A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY


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1 Theoy and Application o Categoie, Vol. 9, No. 16, 014, pp A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY JOHN C. BAEZ AND TOBIAS FRITZ Abtact. We give a new chaacteization o elative entopy, alo known a the Kullback Leible divegence. We ue a numbe o inteeting categoie elated to pobability theoy. In paticula, we conide a categoy FinStat whee an object i a inite et euipped with a pobability ditibution, while a mophim i a meauepeeving unction : X Y togethe with a tochatic ight invee : Y X. The unction can be thought o a a meauement poce, while povide a hypothei about the tate o the meaued ytem given the eult o a meauement. Given thi data we can deine the entopy o the pobability ditibution on X elative to the pio given by puhing the pobability ditibution on Y owad along. We ay that i optimal i thee ditibution agee. We how that any convex linea, lowe emicontinuou uncto om FinStat to the additive monoid [0, ] which vanihe when i optimal mut be a cala multiple o thi elative entopy. Ou poo i independent o all ealie chaacteization, but inpied by the wok o Petz. 1. Intoduction Thi pape give a new chaacteization o the concept o elative entopy, alo known a elative inomation, inomation gain o KullbackLeible divegence. Wheneve we have two pobability ditibution p and on the ame inite et X, we deine the inomation o elative to p a: S, p = x x ln p x x X Hee we et x ln x /p x eual to when p x = 0, unle x i alo zeo, in which cae we et it eual to 0. Relative entopy thu take value in [0, ]. Intuitively peaking, S, p i the expected amount o inomation gained when we dicove the pobability ditibution i eally, when we had thought it wa p. We hould We thank Ryzad Kotecki and Rob Spekken o dicuion and an unintended beneit. TF wa uppoted by Peimete Intitute o Theoetical Phyic though a gant om the John Templeton oundation. Reeach at Peimete Intitute i uppoted by the Govenment o Canada though Induty Canada and by the Povince o Ontaio though the Minity o Reeach and Innovation. JB thank the Cente o Quantum Technologie o thei uppot. Received by the edito and, in evied om, Tanmitted by Tom Leinte. Publihed on Mathematic Subject Claiication: Pimay 94A17, Seconday 6F15, 18B99. Key wod and phae: elative entopy, KullbackLeible divegence, meaue o inomation, categoical pobability theoy. c John C. Baez and Tobia Fitz, 014. Pemiion to copy o pivate ue ganted. 4
2 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 43 think o p a a pio. When we take p to be the uniom ditibution on X, elative entopy educe to the odinay Shannon entopy, up to a ign and an additive contant. The advantage o elative entopy i that it make the ole o the pio explicit. Since Bayeian pobability theoy emphaize the ole o the pio, elative entopy natually lend itel to a Bayeian intepetation [3]. Ou goal hee i to make thi pecie in a mathematical chaacteization o elative entopy. We do thi uing a categoy FinStat whee: an object X, conit o a inite et X and a pobability ditibution x x on that et; a mophim, : X, Y, conit o a meauepeeving unction om X to Y, togethe with a pobability ditibution x xy on X o each element y Y with the popety that xy = 0 unle x = y. We can think o an object o FinStat a a ytem with ome inite et o tate togethe with a pobability ditibution on it tate. A mophim, : X, Y, then conit o two pat. Fit, thee i a deteminitic meauement poce : X Y mapping tate o ome ytem being meaued to tate o a meauement appaatu. The condition that be meauepeeving ay that the pobability that the appaatu wind up in ome tate y Y i the um o the pobabilitie o tate o X leading to that outcome: y = x. x: x=y Second, thee i a hypothei : an aumption about the pobability xy that the ytem being meaued i in the tate x given any meauement outcome y Y. We aume that thi pobability vanihe unle x = y, a we would expect om a hypothei made by omeone who knew the behavio o the meauement appaatu. Suppoe we have any mophim, : X, Y, in FinStat. Fom thi we obtain two pobability ditibution on the tate o the ytem being meaued. Fit, we have the pobability ditibution p: X R given by p x = x x x. 1 Thi i ou pio, given ou hypothei and the pobability ditibution o meauement outcome. Second, we have the tue pobability ditibution : X R. It ollow that any mophim in FinStat ha a elative entopy S, p aociated to it. Thi i the expected amount o inomation we gain when we update ou pio p to. In act, thi way o aigning elative entopie to mophim deine a uncto RE: FinStat [0, ] whee we ue [0, ] to denote the categoy with one object, the nonnegative eal numbe togethe with a mophim, and addition a compoition. Moe peciely, i, : X, Y, i any mophim in FinStat, we deine RE, = S, p
3 44 JOHN C. BAEZ AND TOBIAS FRITZ whee the pio p i deined a in Euation 1. The act that RE i a uncto i nontivial and athe inteeting. It ay that given any compoable pai o meauement pocee: X,, Y, g,t Z, u the elative entopy o thei compoite i the um o the elative entopie o the two pat: REg, t, = REg, t + RE,. We pove that RE i a uncto in Section 3. Howeve, we go much uthe: we chaacteize elative entopy by aying that up to a contant multiple, RE i the uniue uncto om FinStat to [0, ] obeying thee eaonable condition. The it condition i that RE vanihe on mophim, : X, Y, whee the hypothei i optimal. By thi, we mean that Euation 1 give a pio p eual to the tue pobability ditibution on the tate o the ytem being meaued. The econd condition i that RE i lowe emicontinuou. The et P X o pobability ditibution on a inite et X natually ha the topology o an n 1implex when X ha n element. The et [0, ] can be given the topology induced by the uual ode on thi et, and it i then homeomophic to a cloed inteval. Howeve, with thee topologie, the elative entopy doe not deine a continuou unction S : P X P X [0, ], p S, p. The poblem i that S, p = x ln x X x p x and x ln x /p x eual when p x = 0 and x > 0, but 0 when p x = x = 0. So, it tun out that S i only lowe emicontinuou, meaning that it can uddenly jump down, but not up. Moe peciely, i p i, i P X ae euence with p i p, i, then S, p lim in i Si, p i. In Section 3 we give the et o mophim in FinStat a topology, and how that with thi topology, RE map mophim to mophim in a lowe emicontinuou way. The thid condition i that RE i convex linea. In Section 3 we decibe how to take convex linea combination o mophim in FinStat. The uncto RE i convex linea in the ene that it map any convex linea combination o mophim in FinStat to the coeponding convex linea combination o numbe in [0, ]. Intuitively, thi mean that i we lip a pobabilityλ coin to decide whethe to peom one meauement poce o anothe, the expected inomation gained i λ time the expected inomation gain o the it poce plu 1 λ time the expected inomation gain o the econd. Ou main eult i Theoem 3.1: any lowe emicontinuou, convex linea uncto F : FinStat [0, ]
4 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 45 that vanihe on mophim with an optimal hypothei mut eual ome contant time the elative entopy. In othe wod, thee exit ome contant c [0, ] uch that F, = c RE, o any mophim, : X, p Y, in FinStat. Thi theoem, and it poo, wa inpied by eult o Petz [8], who ought to chaacteize elative entopy both in the claical cae dicued hee and in the moe geneal uantum etting. Ou oiginal intent wa meely to expe hi eult in a moe categoytheoetic amewok. Unotunately hi wok contained a law, which we had to epai. A a eult, ou poo i now elcontained. Fo detail, ee the emak ate Theoem 5.. Ou chaacteization o elative entopy implicitly elie on topological categoie and on the opead whoe opeation ae convex linea combination. Howeve, ince thee tuctue ae not tictly neceay o tating o poving ou eult, and they may be unamilia to ome eade, we dicu them only in Appendix A and Appendix B.. The categoie in uetion FinStoch. To decibe the categoie ued in thi pape, we need to tat with a wod on the categoy o inite et and tochatic map. A tochatic map : X Y i dieent om an odinay unction, becaue intead o aigning a uniue element o Y to each element o X, it aign a pobability ditibution on Y to each element o X. Thu x i not a peciic element o Y, but intead ha a pobability o taking on dieent value. Thi i why we ue a wiggly aow to denote a tochatic map. Moe omally:.1. Deinition. Given inite et X and Y, a tochatic map : X Y aign a eal numbe yx to each pai x X, y Y in uch a way that ixing any element x, the numbe yx om a pobability ditibution on Y. We call yx the pobability o y given x. In moe detail, we euie that the numbe yx obey: yx 0 o all x X, y Y, y Y yx = 1 o all x X. Note that we can think o : X Y a a Y Xhaped matix o numbe. A matix obeying the two popetie above i called tochatic. Thi viewpoint i nice becaue it educe the poblem o compoing tochatic map to matix multiplication. It i eay to
5 46 JOHN C. BAEZ AND TOBIAS FRITZ check that multiplying two tochatic matice give a tochatic matix. So, we deine the compoite o tochatic map : X Y and g : Y Z by g zx = y Y g zy yx. Since matix multiplication i aociative and identity matice ae tochatic, thi contuction give a categoy:.. Deinition. Let FinStoch be the categoy o inite et and tochatic map between them. We ae eticting attention to inite et meely to keep the dicuion imple and avoid iue o convegence. It would be inteeting to genealize all ou wok to moe geneal pobability pace. FinPob. Chooe any 1element et and call it 1. A unction : 1 X i jut a point o X. But a tochatic map : 1 X i omething moe inteeting: it i a pobability ditibution on X. We ue the tem inite pobability meaue pace to mean a inite et with a pobability ditibution on it. A we have jut een, thee i a vey uick way to decibe uch a thing within FinStoch: 1 X Thi give a uick way to think about a meauepeeving unction between inite pobability meaue pace! It i imply a commutative tiangle like thi: 1 X Y Note that the hoizontal aow : X Y i not wiggly. The taight aow mean it i an honet unction, not a tochatic map. But a unction can be een a a pecial cae o a tochatic map. So it make ene to compoe a taight aow with a wiggly aow and the eult i, in geneal, a wiggly aow. I we then demand that the above tiangle commute, thi ay that the unction : X Y i meauepeeving. We now wok though the detail. Fit: how can we ee a unction a a pecial cae o a tochatic map? A unction : X Y give a matix o numbe yx = δ y x
6 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 47 whee δ i the Konecke delta. Thi matix i tochatic, and it deine a tochatic map ending each point x X to the pobability ditibution uppoted at x. Given thi, we can ee what the commutativity o the above tiangle mean. I we ue x to tand o the pobability that : 1 X aign to each element x X, and imilaly o y, then the tiangle commute i and only i y = x X δ y x x o in othe wod: y = x: x=y In thi ituation we ay p i puhed owad along, and that i a meauepeeving unction. So, we have ued FinStoch to decibe anothe impotant categoy:.3. Deinition. Let FinPob be the categoy o inite pobability meaue pace and meauepeeving unction between them. Anothe vaiation may be ueul at time: 1 x X Y A commuting tiangle like thi i a meauepeeving tochatic map. In othe wod, give a pobability meaue on X, give a pobability meaue on Y, and : X Y i a tochatic map that i meauepeeving in the ollowing ene: y = x X yx x. FinStat. The categoy we need o ou chaacteization o elative entopy i a bit moe ubtle. In thi categoy, an object i a inite pobability meaue pace: 1 X
7 48 JOHN C. BAEZ AND TOBIAS FRITZ but a mophim look like thi: 1 X Y = = 1 Y The diagam need not commute, but the two euation hown mut hold. The it euation ay that : X Y i a meauepeeving unction. In othe wod, thi tiangle, which we have een beoe, commute: 1 X Y The econd euation ay that i the identity, o in othe wod, i a ection o. Thi euie a bit o dicuion. We can think o X a the et o tate o ome ytem, while Y i a et o poible tate o ome othe ytem: a meauing appaatu. The unction i a meauement poce. One meaue the ytem uing, and i the ytem i in any tate x X the meauing appaatu goe into the tate x. The pobability ditibution give the pobability that the ytem i in any given tate, while give the pobability that the meauing appaatu end up in any given tate ate a meauement i made. Unde thi intepetation, we think o the tochatic map a a hypothei about the ytem tate given the tate o the meauing appaatu. I one meaue the ytem and the appaatu goe into the tate y Y, thi hypothei aet that the ytem i in the tate x with pobability xy. The euation = 1 Y ay that i the meauing appaatu end up in ome tate y Y, ou hypothei aign a nonzeo pobability only to tate o the meaued ytem o which a meauement actually lead to thi tate y:.4. Lemma. I : X Y i a unction between inite et and : Y X i a tochatic map, then = 1 Y i and only o all y Y, xy = 0 unle x = y.
8 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 49 Poo. The condition = 1 Y ay that o any ixed y, y Y, xy = δ y x xy = δ y y. x: x=y x X It ollow that the um at let vanihe i y y. I i tochatic, the tem in thi um ae nonnegative. So, xy mut be zeo i x = y and y y. Conveely, uppoe we have a tochatic map : Y X uch that xy = 0 unle x = y. Then o any y Y we have 1 = xy = xy = δ y x xy x X x: x=y x X while o y y we have 0 = xy = δ y x xy, x: x=y x X o o all y, y Y which ay that = 1 Y. δ y x xy = δ y y, x X It i alo woth noting that = 1 Y implie that i onto: i y Y wee not in the image o, we could not have xy = 1 x X a euied, ince xy = 0 unle x = y. So, the euation = 1 Y alo ule out the poibility that ou meauing appaatu ha extaneou tate that neve aie when we make a meauement. Thi i how we compoe mophim o the above ot: 1 u X Y t g Z = g = u = 1 Y g t = 1 Z We get a meauepeeving unction g : X Z and a tochatic map going back, t: Z X. It i eay to check that thee obey the euied euation: g = u
9 430 JOHN C. BAEZ AND TOBIAS FRITZ g t = 1 Z So, thi way o compoing mophim give a categoy, which we call FinStat, to allude to it ole in tatitical eaoning:.5. Deinition. Let FinStat be the categoy whee an object i a inite pobability meaue pace: 1 a mophim i a diagam X 1 obeying thee euation: and compoition i deined a above. X = = 1 Y FP. We have decibed how to think o a mophim in FinStat a coniting o a meauement poce and a hypothei, obeying two euation: 1 Y X Y We ay the hypothei i optimal i alo = = 1 Y =.
10 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 431 Conceptually, thi ay that i we take the pobability ditibution on ou obevation and ue it to ine a pobability ditibution o the ytem tate uing ou hypothei, we get the coect anwe:. Mathematically, it ay that thi diagam commute: 1 X In othe wod, i a meauepeeving tochatic map. It i eay to check that thi optimality popety i peeved by compoition o mophim. Hence thee i a ubcategoy o FinStat with all the ame object, but only mophim whee the hypothei i optimal:.6. Deinition. Let FP be the ubcategoy o FinStat whee an object i a inite pobability meaue pace 1 Y and a mophim i a diagam X 1 obeying thee euation: X = = 1 Y = The categoy FP wa intoduced by Leinte [5]. He gave it thi name o two eaon. Fit, it i a cloe elative o FinPob, whee a mophim look like thi: 1 Y X Y
11 43 JOHN C. BAEZ AND TOBIAS FRITZ We now explain the imilaitie and dieence between FP and FinPob by tudying the popetie o the ogetul uncto FP FinPob, which end evey mophim, to it undelying meauepeeving unction. Fo a mophim in FP, the condition on ae o tong that they completely detemine it, unle thee ae tate o the meauement appaatu that happen with pobability zeo: that i, unle thee ae y Y with y = 0. To ee thi, note that ay that = xy y = x y Y o any choice o x X. But we have aleady een in Lemma.4 that xy = 0 unle x = y, o the um ha jut one tem, and the euation ay xy y = x whee y = x. We can olve thi o xy unle y = 0. Futhemoe, we have aleady een that evey y Y i o the om x o ome x X. Thu, o a mophim, : X, Y, in FP, we can olve o in tem o the othe data unle thee exit y Y with y = 0. Except o thi pecial cae, a mophim in FP i jut a mophim in FinPob. But in thi pecial cae, a mophim in FP ha a little exta inomation: an abitay pobability ditibution on the invee image o each point y with y = 0. The point i that in FinStat, and thu FP, a hypothei mut povide a pobability o each tate o the ytem given a tate o the meauement appaatu, even o tate o the meauement appaatu that occu with pobability zeo. A moe mathematical way to decibe the ituation i that ou uncto FP FinPob i geneically ull and aithul: the unction FPX,, Y, FinPobX,, Y,, i a bijection i the uppot o i the whole et Y, which i the geneic ituation. The econd eaon Leinte called thi categoy FP i that it i eely omed om an opead called P. Thi i a topological opead whoe nay opeation ae pobability ditibution on the et {1,..., n}. Thee opeation decibe convex linea combination, o algeba o thi opead include convex ubet o R n, moe geneal convex pace [], and even moe. A Leinte explain [5], the categoy FP o moe peciely, an euivalent one i the ee Palgeba among categoie containing an intenal Palgeba. We will not need thi act hee, but it i woth mentioning that Leinte ued thi act to chaacteize entopy a a uncto om FP to [0,. He and the autho then ephaed thi in imple language [1], obtaining a chaacteization o entopy a a uncto om FinPob to [0,. The chaacteization o elative entopy in the cuent pape i a cloely elated eult. Howeve, the poo i completely dieent.
12 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY Chaacteizing entopy The theoem. We begin by tating ou main eult. Then we claiy ome o the tem involved and begin the poo Theoem. Relative entopy detemine a uncto X, RE: FinStat [0, ] Y, S, that i lowe emicontinuou, convex linea, and vanihe on mophim in the ubcategoy FP. Conveely, thee popetie chaacteize the uncto RE up to a cala multiple. In othe wod, i F i anothe uncto with thee popetie, then o ome 0 c we have F, = c RE, o all mophim, in FinStat. Hee we deine a = a = o 0 < a, but 0 = 0 = 0. In the et o thi ection we begin by decibing [0, ] a a categoy and checking that RE i a uncto. Then we decibe what it mean o the uncto RE to be lowe emicontinuou and convex linea, and check thee popetie. We potpone the had pat o the poo, in which we chaacteize RE up to a cala multiple by thee popetie, to Section 4. In what ollow, it will be ueul to have an explicit omula o S,. By deinition, S, = x x ln x x X We have x = y Y xy y, but by Lemma.4, xy = 0 unle x = y, o the um ha jut one tem: x = x x x and we obtain S, = x X x ln x x x x. 3 Functoiality. We make [0, ] into a monoid uing addition, whee we deine addition in the uual way o numbe in [0, and et + a = a + = o all a [0, ]. Thee i thu a categoy with one object and element o [0, ] a endomophim o thi object, with compoition o mophim given by addition. With a light abue o language we alo ue [0, ] to denote thi categoy.
13 434 JOHN C. BAEZ AND TOBIAS FRITZ 3.. Lemma. The map RE: FinStat [0, ] decibed in Theoem 3.1 i a uncto. Poo. Let X, Y, t g Z, u be a compoable pai o mophim in FinStat. Then the unctoiality o RE can be hown by epeated ue o Euation 3: RE g, t = S, t u = x x ln x X x x t x gx u gx = x x ln + x X x x x x X x ln = S, + y y ln t y Y y gy u gy = S, + S, t u = RE, + REg, t. Hee the main tep i, whee we have imply ineted 0 = x x ln 1 x + x x ln x. x t x gx u gx Thi i unpoblematic a long a x > 0 o all x. When thee ae x with x = 0, then we neceaily have x = 0 a well, and both x ln 1 x and x ln x actually vanih, o thi cae i alo ine. In the tep ate, we ue the act that o each y Y, y i the um o x ove all x with x = y. Lowe emicontinuity. Next we explain what it mean o a uncto to be lowe emicontinuou, and pove that RE ha thi popety. Thee i a way to think about emicontinuou uncto in tem o topological categoie, but thi i not eally neceay o ou wok, o we potpone it to Appendix A. Hee we take a moe impleminded appoach. I we ix two inite et X and Y, the et o all mophim, : X, Y, p in FinStat om a topological pace in a natual way. To ee thi, let P X = { : X [0, 1] : x = 1} x X
14 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 435 be the et o pobability ditibution on a inite et X. Thi i a ubet o a initedimenional eal vecto pace, o we give it the ubpace topology. With thi topology, P X i homeomophic to a implex. The et o tochatic map : Y X i alo a ubpace o a initedimenional eal vecto pace, namely the pace o matice R X Y, o we alo give it the ubpace topology. We then give P X P Y R X Y the poduct topology. The et o mophim, : X, Y, p in FinStat can be een a a ubpace o thi, and we give it the ubpace topology. We then ay: 3.3. Deinition. A uncto F : FinStat [0, ] i lowe emicontinuou i o any euence o mophim, i : X, i Y, i that convege to a mophim, : X, Y,, we have F, lim in i F, i. We could ue net intead o euence hee, but it would make no dieence. We can then check anothe pat o ou main theoem: 3.4. Lemma. The uncto RE: FinStat [0, ] decibed in Theoem 3.1 i lowe emicontinuou. Poo. Suppoe that, i : X, i Y, i i a euence o mophim in FinStat that convege to, : X, Y,. We need to how that S, lim in i Si, i i. I thee i no x X with x x x = 0 then thi i clea, ince all the elementay unction involved in the deinition o elative entopy ae continuou away om 0. I all x X with x x = 0 alo atiy x = 0, then S, i till inite ince none o thee x contibute to the um o S. In thi cae S i, i i may emain abitaily lage, even ininite a i. But the ineuality S, lim in i Si, i i emain tue. The ame agument applie i thee ae x X with x = 0, which implie x = 0. Finally, i thee ae x X with x x = 0 but x x > 0, then S, =. The above ineuality i till valid in thi cae. That lowe emicontinuity o elative entopy i an impotant popety wa aleady known to Petz; ee the cloing emak in [8]. Convex lineaity. Next we explain what it mean to ay that elative entopy give a convex linea uncto om FinPob to [0, ], and we pove thi i tue. In geneal, convex linea uncto go between convex categoie. Thee ae topological categoie euipped with an action o the opead P dicued by Leinte [5]. Since we do not need the geneal theoy hee, we potpone it to Appendix B.
15 436 JOHN C. BAEZ AND TOBIAS FRITZ Fit, note that thee i a way to take convex linea combination o object and mophim in FinPob. Let X, p and Y, be inite et euipped with pobability meaue, and let λ [0, 1]. Then thee i a pobability meaue λp 1 λ on the dijoint union X + Y, whoe value at a point x i given by { λp x i x X λp 1 λ x = 1 λ x i x Y. Given a pai o mophim in FinPob, thee i a uniue mophim : X, p X, p, g : Y, Y, λ 1 λg : X + Y, λp 1 λ X + Y, λp 1 λ that etict to on X and to g on Y. A imila contuction applie to FinStat. Given a pai o mophim X, p X, p Y, t g Y, in FinStat, we deine thei convex linea combination to be X + Y, λp 1 λ t λ 1 λg X + Y, λp 1 λ whee t: X + Y X + Y i the tochatic map which etict to on X and t on Y. A a tochatic matix, it i o blockdiagonal om. It i ight invee to λ 1 λg by contuction. We may alo deine convex linea combination o object and mophim in the categoy [0, ]. Since thi categoy ha only one object, thee i only one way to deine convex linea combination o object. Mophim in thi categoy ae element o the et [0, ]. We have aleady made thi et into a monoid uing addition. We can alo intoduce multiplication, deined in the uual way o numbe in [0,, and with 0a = a0 = 0 o all a [0, ]. Thi give meaning to the convex linea combination λa + 1 λb o two mophim a, b in [0, ]. Fo moe detail, ee Appendice A and B.
16 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY Deinition. A uncto F : FinStat [0, ] i convex linea i it peeve convex combination o object and mophim. Fo object thi euiement i tivial, o all thi eally mean i that o any pai o mophim, and g, t in FinStat and any λ [0, 1], we have F λ, 1 λg, t = λf, + 1 λf g, t Lemma. The uncto RE: FinStat [0, ] decibed in Theoem 3.1 i convex linea. Poo. Thi ollow om a diect computation: REλ, 1 λg, t = Sλp 1 λ, λ p 1 λt = λp x λp x ln + 1 λ y 1 λ x X x x λp y ln x t y Y y gy 1 λ gy = λ p x p x ln + 1 λ y ln x X x x p x t y Y y gy y = λsp, p + 1 λs, t = λ RE, + 1 λ REg, t 4. Poo o the theoem Now we pove the main pat o Theoem Lemma. Suppoe that a uncto F : FinStat [0, ] i lowe emicontinuou, convex linea, and vanihe on mophim in the ubcategoy FP. Then o ome 0 c we have F, = c RE, o all mophim, in FinStat. Poo. Let F : FinStat [0, ] be any uncto atiying thee hypothee. By unctoiality and the act that 0 i the only mophim in [0, ] with an invee, F vanihe on iomophim. Thu, given any commutative uae in FinStat whee the vetical
17 438 JOHN C. BAEZ AND TOBIAS FRITZ mophim ae iomophim: X, p Y, X, p Y, unctoiality implie that F take the ame value on the top and bottom mophim: F, = F,. So, in what ollow, we can eplace an object by an iomophic object without changing the value o F on mophim om o to thi object. Given any mophim in FinStat, complete it to a diagam o thi om: X, p Y,! Y! X 1, 1 Hee 1 denote any oneelement et euipped with the uniue pobability meaue 1, and! X : X 1 i the uniue unction, which i automatically meauepeeving ince p i aumed to be nomalized. Since thi diagam commute, and the mophim on the lowe ight lie in FP, we obtain F X, p Y, = F X, p 1, 1.! X In othe wod: the value o F on a mophim depend only on the two ditibution p and living on the domain o the mophim. Fo thi eaon, it i enough to pove the claim only o thoe mophim whoe codomain i 1, 1. We now conide the amily o ditibution α = α, 1 α, on a twoelement et = {0, 1}, and conide the unction gα = F, 1 1, 1 α 4
18 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 439 o α [0, 1]. Note that o all β [0, 1, thi uae in FinStat commute: 3, 1, 0, , 1 β 1 0, α1 β 1 αβ, 1, 0 α, 1, 0 αβ 1, 1 whee the let vetical mophim i in FP, while the top hoizontal mophim i the convex linea combination 1, 1 β 1, ,1. Applying the unctoiality and convex lineaity o F to thi uae, we thu obtain the euation gαβ = gα + gβ. 5 We claim that all olution o thi euation ae o the om gα = c ln α o ome c [0, ]. Fit we how thi o α 0, 1]. I gα < o all α 0, 1], thi euation i Cauchy unctional euation in it multiplicativetoadditive om, and it i known [7] that any olution with g meauable i o the deied om o ome c <. By ou hypothee on F, g i lowe emicontinuou, hence meauable. Thu, o ome c < we have gα = c ln α o all α 0, 1]. I gα = o ome α 0, 1], then Euation 5 implie that gβ = o all β < α. Since it alo implie that gβ = 1 gβ, we conclude that then gβ = o all β 0, 1. Thu, i we take c = we again have gα = c ln α o all α 0, 1]. Next conide α = 0. I c > 0, then g0 = g0 + g 1 how that we neceaily have g0 =. I c = 0, then lowe emicontinuity implie g0 = 0. In both cae, the euation gα = c ln α alo hold o α = 0. In what ollow, chooing the value o c that make gα = c ln α, we hall pove that the euation F X, p! X 1, 1 = c Sp, hold o any two pobability ditibution p and on any inite et X. Uing Euation 3, it uice to how that F X, p 1, 1 = c px p x ln. 6! X x x X
19 440 JOHN C. BAEZ AND TOBIAS FRITZ We pove thi o moe and moe geneal cae in the ollowing eie o lemma. We tat with the geneic cae, whee c < and the pobability ditibution ha ull uppot. In Lemma 4.4 we teat all cae with 0 < c <. In Lemma 4.5 we teat the cae c = 0, and in Lemma 4.1 we teat the cae c =, which eem much hade than the et. 4.. Lemma. Euation 6 hold i c < and the uppot o i all o X. Poo. Chooe α 0, 1 uch that α < x o all x X. The deciive tep i to conide the commutative uae X + X, p 0 1 X,1 X X, p! X +! X t! X, 1, 0 α 1, 1 whee the tochatic matice and t ae given by = α p α 0 pn n 1 α p α 0 pn n, t = p 1 1 αp 1 1 α. p n. n αp n 1 α The econd column o t i only elevant o commutativity. The let vetical mophim i in FP, while we aleady know that the lowe hoizontal mophim evaluate to gα = c ln α unde the uncto F. Hence the diagonal o the uae get aigned the value c ln α unde F. On the othe hand, the uppe hoizontal mophim i actually a convex linea combination o mophim., 1, 0 α px x 1, 1,
20 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 441 one o each x X, with the pobabilitie p x a coeicient. Thu, compoing thi with the ight vetical mophim we get a mophim to which F aign the value c p x ln α p x + F X, p 1, 1. x x X! X Thu, we obtain c p x ln α p x + F x x X X, p! X 1, 1 = c ln α and becaue c <, we can impliy thi to F X, p! X 1, 1 = c x X p x ln Thi i the deied eult, Euation Lemma. Euation 6 hold i c < and uppp upp. Poo. Thi can be educed to the peviou cae by conideing the commutative tiangle px x X, p! X 1, 1 upp, p! upp in which p = p upp and = upp, and the vetical mophim conit o any map X upp that etict to the identity on upp and, a it tochatic ight invee, the incluion upp X. Thi mophim lie in FP Lemma. Euation 6 hold i 0 < c <. Poo. We aleady know by Lemma 4.3 that thi hold when uppp upp, o aume othewie. Ou tak i then how that F X, p! X 1, 1 =.
21 44 JOHN C. BAEZ AND TOBIAS FRITZ To do thi, chooe x X with p x > 0 = x, and conide the commutative tiangle X + 1, p 0 0! X+1 1, 1 X, p! X in which map X to itel by the identity and end the uniue element o 1 to x. Thi unction ha a onepaamete amily o tochatic ight invee, and we take the aow : X X + 1 to be any element o thi amily. To contuct thee tochatic ight invee, let Y = X {x}. Thi et i nonempty becaue the pobability ditibution i uppoted on it. I p x < 1 let be the pobability ditibution on Y given by = 1 p Y, 1 p x while i p x = 1 let be an abitay pobability ditibution on Y. Fo any α [0, 1], the convex linea combination 1 p x Y, 1 Y 1 Y Y, p x, 1, 0 α 1, 1 7 i a mophim in FinStat. Thee i a natual iomophim om it domain to that o the deied mophim, : and imilaly o it codomain: 1 p x Y, p x, 1, 0 = X + 1, p 0 1 p x Y, p x 1, 1 = X, p. Compoing 7 with thee oe and at, we obtain the deied mophim X + 1, p 0 X, p. Uing convex lineaity and the act that F vanihe on iomophim, 7 implie that F, = p x c ln α. Applying F to ou commutative tiangle, we thu obtain F X + 1, p 0 0! X+1 1, 1 = p x c ln α + F X, p! X 1, 1.
22 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 443 Since p x, c > 0, the it tem on the ighthand ide depend on α, but no othe tem do. Thi i only poible i both othe tem ae ininite. Thi pove a wa to be hown. F X, p 4.5. Lemma. Euation 6 hold i c = 0.! X 1, 1 =, Poo. That 6 hold in thi cae i a imple coneuence o lowe emicontinuity: appoximate by a amily o pobability ditibution whoe uppot i all o X. By Lemma 4.4, F map all the eulting mophim to 0. Thu, the ame mut be tue o the oiginal. To conclude the poo o Lemma 4.1, we need to how Euation 6 hold i c =. To do thi, it uice to aume c = and how that F X, p! X 1, 1 = wheneve p. The eaoning in the peviou lemma will not help u now, ince in Lemma 4. we needed c <. A we hall ee in Popoition 5.1, the poo o c = mut ue lowe emicontinuity. Howeve, ince lowe emicontinuity only poduce an uppe bound on the value o F at a limit point, it will have to be ued in poving the contapoitive tatement: i F i inite on ome mophim o the above om with p, then it i inite on ome mophim o the om 4. Now in ode to ine that the value o F at the limit point o a conveging amily o ditibution i inite, it i not enough to know that the value o F i inite at each element o the amily: one need a uniom bound. The need to deive uch a uniom bound i the eaon o the complexity o the ollowing agument. In what ollow we aume that p and ae pobability ditibution on X with p and F X, p! X 1, 1 <. We develop a eie o coneuence culminating in Lemma 4.1, in which we ee that gα i inite o ome α < 1. Thi implie c <, thu demontating the contapoitive o ou claim that Euation 6 hold i c = Lemma. Thee exit α, β [0, 1] with α β uch that i inite. hα, β = F, α β 1, 1 8
23 444 JOHN C. BAEZ AND TOBIAS FRITZ Poo. Chooe ome y X with p y y, and deine : X by { 1 i x = y x = 0 i x y. Put β = 1 y. Then ha a tochatic ight invee given by x xj = β 1 δ xy i j = 0 δ xy i j = 1 whee, i β = 0, we intepet the action a oming an abitaily choen pobability ditibution on X {y}. Setting α = 1 p y, we have a commutative tiangle X, p! X β 1, 1, α and the claim ollow om unctoiality Lemma. hα, 1 i inite o ome α < 1. Poo. Chooe α, β a in Lemma 4.6. Conide the commutative uae 4, 1 α 1 β 0, 1 0, 1, 3 1 0, 0 1, 3 1 t 1 β, α+β 1, 1
24 with the tochatic matice A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 445 = β 0 1 β 0 0 β 0 1 β = β β, t = The ight vetical mophim in thi uae lie in FP, o F vanihe on thi. The top hoizontal mophim i a convex linea combination β 1, α 1, 1 1 β, β 1, 1, whee the econd tem i in FP. Thu, by convex lineaity and Lemma 4.6, F o the top hoizontal mophim eual 1hα, β <. By unctoiality, F i 1 hα, β on the compoite o the top and ight mophim. Thi implie that the value o F on the othe two mophim in the uae mut alo be inite. Let u compute F o thei compoite in anothe way. By deinition, F o the bottom hoizontal mophim i h α+β, β. The let vetical mophim i a convex linea combination α + β 1, α α+β 1, 1 α β 1, 1 α α β 1, 1. By unctoiality and convex lineaity, F on the compoite o thee two mophim i thu α + β α h α + β, 1 + α β 1 α h α β, 1 α + β + h, β.. Compaing thee computation, we obtain α hα, β = α + β h α + β, 1 1 α + α β h α β, 1 α + β + h, β. 9 Thi how that each tem on the ighthand ide mut be inite. Note that the coeicient in ont o thee tem do not vanih, ince α β. I α < β then we can take α = α, α+β o that α < 1, and the it tem on the ighthand ide give hα, 1 <. I α > β we can take α = 1 α, o that α β α < 1, and the econd tem on the ighthand ide give that hα, 1 <.
25 446 JOHN C. BAEZ AND TOBIAS FRITZ 4.8. Lemma. Fo α β 1, we have hβ, 1 hα, 1. Poo. By the intemediate value theoem, thee exit γ [0, 1] with γα + 1 γ1 α = β. Now let α γ tand o the ditibution on 4 with weight αγ, α1 γ, 1 αγ, 1 α1 γ. The euation above guaantee that the let vetical mophim in thi uae i welldeined: 4, α γ 0, 1 0, 3 1, α 0, 3 0 1, 1 t 1 whee we take: =, β γ 0 1 γ 0 0 γ 0 1 γ 1, t = 1, 1 γ γ 0 γ 1 γ 0 The uae commute and the uppe hoizontal mophim i in FP, o the value o F on the bottom hoizontal mophim i bounded by the value o F on the ight vetical one, a wa to be hown. In the peceding lemma we ae not yet claiming that hα, 1 i inite. We how thi o α = 1 in Lemma 4.10, and o all α 0, 1 in Lemma 4.11, whee we actually obtain 4 a uniom bound Lemma. hα, 1 = h1 α, 1 o all α [0, 1].
26 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 447 Poo. Apply unctoiality to the commutative tiangle, α , 1, α whee the vetical mophim i in FP Lemma. h 1 4, 1 <. Poo. We ue 9 with β = 1 : h α, 1 = + α α α h 1 + α, 1 α h 3 α, α + h, 1, 4 10 which we will apply o α < 1. On the ighthand ide hee, the it agument o h in the 1 econd tem can be eplaced by, thank to Lemma 4.9. Then the it agument in 3 α all thee tem on the ighthand ide ae in [0, 1 ], with the mallet in the it tem, o Lemma 4.8 tell u that h α, 1 α 4h 1 + α, 1. Now with α 0 = 1, the euence ecuively deined by α 4 n+1 = αn 1+α n inceae and convege to 1. In paticula we can ind n with α < α n < 1, whee α i choen a in Lemma 4.7. Uing that eult togethe with Lemma 4.8, we obtain 1 h 4, 1 4 n h α n, 1 4 n h α, 1 < Lemma. Thee i a contant B < uch that hα, 1 B h 1, 1 o all α 0, 1. 4
27 448 JOHN C. BAEZ AND TOBIAS FRITZ Poo. By the ymmety in Lemma 4.9, it i uicient to conide α 0, 1]. By Lemma 4.8, we may ue the bound B = 1 o all α [ 1, 1 ]. It thu emain to ind a 4 choice o B that wok o all α 0, 1, and we aume α to lie in thi inteval om now 4 on. We eue Euation 10. Both the econd and the thid tem on the ighthand ide have thei it agument o h in the inteval [ 1, 3 ], o we can apply Lemma 4.8 and to obtain h α, 1 α + 1 α h 1 + α, α h 4, 1. To ind a implelooking uppe bound, we bound the ighthand ide om above by α applying Lemma 4.8 in ode to eplace the agument by jut α, and at the ame 1+α time ue α 0, 1 in ode to bound the coeicient o both tem by α and α 7: h α, 1 34 h α, h 4, 1. I we put α = n o n, then we can apply thi ineuality epeatedly until only tem o the om h 1, 1 ae let. Thi eult in a geometic eie: 4 h n, 1 3 n n 3 k h 4 4 4, 1. whoe convegence a n implie the exitence o a contant B < with k=0 h n, 1 B h 1 4, 1 o all n. The peent lemma then ollow with the help o Lemma Lemma. Euation 6 hold i c =. Poo. By Lemma 4.11 and the lowe emicontinuity o h, we ee that g 1 = h0, 1 < Thi implie that the contant c with gα = c ln α ha c <. Recall that we have hown thi unde the aumption that thee exit pobability ditibution p and on a inite et X with p and F X, p! X 1, 1 So, taking the contapoitive, we ee that i c =, then F X, p! X 1, 1 <. = wheneve p and ae ditinct pobability ditibution on X. Thi pove Euation 6 except in the cae whee p =. But in that cae, both ide vanih, ince on the let we ae taking F o a mophim in FP, and on the ight we obtain 0 = 0.
28 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY Counteexample and ubtletie One might be tempted to think that ou Theoem 3.1 alo hold i one elaxe the lowe emicontinuity aumption to meauability, upon euipping the hompace o both FinStat and [0, ] with thei σalgeba o Boel et. Fo [0, ], thi σalgeba i the uual Boel σalgeba: the et o the om a, ae open and hence meauable, the et o the om [0, b] ae cloed and hence meauable, and theeoe all halopen inteval a, b] ae meauable, and thee geneate the tandad Boel σalgeba. Howeve, o Theoem 3.1, mee meauability o the uncto F i not enough: 5.1. Popoition. Thee i a uncto FinStat [0, ] that i convex linea, meauable on hompace, and vanihe on FP, but i not a cala multiple o elative entopy. Poo. We claim that one uch uncto G: FinStat [0, ] i given by { 0 i uppp = upp, G X, p Y, = i uppp upp. Thi G clealy vanihe on FP. Since taking the uppot o a pobability ditibution i a lowe emicontinuou and hence meauable unction, the et o all mophim obeying uppp = upp i alo meauable, and hence G i meauable. Concening unctoiality, o a compoable pai o mophim X, p Y, t g Z,, we have uppp = upp, upp = uppt uppp = upp t. Thi pove unctoiality. A imila agument pove convex lineaity. A a meaue o inomation gain, thi uncto G i not had to undetand intuitively: we gain no inomation wheneve the et o poible outcome i peciely the et that we expected; othewie, we gain an ininite amount inomation. Since the collection o all uncto atiying ou hypothee i cloed unde um and cala multiple and alo contain the elative entopy uncto, we actually obtain a whole amily o uch uncto. Fo example, anothe one o thee uncto i G : FinStat [0, ] given by G X, p Y, = { Sp, i uppp = upp, i uppp upp. Ou oiginal idea wa to ue the wok o Petz [8, 9] to pove Theoem 3.1. Howeve, a it tuned out, thee i a gap in Petz agument. Although hi pupoted chaacteization concen the uantum veion o elative entopy, the it pat o hi poo in [8] teat the claical cae. I hi poo wee coect, it would pove thi:
29 450 JOHN C. BAEZ AND TOBIAS FRITZ 5.. Unpoved Theoem. The elative entopy Sp, o pai o pobability meaue on the ame inite et uch that ha ull uppot i chaacteized up to a multiplicative contant by thee popetie: a Conditional expectation law. Suppoe : X Y i a unction and : Y X a tochatic map with = 1 Y. Given pobability ditibution p and on X, and auming that ha ull uppot and =, we have Sp, = S p, + Sp, p. 11 b Invaiance. Given any bijection : X Y and pobability ditibution p, on X uch that ha ull uppot i.e. it uppot i all o X, we have S p, = Sp,. c Convex lineaity. Given pobability ditibution p, on X and p, on Y uch that and have ull uppot, and given λ [0, 1], we have Sλp 1 λp, λ 1 λ = λsp, + 1 λsp,. d Nilpotence. Fo any pobability ditibution p with ull uppot on a inite et, Sp, p = 0. e Meauability popety. The unction p, Sp, i meauable on the pace o pai o pobability ditibution on X uch that ha ull uppot. Note that [8] ue the oppoite odeing o the two agument o S. The poblem with thi theoem i the ange o applicability o Euation 11: what i thi omula uppoed to mean when p doe not have ull uppot? Ate all, Sp, i aumed to be deined only when the econd agument ha ull uppot, but thi need not be the cae o p, given the aumption made in the tatement o the conditional expectation popety. Note that ha ull uppot, o the tem S p, i ine. One can ty to coect thi poblem by auming that the conditional expectation popety hold only i p ha ull uppot a well. Howeve, thi mean that the poo o Petz Lemma 1 i valid only when uing hi notation p 3 > 0, which implie that hi Euation 5 i known to hold only o p > 0 and p 3 > 0. Upon ollowing the thead o Petz agument, one ind that hi Euation 6 ha been poven to ollow om hi aumption only o x 0, 1 and u 0, 1. Howeve, the olution o that unctional euation in the eeence he point to cucially ue the aumption that the unctional euation alo hold in cae that x = 0 o u = 0. Thi i the gap in Petz poo.
30 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 451 In act, i one allow S to take on ininite value, then the above claical veion o Petz theoem i not even coect, i one ue the intepetation that 11 i to be applied only when p ha ull uppot. The counteexample i imila to ou uncto G om above: { S Sp, i p ha ull uppot, p, = othewie. 6. Concluion The theoem hee, and ou ealie chaacteization o entopy [1], can be een a pat o a pogam o demontating that mathematical tuctue that ae ocially impotant ae alo categoically natual. Tom Leinte, whoe wod we uote hee, ha caied thi owad to a categoical explanation o Lebegue integation [6]. It would be inteeting to genealize ou eult on entopy and elative entopy om inite et to geneal meaue pace, whee integal eplace um. It would be even moe inteeting to do thi uing a categoytheoetic appoach to integation. It would alo be good to expe ou theoem moe conciely. A noted in Appendix B, convex linea combination ae opeation in a topological opead P. We can deine convex algeba, that i, algeba o P, in any ymmetic monoidal topological categoy. The categoy [0, ] with the uppe topology on it et o mophim i a convex algeba in TopCat, the lage topological categoy o topological categoie. We believe, but have not poved, that FinStat i a weak convex algeba in TopCat. Thi would mean that the axiom o a convex algeba hold up to coheent natual iomophim [4]. I thi i tue, the elative entopy RE: FinStat [0, ] hould be, up to a contant acto, the uniue map o weak convex algeba that vanihe on mophim in FP. Leinte [5] ha hown that FP i alo a weak convex algeba in CatTop. In act, it i the ee uch thing on an intenal convex algeba. So, it eem that both entopy and elative entopy emege natually om a categoytheoetic examination o convex lineaity. A. Semicontinuou uncto In Section 3 we explained what it meant o elative entopy to be a emicontinuou uncto. A moe ophiticated way to think about emicontinuou uncto ue topological categoie. Thi euie that we put a nontandad topology on [0, ], the ocalled uppe topology. A topological categoy i a categoy intenal to Top, and a continuou uncto i a uncto intenal to Top. In othe wod:
31 45 JOHN C. BAEZ AND TOBIAS FRITZ A.1. Deinition. A topological categoy C i a mall categoy whee the et o object C 0 and the et o mophim C 1 ae euipped with the tuctue o topological pace, and the map aigning to each mophim it ouce and taget:, t: C 1 C 0 the map aigning to each object it identity mophim i: C 0 C 1 and the map ending each pai o compoable mophim to thei compoite : C 1 C0 C 1 C 1 ae continuou. Given topological categoie C and D, a continuou uncto i a uncto F : C D uch that the map on object F 0 : C 0 D 0 and the map on mophim F 1 : C 1 D 1 ae continuou. We now explain how FinStoch and FinStat ae topological categoie. Stictly peaking, in ode o thi to wok, we need to deal with ize iue. One appoach i to let the object o Top be lage et living in a highe Gothendieck univee, which allow u to talk about the et o all object o mophim o FinStat o FinStoch. Anothe i to eplace each o thee categoie by it keleton, which i an euivalent mall categoy. Fom now on, we aume that one o thee thing ha been done. Fo FinStoch, we put the dicete topology on it et o object FinStoch 0. Each homet FinStochX, Y i a ubet o the Euclidean pace R X Y, and we put the ubpace topology on thi homet; o example, FinStoch1, Y, the et o all pobability ditibution on Y, i topologized a a implex. In thi way, FinStoch become a categoy eniched ove Top, and in paticula intenal to Top. A o FinStat, the identiication FinStat 0 = {X, p X FinStoch 0, p FinStoch1, X} FinStoch 0 FinStoch 1 induce a topology on FinStat 0. In thi topology, a net X λ, p λ λ Λ convege to X, p i and only i eventually X λ = X, and p λ p o thoe λ with X λ = X. Similaly, evey mophim in FinStat conit o a pai o mophim in FinStoch atiying cetain condition, and the eulting incluion FinStat 1 FinStoch 1 FinStoch 1 can be ued to deine a topology on FinStat 1. We omit the veiication that thee topologie make FinStat into a topological categoy. Thee i a topology on [0, ] whee the open et ae thoe o the om a, ], togethe with the whole pace and the empty et. Thi i called the uppe topology. With thi
32 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 453 topology, a unction ψ : A [0, ] om any topological pace A i continuou i and only ψ i lowe emicontinuou, meaning ψa lim in λ ψaλ o evey convegent net a λ A. It i eay to check that thi topology on [0, ] make addition continuou. In hot, [0, ] with it uppe topology i a topological monoid unde addition. We thu obtain a topological categoy with one object and [0, ] a it topological monoid o endomophim. By abue o notation we alo call thi topological categoy imply [0, ]. Thi let u tate Lemma 3.4 in a dieent way: A.. Lemma. I [0, ] i viewed a a topological categoy uing the uppe topology, the uncto RE: FinStat [0, ] i continuou. On the othe hand, i we give the monoid [0, ] the le exotic topology whee it i homeomophic to a cloed inteval, then thi uncto i not continuou. Having gone thi a, we cannot eit pointing out that [0, ] with it uppe topology i alo a topological ig. Recall that a ig i a ing without negative : a et euipped with an addition making it into a commutative monoid and a multiplication making it into a monoid, with multiplication ditibuting ove addition. In othe wod, it i a monoid in the monoidal categoy o commutative monoid. A topological ig i a ig with a topology in which addition and multiplication ae continuou. To make [0, ] into a ig, we deine addition a beoe, deine multiplication in the uual way o numbe in [0,, and et 0a = a0 = 0 o all a [0, ]. One can veiy that multiplication i continuou: but again, the key point i that we need to ue the uppe topology, ince a uddenly jump om to 0 a a eache zeo. Thu: A.3. Lemma. With it uppe topology, [0, ] i a topological ig. Moe impotant now i that [0, ] i a module ove the ig [0,, whee addition and multiplication in the latte ae deined a uual and we deine the action o [0, on [0, ] uing multiplication, with the povio that 0 a = 0 even when a =. And hee we ee: A.4. Lemma. The topological monoid [0, ] with it uppe topology become a topological module ove the ig [0, with it uual topology. B. Convex algeba We deine the monad o convex et to be the monad on Set ending any et X to the et o initelyuppoted pobability ditibution on X. Fo example, thi monad end
33 454 JOHN C. BAEZ AND TOBIAS FRITZ {1,..., n} to the et P n = {p [0, 1] n : n p i = 1} which can be identiied with the n 1implex. Thi monad i initay, o can be thought about in a ew dieent way. Fit, a initay monad can thought o a a initay algebaic theoy. The monad o convex et can be peented by a amily λ λ [0,1] o binay opeation, ubject to the euation i=1 x 0 y = x, x λ x = x, x λ y = y 1 λ x, x µ y λ z = x λµ y λ1 µ 1 λµ z Fo λ = µ = 1, the action λ1 µ in the lat euation may be taken to be an abitay 1 λµ numbe in [0, 1]. See [] o moe detail on how to deive thi peentation om the monad. A initay algebaic theoy can alo be thought o a an opead with exta tuctue. In a ymmetic opead O, one ha o each bijection σ : {1,..., n} {1,..., n} an induced map σ : O n O n. In a initay algebaic theoy, one ha the ame thing o abitay unction between inite et, not jut bijection. In othe wod, a initay algebaic theoy amount to a nonymmetic opead O togethe with, o each unction θ : {1,..., m} {1,..., n} between inite et, an induced map θ : O m O n, atiying uitable axiom. B.1. Deinition. The undelying ymmetic opead o the monad o convex et i called the opead o convex algeba and denoted P. An algeba o P i called a convex algeba. The pace o nay opeation o thi opead i P n, the pace o pobability ditibution on {1,..., n}. The compoition o opeation wok a ollow. Given pobability ditibution p P n and i P ki o each i {1,..., n}, we obtain a pobability ditibution p 1,..., n P k1 + +k n, namely p 1,..., n = p , p 1 1k1,... p n n1,..., p n nkn. The map θ : P m P n can be deined by puhowad o meaue. An algeba o the algebaic theoy o convex algeba i an algeba X o the opead with the uthe
34 A BAYESIAN CHARACTERIZATION OF RELATIVE ENTROPY 455 popety that the uae P m X n 1 θ P m X m θ 1 P n X n X commute o all θ : {1,..., m} {1,..., n}, whee the unlabelled aow ae given by the convex algeba tuctue o X. Note that P i natually a topological opead, whee the topology on P n i the uual topology on the n 1implex. In thi pape we have implicitly been uing algeba o P in vaiou topological categoie E with inite poduct. We call thee convex algeba in E. Hee ae ome example: Any convex ubet o R n i a convex algeba in Top. The additive monoid [0, ] with it uppe topology become a convex algeba in Top i we deine convex linea combination by teating [0, ] a a topological module o the ig [0, a in Lemma A.4. We mut euip [0, ] with it uppe topology o thi to wok, becaue the convex linea combination λ + 1 λ a eual when λ > 0, but uddenly jump down to a when λ eache zeo. The categoy CatTop o mall topological categoie and continuou uncto i itel a lage topological categoy. I we egad [0, ] with it uppe topology a a oneobject topological categoy a in Appendix A, then it become a convex algeba in CatTop thank to the peviou emak. The categoie FinPob, FinStat hould be weak convex algeba in CatTop, though we have not caeully checked thi. By thi, we mean that axiom o an algeba o the opead P hold up to coheent natual iomophim, in the ene made pecie by Leinte [4]. Similaly, Leinte ha hown that FP i a weak convex algeba in CatTop. In act, it i euivalent to the ee convex algeba in CatTop on an intenal convex algeba [5]. Reeence [1] J. Baez, T. Fitz and T. Leinte, A chaacteization o entopy in tem o inomation lo, Entopy , Alo available a axiv: [] T. Fitz, Convex pace I: deinition and example, available a axiv: