize the notion of a bag. A bag (or multiet, a it i ometime called) i a collection of element in which repetition i allowed. It i like a et inamuch a it element are unordered but like a lit inamuch a an element can occur more than once. We write a,..., a N for the bag obtained from the lit a,..., a N by removing information about the ordering. Here are three equivalent condition on the joint ditribution of a equence of random variable z,..., z N, any of which can be taken a the definition of exchangeability.. For any bag B of ize N, and for any example a,..., a N, Pr (z = a &... & z N = a N z,..., z N = B) i equal to the probability that ucceive random drawing from the bag B without replacement produce firt a N, then a N, and o on, until the lat element remaining in the bag i a. 2. For any n, n N, z n i independent of z n+,..., z N given the bag z,..., z n and for any bag B of ize n, Pr (z n = a z,..., z n = B) = k n, (7) where k i the number of time a occur in B. 3. For any bag B of ize N, and for any example a,..., a N, Pr (z = a &... & z N = a N z,..., z N = B) { n! n k! = N! if B = a,..., a N 0 if B a,..., a N, (8) where k i the number of ditinct value among the a i, and n,..., n k are the repective number of time they occur. (If the a i are all ditinct, the expreion n! n k!/(n!) reduce to /(N!).) 0

14 z z 2 z N z N z z, z 2 z,..., z N z,..., z N Figure 3: Backward probabilitie, tep by tep. The two arrow backward from each bag z,..., z n ymbolize drawing an example z n out at random, leaving the maller bag z,..., z n. The probabilitie for the reult of the drawing are given by (7). Reader familiar with Baye net [4] will recognize thi diagram a an example; conditional on each variable, a joint probability ditribution i given for it children (the variable to which arrow from it point), and given the variable, it decendant are independent of it ancetor. We leave it to the reader to verify that thee three condition are equivalent to each other. The econd condition, which we will emphaize, i repreented pictorially in Figure 3. The backward-looking condition are alo equivalent to the definition of exchangeability uing permutation given on p. 8. Thi equivalence i elementary in the cae where every poible equence of value a,..., a n ha poitive probability. But complication arie when thi probability i zero, becaue the conditional probability on the left-hand ide of (8) i then defined only with probability one by the joint ditribution. We do not explore thee complication here. 3.3 The betting interpretation of exchangeability The framework for probability developed in [24] formalize claical reult of probability theory, uch a the law of large number, a theorem of game theory: a bettor can multiply the capital he rik by a large factor if thee reult do not hold. Thi allow u to expre the empirical interpretation of given probabilitie in term of betting, uing what we call Cournot principle: the odd determined by the probabilitie will not allow a bettor to multiply the capital he or he rik by a large factor [23]. By applying thi idea to the equence of probabilitie (7), we obtain a betting interpretation of exchangeability. Think of Joe and Bill a two player in a game that move backward from point N in Figure 3. At each tep, Joe provide new information and Bill bet. Deignate by K N the total capital Bill rik. He begin with thi capital at N, and at each tep n he bet on what z n will turn out to be. When he bet at tep n, he cannot rik loing more than he ha at that point (becaue he i not riking more than K N in the whole game), but otherwie he can bet a much a he want for or againt each poible value a for z n at the odd (k/n) : ( k/n), where k i the number of element in the

15 current bag equal to a. For brevity, we write B n for the bag z,..., z n, and for implicity, we et the initial capital K N equal to $. Thi give the following protocol: The Backward-Looking Betting Protocol Player: Joe, Bill K N :=. Joe announce a bag B N of ize N. FOR n = N, N,..., 2, Bill bet on z n at odd et by (7). Joe announce z n B n. K n := K n + Bill net gain. B n := B n \ z n. Contraint: Bill mut move o that hi capital K n will be nonnegative for all n no matter how Joe move. Our betting interpretation of exchangeability i that Bill will not multiply hi initial capital K N by a large factor in thi game. The permutation definition of exchangeability doe not lead to an equally imple betting interpretation, becaue the probabilitie for z,..., z N to which the permutation definition refer are not determined by the mere aumption of exchangeability. 3.4 A law of large number for exchangeable equence A we noted when we tudied Fiher prediction interval in 2..3, the validity of on-line prediction require more than having a high probability of a hit for each individual prediction. We alo need a law of large number, o that we can conclude that a high proportion of the high-probability prediction will be correct. A we how in A.3, the ucceive hit in the cae of Fiher region predictor are independent, o that the uual law of large number applie. What can we ay in the cae of conformal prediction under exchangeability? Suppoe z,..., z N are exchangeable, drawn from an example pace Z. In thi context, we adopt the following definition. An event E i an n-event, where n N, if it happening or failing i determined by the value of z n and the value of the bag z,..., z n. An n-event E i ɛ-rare if Pr(E z,..., z n ) ɛ. (9) The left-hand ide of the inequality (9) i a random variable, becaue the bag z,..., z n i random. The inequality ay that thi random variable never exceed ɛ. A we will ee in the next ection, the ucceive error for a conformal predictor are ɛ-rare n-event. So the validity of conformal prediction follow from the following informal propoition. 2

16 Informal Propoition Suppoe N i large, and the variable z,..., z N are exchangeable. Suppoe E n i an ɛ-rare n-event for n =,..., N. Then the law of large number applie; with very high probability, no more than approximately the fraction ɛ of the event E,..., E N will happen. In Appendix A, we formalize thi informal propoition in two way: claically and game-theoretically. The claical approach appeal to the claical weak law of large number, which tell u that if E,..., E N are mutually independent and each have probability exactly ɛ, and N i ufficiently large, then there i a very high probability that the fraction of the event that happen will be cloe to ɛ. We how in A. that if (9) hold with equality, then E n are mutually independent and each of them ha unconditional probability ɛ. Having the inequality intead of equality mean that the E n are even le likely to happen, and thi will not revere the concluion that few of them will happen. The game-theoretic approach i more traightforward, becaue the gametheoretic verion law of large number doe not require independence or exact level of probability. In the game-theoretic framework, the only quetion i whether the probabilitie pecified for ucceive event are rate at which a bettor can place ucceive bet. The Backward-Looking Betting Protocol ay that thi i the cae for ɛ-rare n-event. A Bill move through the protocol from N to, he i allowed to bet againt each error E n at a rate correponding to it having probability ɛ or le. So the game-theoretic weak law of large number ([24], pp ) applie directly. Becaue the game-theoretic framework i not well known, we tate and prove thi law of large number, pecialized to the Backward-Looking Betting Protocol, in A.2. 4 Conformal prediction under exchangeability We are now in a poition to tate the conformal algorithm under exchangeability and explain why it produce valid neted prediction region. We ditinguih two cae of on-line prediction. In both cae, we oberve example z,..., z N one after the other and repeatedly predict what we will oberve next. But in the econd cae we have more to go on when we make each prediction.. Prediction from old example alone. Jut before oberving z n, we predict it baed on the previou example, z,..., z n. 2. Prediction uing feature of the new object. Each example z i conit of an object x i and a label y i. In ymbol: z i = (x i, y i ). We oberve in equence x, y,..., x N, y N. Jut before oberving y n, we predict it baed on what we have oberved o far, x n and the previou example z,..., z n. Prediction from old example may eem relatively unintereting. It can be conidered a pecial cae of prediction uing feature x n of new example the cae in which the x n provide no information, and thi pecial cae we may 3

17 have too little information to make ueful prediction. But it implicity make prediction with old example alone advantageou a a etting for explaining the conformal algorithm, and a we will ee, it i then traightforward to take account of the new information x n. Conformal prediction require that we firt chooe a nonconformity meaure, which meaure how different a new example i from old example. In 4., we explain how nonconformity meaure can be obtained from method of point prediction. In 4.2, we tate and illutrate the conformal algorithm for predicting new example from old example alone. In 4.3, we generalize to prediction with the help of feature of a new example. In 4.4, we explain why conformal prediction produce the bet poible valid neted prediction region under exchangeability. Finally, in 4.5 we dicu the implication of the failure of the aumption of exchangeability. For ome reader, the implicity of the conformal algorithm may be obcured by it generality and the cope of our preliminary dicuion of nonconformity meaure. We encourage uch reader to look firt at 4.2., 4.3., and 4.3.2, which provide largely elf-contained account of the algorithm a it applie to ome mall dataet. 4. Nonconformity meaure The tarting point for conformal prediction i what we call a nonconformity meaure, a real-valued function A(B, z) that meaure how different an example z i from the example in a bag B. The conformal algorithm aume that a nonconformity meaure ha been choen. The algorithm will produce valid neted prediction region uing any real-valued function A(B, z) a the nonconformity meaure. But the prediction region will be efficient (mall) only if A(B, z) meaure well how different z i from the example in B. A method ẑ(b) for obtaining a point prediction ẑ for a new example from a bag B of old example uually lead naturally to a nonconformity meaure A. In many cae, we only need to add a way of meauring the ditance d(z, z ) between two example. Then we define A by A(B, z) := d(ẑ(b), z). (0) The prediction region produced by the conformal algorithm do not change when the nonconformity meaure A i tranformed monotonically. If A i nonnegative, for example, replacing A with A 2 will make no difference. Conequently, the choice of the ditance meaure d(z, z ) i relatively unimportant. The important tep in determining the nonconformity meaure A i chooing the point predictor ẑ(b). To be more concrete, uppoe the example are real number, and write z B for the average of the number in B. If we take thi average a our point predictor ẑ(b), and we meaure the ditance between two real number by the abolute value of their difference, then (0) become A(B, z) := z B z. () 4

18 If we ue the median of the number in B intead of their average a ẑ(b), we get a different nonconformity meaure, which will produce different prediction region when we ue the conformal algorithm. On the other hand, a we have already aid, it will make no difference if we replace the abolute difference d(z, z ) = z z with the quared difference d(z, z ) = (z z ) 2, thu quaring A. We can alo vary () by including the new example in the average: A(B, z) := (average of z and all the example in B) z. (2) Thi reult in the ame prediction region a (), becaue if B ha n element, then (average of z and all the example in B) z = nz B + z n + z = n n + z B z, and a we have aid, conformal prediction region are not changed by a monotonic tranformation of the nonconformity meaure. In the numerical example that we give in 4.2. below, we ue (2) a our nonconformity meaure. When we turn to the cae where feature of a new object help u predict a new label, we will conider, among other, the following two nonconformity meaure: Ditance to the nearet neighbor for claification. Suppoe B = z,..., z n, where each z i conit of a number x i and a nonnumerical label y i. Again we oberve x but not y for a new example z = (x, y). The nearetneighbor method find the x i cloet to x and ue it label y i a our prediction of y. If there are only two label, or if there i no natural way to meaure the ditance between label, we cannot meaure how wrong the prediction i; it i imply right or wrong. But it i natural to meaure the nonconformity of the new example (x, y) to the old example (x i, y i ) by comparing x ditance to old object with the ame label to it ditance to old object with a different label. For example, we can et A(B, z) : = min{ x i x : i n, y i = y} min{ x i x : i n, y i y} ditance to z nearet neighbor in B with the ame label = ditance to z nearet neighbor in B with a different label. (3) Ditance to a regreion line. Suppoe B = (x, y ),..., (x l, y l ), where the x i and y i are number. The mot common way of fitting a line to uch pair of number i to calculate the average x l := l x j and y l := j= l y j, j= 5

19 and then the coefficient l j= b l = j x l )y j l j= (x j x l ) 2 and a l = y l b l x l. Thi give the leat-quare line y = a l + b l x. The coefficient a l and b l are not affected if we change the order of the z i ; they depend only on the bag B. If we oberve a bag B = z,..., z n of example of the form z i = (x i, y i ) and alo x but not y for a new example z = (x, y), then the leat-quare prediction of y i ŷ = a n + b n x. (4) We can ue the error in thi prediction a a nonconformity meaure: A(B, z) := y ŷ = y (a n + b n x). We can obtain other nonconformity meaure by uing other method to etimate a line. Alternatively, we can include the new example a one of the example ued to etimate the leat quare line or ome other regreion line. In thi cae, it i natural to write (x n, y n ) for the new example. Then a n and b n deignate the coefficient calculated from all n example, and we can ue y i (a n + b n x i ) (5) to meaure the nonconformity of each of the (x i, y i ) with the other. In general, the incluion of the new example implifie the implementation or at leat the explanation of the conformal algorithm. In the cae of leat quare, it doe not change the prediction region. 4.2 Conformal prediction from old example alone Suppoe we have choen a nonconformity meaure A for our problem. Given A, and given the aumption that the z i are exchangeable, we now define a valid prediction region γ ɛ (z,..., z n ) Z, where Z i the example pace. We do thi by giving an algorithm for deciding, for each z Z, whether z hould be included in the region. For implicity in tating thi algorithm, we proviionally ue the ymbol z n for z, a if we were auming that z n i in fact equal to z. 6

20 The Conformal Algorithm Uing Old Example Alone Input: Nonconformity meaure A, ignificance level ɛ, example z,..., z n, example z, Tak: Decide whether to include z in γ ɛ (z,..., z n ). Algorithm:. Proviionally et z n := z. 2. For i =,..., n, et α i := A( z,..., z n \ z i, z i ). 3. Set p z := number of i uch that i n and α i α n n 4. Include z in γ ɛ (z,..., z n ) if and only if p z > ɛ.. If Z ha only a few element, thi algorithm can be implemented in a brute-force way: calculate p z for every z Z. If Z ha many element, we will need ome other way of identifying the z atifying p z > ɛ. The number p z i the fraction of the example in z,..., z n, z that are at leat a different from the other a z i, in the ene meaured by A. So the algorithm tell u to form a prediction region coniting of the z that are not among the fraction ɛ mot out of place when they are added to the bag of old example. The definition of γ ɛ (z,..., z n ) can be framed a an application of the widely accepted Neyman-Pearon theory for hypothei teting and confidence interval [7]. In the Neyman-Pearon theory, we tet a hypothei H uing a random variable T that i likely to be large if H i fale. Once we oberve T = t, we calculate p H := Pr(T t H). We reject H at level ɛ if p H ɛ. Becaue thi happen under H with probability no more than ɛ, we can declare ɛ confidence that the true hypothei H i among thoe not rejected. Our procedure make thee choice of H and T : The hypothei H ay the bag of the firt n example i z,..., z n, z. The tet tatitic T i the random value of α n. Under H i.e., conditional on the bag z,..., z n, z, T i equally likely to come out equal to any of the α i. It oberved value i α n. So p H = Pr(T α n z,..., z n, z ) = p z. Since z,..., z n are known, rejecting the bag z,..., z n, z mean rejecting z n = z. So our ɛ confidence i in the et of z for which p z > ɛ. The region γ ɛ (z,..., z n ) for ucceive n are baed on overlapping obervation rather than independent obervation. But the ucceive error are ɛ-rare n-event. The event that our nth prediction i an error, z n / γ ɛ (z,..., z n ), i the event p zn ɛ. Thi i an n-event, becaue the value of p zn i determined by z n and the bag z,..., z n. It i ɛ-rare becaue it 7

21 i the event that α n i among a fraction ɛ or fewer of the α i that are trictly larger than all the other α i, and thi can have probability at mot ɛ when the α i are exchangeable. So it follow from Informal Propoition ( 3.4) that we can expect at leat ɛ of the γ ɛ (z,..., z n ), n =,..., N, to be correct Example: Predicting a number with an average In 2., we dicued Fiher 95% prediction interval for z n baed on z,..., z n, which i valid under the aumption that the z i are independent and normally ditributed. We ued it to predict z 20 when the firt 9 z i are 7, 20, 0, 7, 2, 5, 9, 22, 7, 9, 4, 22, 8, 7, 3, 2, 8, 5, 7. Taking into account our knowledge that the z i are all integer, we arrived at the 95% prediction that z 20 i an integer between 0 to 23, incluive. What can we predict about z 20 at the 95% level if we drop the aumption of normality and aume only exchangeability? To produce a 95% prediction interval valid under the exchangeability aumption alone, we reaon a follow. To decide whether to include a particular value z in the interval, we conider twenty number that depend on z: Firt, the deviation of z from the average of it and the other 9 number. Becaue the um of the 9 i 34, thi i 34 + z z 20 = 34 9z. (6) 20 Then, for i =,..., 9, the deviation of z i from thi ame average. Thi i 34 + z z i = z 20z i. (7) Under the hypothei that z i the actual value of z n, thee 20 number are exchangeable. Each of them i a likely a the other to be the larget. So there i at leat a 95% (9 in 20) chance that (6) will not exceed the larget of the 9 number in (7). The larget of the 9 z i being 22 and the mallet 0, we can write thi condition a 34 9z max { 34 + z (20 22), 34 + z (20 0) }, (8) which reduce to 0 z Taking into account that z 20 i an integer, our 95% prediction i that it will be an integer between 0 and 23, incluive. Thi i exactly the ame prediction we obtained by Fiher method. We have lot nothing by weakening the aumption that the z i are independent and normally ditributed to the aumption that they are exchangeable. But we are till baing our prediction region on the average of old example, which i an optimal etimator in variou repect under the aumption of normality. 8

22 4.2.2 Are we complicating the tory unnecearily? The reader may feel that we are vacillating about whether to include the new example in the bag with which we are comparing it. In our tatement of the conformal algorithm, we define the nonconformity core by α i := A( z,..., z n \ z i, z i ), (9) apparently ignaling that we do not want to include z i in the bag to which it i compared. But then we ue the nonconformity meaure A(B, z) := (average of z and all the example in B) z, which eem to put z back in the bag, reducing (9) to n j= α i = z j z i. n We could have reached thi point more eaily by writing α i := A( z,..., z n, z i ) (20) in the conformal algorithm and uing A(B, z) := z B z. The two way of defining nonconformity core, (9) and (20), are equivalent, inamuch a whatever we can get with one of them we can get from the other by changing the nonconformity meaure. In thi cae, (20) might be more convenient. But we will ee other cae where (9) i more convenient. We alo have another reaon for uing (9). It i the form that generalize, a we will ee in 5, to on-line compreion model. 4.3 Conformal prediction uing a new object Now we turn to the cae where our example pace Z i of the form Z = X Y. We call X the object pace, Y the label pace. We oberve in equence example z,..., z N, where z i = (x i, y i ). At the point where we have oberved z,..., z n, x n = (x, y ),..., (x n, y n ), x n, we want to predict y n by giving a prediction region Γ ɛ (z,..., z n, x n ) Y that i valid at the ( ɛ) level. A in the pecial cae where the x i are abent, we tart with a nonconformity meaure A(B, z). We define the prediction region by giving an algorithm for deciding, for each y Y, whether y hould be included in the region. For implicity in tating thi algorithm, we proviionally ue the ymbol z n for (x n, y), a if we were auming that y n i in fact equal to y. 9

23 The Conformal Algorithm Input: Nonconformity meaure A, ignificance level ɛ, example z,..., z n, object x n, label y Tak: Decide whether to include y in Γ ɛ (z,..., z n, x n ). Algorithm:. Proviionally et z n := (x n, y). 2. For i =,..., n, et α i := A( z,..., z n \ z i, z i ). 3. Set p y := #{i =,..., n α i α n } n 4. Include y in Γ ɛ (z,..., z n, x n ) if and only if p y > ɛ. Thi differ only lightly from the conformal algorithm uing old example alone (p. 7). Now we write p y intead of p z, and we ay that we are including y in Γ ɛ (z,..., z n, x n ) intead of aying that we are including z in γ ɛ (z,..., z n ). To ee that thi algorithm produce valid prediction region, it uffice to ee that it conit of the algorithm for old example alone together with a further tep that doe not change the frequency of hit. We know that the region the old algorithm produce, γ ɛ (z,..., z n ) Z, (2) contain the new example z n = (x n, y n ) at leat 95% of the time. Once we know x n, we can rule out all z = (x, y) in (2) with x x n. The y not ruled out, thoe uch that (x n, y) i in (2), are preciely thoe in the et. Γ ɛ (z,..., z n, x n ) Y (22) produced by our new algorithm. Having (x n, y n ) in (2) ɛ of the time i equivalent to having y n in (22) ɛ of the time Example: Claifying iri flower In 936 [], R. A. Fiher ued dicriminant analyi to ditinguih different pecie of iri on the bai of meaurement of their flower. The data he ued included meaurement by Edgar Anderon of flower from 50 plant each of two pecie, iri etoa and iri vericolor. Two of the meaurement, epal length and petal width, are plotted in Figure 4. To illutrate how the conformal algorithm can be ued for claification, we have randomly choen 25 of the 00 plant. The epal length and pecie for the firt 24 of them are lited in Table 2 and plotted in Figure 5. The 25th plant in the ample ha epal length 6.8. On the bai of thi information, would you claify it a etoa or vericolor, and how confident would you be in the claification? Becaue 6.8 i the longet epal in the ample, nearly any reaonable method will claify the plant a vericolor, and thi i in fact the correct anwer. But the appropriate level of confidence i not o obviou. 20