A Convenient Category for Higher-Order Probability Theory

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by  Chris Heunen

A Convenient Category for
Higher-Order Probability Theory
Chris Heunen Ohad Kammar Sam Staton Hongseok Yang
University of Edinburgh, UK University of Oxford, UK University of Oxford, UK University of Oxford, UK
Abstract—Higher-order probabilistic programming 1 (defquery Bayesian-linear-regression
languages allow programmers to write sophisticated
2 (let [f (let [s (sample (normal 0.0 3.0))
models in machine learning and statistics in a succinct
3 b (sample (normal 0.0 3.0))]
and structured way, but step outside the standard
7 4 (fn [x] (+ (* s x) b)))]
measure-theoretic formalization of probability theory.
1
Programs may use both higher-order functions and 5 (observe (normal (f 1.0) 0.5) 2.5)
0
continuous distributions, or even define a probability 6 (observe (normal (f 2.0) 0.5) 3.8)
2
distributiononfunctions.Butstandardprobabilitythe- 7 (observe (normal (f 3.0) 0.5) 4.5)

n ory cannot support higher-order functions, that is, the 8 (observe (normal (f 4.0) 0.5) 6.2)
a category of measurable spaces is not cartesian closed. 9 (observe (normal (f 5.0) 0.5) 8.0)
J Here we introduce quasi-Borel spaces. We show that 10 (predict :f f)))
1 these spaces: form a new formalization of probability
1 theory replacing measurable spaces; form a cartesian
closed category and so support higher-order functions;
] formanextensionalcategoryandsosupportgoodproof
L principles for equational reasoning; and support con-
P tinuous probability distributions. We demonstrate the
. useofquasi-Borelspacesforhigher-orderfunctionsand
s probabilityby:showingthatawell-knownconstruction
c
of probability theory involving random functions gains
[
a cleaner expression; and generalizing de Finetti’s the-

2 orem, that is a crucial theorem in probability theory,
v to quasi-Borel spaces.
7
4 I. Introduction
5
2 Toexpressprobabilisticmodelsinmachinelearningand Fig.1. BayesianlinearregressioninAnglican.Theprogramdefines
0 statistics in a succinct and structured way, it pays to use a probability distribution on functions R → R. It first samples a
randomlinearfunctionfbyrandomlyselectingslopesandintercept
.
1 higher-order programminglanguages,suchasChurch[16], b.Itthenadjuststheprobabilitydistributionofthefunctiontobetter
0 Venture [24], or Anglican [37]. These languages support describefiveobservations(1.0,2.5),(2.0,3.8),(3.0,4.5),(4.0,6.2)and
7 (5.0,8.0)byposteriorcomputation.Inthegraph,eachlinehasbeen
advanced features from both programming language the-
1 sampledfromtheposteriordistributionoverfunctions.
ory and probability theory, while providing generic infer-
:
v encealgorithmsforansweringprobabilisticqueries,suchas
Xi marginalization and posterior computation, for all models We introduce a new formalization of probability theory
written in the language. As a result, the programmer that accommodates higher-order functions. The main no-
r
a can succinctly express a sophisticated probabilistic model tion replacing a measurable space is a quasi-Borel space:
and explore its properties while avoiding the nontrivial a set X equipped with a collection of functions MX ⊆
busywork of designing a custom inference algorithm. [R→X] satisfying certain conditions (Def. 7).Intuitively,
This exciting development comes at a foundational MX isthesetofrandomvariablesoftypeX.HereRmeans
price. Programs in these languages may combine higher- that the randomness of random variables in MX comes
order functions and continuous distributions, or even de- from (a probability distribution on) R, one of the best
fine a probability distribution on functions. But the stan- behaving measurable spaces. Thus the primitive notion
dardmeasure-theoreticformalizationofprobabilitytheory shifts from measurable subset to random variable, which
cannot support higher-order functions, as the category of istraditionallyaderivednotion.Forrelatedideassee§IX.
measurablespacesisnotcartesianclosed[1].Forinstance, Quasi-Borel spaces have good properties and structure.
theAnglicanimplementationofBayesianlinearregression • The category of quasi-Borel spaces is extensional,
in Figure 1 goes beyond the standard measure-theoretic since a morphism is just a structure-preserving func-
foundationofprobabilitytheory,asitdefinesaprobability tion (§III), and so it supports good proof principles
distribution on functions R→R. for equational reasoning (in contrast to [34, §8]).
• The category of quasi-Borel spaces is cartesian closed Definition 2. A probability measure on R is a function
(§IV), so that it becomes a setting to study probabil- µ: ΣR →[0,1] satisfying µ(R)=1 and µ(USi)=Pµ(Si)
ity distributions on higher-order functions. for any countable sequence of disjoint Borel sets S .
i
• There is a natural notion of probability measure on
The natural generalization gives measurable spaces.
quasi-Borel spaces (Def. 10). The space of all proba-
bilitymeasuresisagainaquasi-Borelspace,andforms Definition 3. A σ-algebra on a set X is a nonempty
thebasisforacommutativemonad onthecategoryof familyofsubsetsofX thatisclosedundercomplementsand
quasi-Borelspaces(§V).Thusquasi-Borelspacesform countableunions.Ameasurablespaceisapair(X,Σ )of
X
semantics for a probabilistic programming language a set X and a σ-algebra Σ on it. A probability measure
X
in the monadic style [26]. on a measurable space X is a function µ: Σ → [0,1]
X
U P
We also illustrate the use of quasi-Borel spaces. satisfying µ(X) = 1 and µ( Si) = µ(Si) for any
countable sequence of disjoint sets S ∈Σ .
• Bayesian regression (§VI). Quasi-Borel spaces are a i X
natural setting for understanding programs such as TheBorelsetsoftherealsformaleadingexampleofaσ-
the one in Figure 1: the prior (Lines 2–4) defines algebra.Otherimportantexamplesarecountablesetswith
a probability distribution over functions f, i.e. a theirdiscreteσ-algebra,whichcontainsallsubsets.Wecan
measure on RR, and the posterior (illustrated in characterize these spaces as standard Borel spaces, but
the graph), is again a probability measure on RR, first introduce the appropriate structure-preserving maps.
conditioned by the observations (Lines 5–9).
• Randomization (§VII). A key idea of categorical logic Definition 4. Let (X,ΣX) and (Y,ΣY) be measurable
is that ∀∃ statements should become statements spaces. A measurable function f: X → Y is a function
about quotients of objects. The structure of quasi- such that f-1(U)∈ΣX when U ∈ΣY.
Borel spaces allows us to rephrase a crucial random-
Thus a measurable function f: X → Y lets us push-
izationlemmainthisway.Classically,itsaysthatev-
forward a probability measure µ on X to a probability
ery probability kernel arises from a random function. measure f µ on Y by (f µ)(U) = µ(f-1(U)). Measurable
∗ ∗
In the setting of quasi-Borel spaces, it says that the
spaces and measurable functions form a category Meas.
space of probability kernels P(R)X is a quotient of Real-valued measurable functions f: X → R can be
thespaceofrandomfunctions,P(RX)(Theorem25).
integrated with respect to a probability measure µ on
Notice that the higher-order structure of quasi-Borel
(X,Σ ). The integral of a nonnegative function f is
X
spaces allows us to succinctly state this result.
Z (cid:18) (cid:19)
• De Finetti’s theorem (§VIII). Probability theorists fdµd=ef supX µ(U )· inf f(x) ,
i
often encounter problems when working with arbi- X {Ui} i x∈Ui
trary probability measures on arbitrary measurable
spaces. Quasi-Borel spaces allow us to better manage where{Ui}rangesoverthefinitepartitionsofX consisting
the source of randomness. For example, de Finetti’s of measurable subsets. When f may be negative, the
theoremisafoundationalresultinBayesianstatistics integraloff isdefinedifthoseofmin(0,f)andmin(0,−f)
which says that all exchangeable random sequences are finite, in which case
can be generated from independent and identically Z (cid:18)Z (cid:19) (cid:18)Z (cid:19)
def
distributedsamples.Thetheoremisknowntoholdfor fdµ = min(0,f)dµ − min(0,−f)dµ .
standard Borel spaces [8] or measurable spaces that X X X
arise from good topologies [19], but not for arbitrary When it is convenient to make the integrated variable ex-
R R
measurable spaces [9]. We show that it holds for all plicit, we write f(x)dµ for (λx.f(x)·[x∈U])dµ,
x∈U X
quasi-Borel spaces (Theorem 28). where U ∈ ΣX is a measurable subset and [ϕ] has the
value 1 if ϕ holds and 0 otherwise.
All of this is evidence that quasi-Borel spaces form a
convenient category for higher-order probability theory.
A. Standard Borel spaces
II. Preliminaries on probability measures and
Proposition 5 (e.g. [23], App. A1). For a measurable
measurable spaces
space (X,Σ ) the following are equivalent:
X
Definition1. TheBorelsetsformthefamilyΣRofsubsets • (X,ΣX) is a retract of (R,ΣR), that is, there exist
of R inductively defined as follows: measurable X −→f R−→g X such that g◦f =id ;
X
• intervals (a,b) are Borel sets; • (X,ΣX)iseithermeasurablyisomorphicto(R,ΣR)or
• complements of Borel sets are Borel; countable and discrete;
• countable unions of Borel sets are Borel. • X hasacompletemetricwithacountabledensesubset
such that Σ is freely generated by the open sets.
The Borel sets play a crucial role in probability theory X
because of the tight connection between the notion of When (X,Σ ) satisfies any of the above conditions,
X
probability measure and the axiomatization of Borel sets. we call it standard Borel space. These spaces play an
important role in probability theory because they en- of measurable functions R → 2, which are exactly the
joy properties that do not hold for general measurable characteristic functions of the Borel sets (Def. 1).
spaces, such as the existence of conditional probability
Beforewecontinue,weremarkthatthenotionofquasi-
kernels[23],[28]anddeFinetti’stheoremforexchangeable
Borel space is invariant under replacing R with a different
random processes [9].
uncountable standard Borel space.
Besides R, another popular uncountable standardBorel
space is (0,1) with the σ-algebra {U ∩(0,1) | U ∈ΣR}. Proposition 9. For an uncountable standard Borel space
As the above proposition indicates, these spaces are iso- (Ω,Σ ), a set X, and a set N of functions Ω→X,
Ω
morphic by, for instance, λr.(1+1e−r) :R→(0,1). (cid:0)X,{α◦f | f: R→Ω measurable, α∈N}(cid:1)
B. Failure of cartesian closure
is a quasi-Borel space if and only if:
Proposition 6 (Aumann, [1]). The category Meas is not • α◦f ∈N if α∈N and f: Ω→Ω is measurable;
cartesian closed: there is no space of functions R→R. • α∈N if α: Ω→X is constant;
U
Specifically, the evaluation function • if Ω = i∈NSi, with each set Si ∈ ΣΩ, and
α ,α ,...∈N, then β is in N, where β(r)=α (r) if
1 2 i
ε: Meas(R,R)×R→R with ε(f,r)=f(r) r ∈S .
i
is never measurable (Meas(R,R)×R,Σ⊗ΣR)→(R,ΣR) Probability theory typically considers a basic probabil-
regardless of the choice of σ-algebra Σ on Meas(R,R). itymeasureonthesamplespaceΩ.Eachrandomvariable,
Here, Σ ⊗ ΣR is the product σ-algebra, generated by that is each measurable function Ω → X, then induces a
rectangles (U ×V) for U ∈Σ and V ∈ΣR. probability measure on X by pushing forward the basic
measure.Quasi-Borelspacestakethisideaasanaxiomatic
III. Quasi-Borel spaces notion of probability measure.
The typical situation in probability theory is that there Definition 10. A probability measure on a quasi-Borel
is a fixed measurable space (Ω,ΣΩ), called sample space, space(X,MX)isapair(α,µ)ofα∈MX andaprobability
from which all randomness originates, and that obser- measure µ on R (as in Def. 2).
vations are made in terms of random variables, which
are pairs (X,f) of a measurable space of observations A. Morphisms and integration
(X,Σ ) and a measurable function f: Ω → X. From Definition 11. A morphism of quasi-Borel spaces
X
thisperspective,thenotionofmeasurablefunctionismore (X,M ) → (Y,M ) is a function f: X → Y such that
X Y
(cid:0) (cid:1)
important than the notion of measurable space. In some f ◦α ∈ M if α ∈ M . Write QBS (X,M ),(Y,M )
Y X X Y
ways,theσ-algebraΣ isonlyusedasanintermediaryto for the set of morphisms from (X,M ) to (Y,M ).
X X Y
restrain the class of measurable functions Ω→X.
In particular, elements of M are precisely morphisms
spaWcee.nInowdouisnegtshoi,swideeaassausmaebtahsaist ofourrosuarmnpelewspnaoctieonΩoisf (R,MR)→(X,MX), so MX =XQBS(cid:0)(R,MR),(X,MX)(cid:1).
Morphismscomposeasfunctions,andidentityfunctions
the real numbers, which makes probabilities behave well.
are morphisms, so that quasi-Borel spaces and their mor-
Definition7. Aquasi-BorelspaceisasetX togetherwith phismsformacategoryQBS.Also,Def.11isindependent
a set M ⊆[R→X] satisfying: of our choice of R as the sample space.
X
• α◦f ∈MX if α∈MX and f: R→R is measurable; Proposition12. ConsideranuncountablestandardBorel
•• iαf ∈RM=XUifi∈αN:SRi,→wXithis ceoancshtanste;t Si Borel, and sfupnaccetio(Ωns,ΣΩΩ→). FXoirsiu∈ch1t,h2a,tlet Xi be a set and Ni a set of
α ,α ,...∈M ,thenβisinM ,whereβ(r)=α (r)
fo1r r2∈Si. X X i Mi =(cid:0)Xi,{α◦f | f: R→Ω measurable, α∈Ni}(cid:1)
The name ‘quasi-Borel space’ is motivated firstly by are quasi-Borel spaces. A function g: X1 → X2 is a
analogytoquasi-topologicalspaces(see§IX),andsecondly morphism (X1,M1) → (X2,M2) if and only if g◦α ∈ N2
in recognition of the intimate connection to the standard for all α∈N1.
Borel space R (see also Prop. 14(2)).
Morphisms between quasi-Borel spaces are analogous
Example 8. For every measurable space (X,Σ ), let to measurable functions between measurable spaces. The
X
M bethesetofmeasurablefunctionsR→X.ThusM crucial properties of measurable functions are that they
ΣX ΣX
is the set of X-valued random variables. In particular, R workwellwithprobabilitymeasures:wecanpush-forward
itselfcanbeconsideredasaquasi-Borelspace,withMR the these measures, and integrate over them. Morphisms of
set of measurable functions R→R. quasi-Borel spaces also support these constructions.
For another example, the two-element discrete space 2 • Pushing forward: if f: X → Y is a morphism and
can be considered as a quasi-Borel space, with M the set (α,µ) is a probability measure on X then f ◦ α is
2
by definition in M and so (f ◦α,µ) is a probability Proposition 14(1) means there is an adjunction
Y
measure on Y.
• Integrating:Iff: X →Risamorphismofquasi-Borel (cid:114)(cid:114) L
spaces and (α,µ) is a probability measure on X, the Meas ⊥ (cid:50)(cid:50)QBS
integral of f with respect to (α,µ) is R
Z Z
fd(α,µ)d=ef (f ◦α)dµ. (1) whereL(X,MX)=(X,ΣMX)andR(X,ΣX)=(X,MΣX).
R Proposition 14(2) means that the functor R is full and
So integration formally reduces to integration on R. faithful when restricted to standard Borel spaces. Equiv-
alently, L(R(X,Σ ))=(X,Σ ), that is Σ =Σ for
B. Relationship to measurable spaces X X X MΣX
standard Borel spaces (X,Σ ).
X
IfweregardasubsetS ⊆X asitscharacteristicfunction
χ : X →2, then we can regard a σ-algebra on a set X as
S IV. Products, coproducts and function spaces
a set of characteristic functions F ⊆ [X → 2] satisfying
X
certainconditions.Thusameasurablespace(Def.3)could
Quasi-Borel spaces support products, coproducts, and
equivalentlybedescribedasapair(X,F )ofasetX and
X function spaces. These basic constructions form the basis
a collection F ⊆ [X → 2] of characteristic functions.
X for interpreting simple type theory in quasi-Borel spaces.
Moreover, from this perspective, a measurable function
fssuh:cifh(tXtt,hhFeaXtem)χp◦→hfas(∈iYs,FfFrXoYm)ificχsha∈srimaFcpYtle.yrTisathiucfusfnuqcnutcaiotsinio-nBfso:rXeXl→sp→a2cteYos Pqisuraaospqi-uoBasosiirt-eiBolosnprea1lc5essp(Paincreod,dewxuechdtesrb)ey.QaIfs(eXXtiI,i,MsthtXheien)is(∈eQItiipsXroaidf,uaMcmtΠ,iliayXnoid)f
random variables R→X. i i
1) Quasi-Borel spaces as structured measurable spaces: M d=efnf: R→Q X | ∀i.(π ◦f)∈M o.
A subset S ⊆ X is in the σ-algebra Σ of a measurable ΠiXi i i i Xi
X
space (X,Σ ) if and only if its characteristic function
X Q
The projections X → X are morphisms, and provide
X → 2 is measurable. With this in mind, we define a i i i
the structure of a categorical product in QBS.
measurable subset of a quasi-Borel space (X,M ) to be a
X
subsetS ⊆X suchthatthecharacteristicfunctionX →2 Proposition16(Coproducts). If(X ,M ) isafamily
is a morphism of quasi-Borel spaces. of quasi-Borel spaces indexed by a ciountXaiblie∈Iset I, then
‘ ‘
( X ,M ) is a quasi-Borel space, where X is the
Proposition 13. The collection of all measurable subsets i i qiXi i i
disjoint union of sets,
of a quasi-Borel space (X,M ) is characterized as
X
ΣMX d=ef{U | ∀α∈MX.α-1(U)∈ΣR} (2) MqiXi d=ef{λr.(f(r),αf(r)(r))|f: R→I is measurable,
and forms a σ-algebra. (α ∈M ) },
i Xi i∈image(f)
Thus we can understand a quasi-Borel space as a mea-
and I carries the discrete σ-algebra. This space has the
surablespace(X,Σ )equippedwithaclassofmeasurable
X universal property of a coproduct in the category QBS.
functions M ⊆ [R → X] generating the σ-algebra by
X
ΣX =ΣMX as in (2). Proof notes. The third condition of quasi-Borel spaces is
Moreover, every morphism (X,MX) → (Y,MY) is also needed here. It is a crucial step in showing that for an
a measurable function (X,ΣMX) → (Y,ΣMY) (but the I-indexed family of morphisms (fi: Xi → Z)i∈I, the
converse does not hold in general). copairing [f ] : ‘ X →Z is again a morphism.
i i∈I i∈I i
A probability measure (α,µ) on a quasi-Borel space
(X,M ) induces a probability measure α µ on the un-
X ∗ Proposition 17 (Function spaces). If (X,M ) and
X
derlying measurable space. Integration as in (1) matches
(Y,M ) are quasi-Borel spaces, so is (YX,M ), where
the standard definition for measurable spaces. Y YX
YX d=efQBS(X,Y) is the set of morphisms X →Y, and
2) AnadjunctionembeddingstandardBorelspaces: Un-
der some circumstances morphisms of quasi-Borel spaces
coincide with measurable functions. MYX d=ef{α: R→YX |uncurry(α)∈QBS(R×X,Y)}.
Proposition 14. Let (Y,Σ ) be a measurable space.
Y The evaluation function YX ×X →Y is a morphism and
1) If(X,MX)isaquasi-Borelspace,afunctionX →Y hastheuniversalpropertyofthefunctionspace.ThusQBS
is a measurable function (X,ΣMX)→(Y,ΣY) if and is a cartesian closed category.
only if it is a morphism (X,M )→(Y,M ).
X ΣY
2) If(X,Σ )isastandardBorelspace,afunctionX → Proof notes. The only difficult part is showing that
X
Y is a morphism (X,MΣX)→(Y,MΣY) if and only (YX,MYX) satisfies the third condition of quasi-Borel
if it is a measurable function (X,Σ )→(Y,Σ ). spaces. Prop. 16 is useful here.
X Y
A. Relationship with standard Borel spaces and then the conditions are understood as expressions in
a cartesian closed category.
Recall that standard Borel spaces can be thought of as
a full subcategory of the quasi-Borel spaces, that is, the
B. Kernels and the Giry monad
functorR: Meas→QBSisfullandfaithful(Prop.14(2))
We recall the notion of probability kernel, which is a
when restricted to the standard Borel spaces. This full
measurable family of probability measures.
subcategory has the same countable products, coproducts
and function spaces (whenever they exist). We may thus Definition 19. Let (X,Σ ) and (Y,Σ ) be measurable
X Y
regard quasi-Borel spaces as a conservative extension of spaces. A probability kernel from X to Y is a function
standard Borel spaces that supports simple type theory. k :X×Σ →[0,1] such that k(x,−) is a probability mea-
Y
sure for all x ∈ X (Def. 3), and k(−,U) is a measurable
Proposition 18. The functor R(X,Σ )=(X,M ):
X ΣX function for all U ∈Σ (Def. 4).
1) preserves products of standard Borel spaces: Y
Q Q
R( X ) = R(X ), where (X ,Σ ) is a We can classify probability kernels as follows. Let
i i i i i Xi i∈I
countable family of standard Borel spaces; G(X) be the set of probability measures on (X,Σ ).
X
2) preserves spaces of functions between standard Borel We can equip this set with the σ-algebra generated by
spaces whenever they exist: if (Y,Σ ) is countable {µ∈G(X) | µ(U)<r},forU ∈Σ andr ∈[0,1],toform
Y X
and discrete, and (X,Σ ) is standard Borel, then ameasurablespace(G(X),Σ ).Ameasurablefunction
X G(X)
R(XY)=R(X)R(Y); X →G(Y) amounts to a probability kernel from X to Y.
3) preserves coproducts of standard Borel spaces: The construction G has the structure of a monad, as
‘ ‘
R( X ) = R(X ), where (X ,Σ ) is a firstdiscussedbyGiryandLawvere[15].Acomputational
i i i i i Xi i∈I
countable family of standard Borel spaces. intuition is that G(X) is a space of probabilistic compu-
tations over X, and this provides a semantic foundation
Consequently, a standard programming language se-
for a first-order probabilistic programming language (see
mantics in standard Borel spaces can be conservatively
e.g. [34]). The unit η: X → G(X) lets η(x) be the Dirac
embedded in quasi-Borel spaces, allowing higher-order
measureonx,withη(x)(U)=1ifx∈U,andη(x)(U)=0
functionswhilepreservingallthetypetheoreticstructure.
if x 6∈ U. If µ ∈ G(X) and k is a measurable function
We note, however, that in light of Prop. 6, the quasi-
X → G(Y), then (µ>>= k) is the measure in G(Y) with
BorelspaceRRdoesnotcomefromastandardBorelspace. R G
(µ>>= k)(U)= k(x)(U)dµ.
Moreover,theleftadjointL: QBS→Measdoesnotpre- G x∈X
serveproductsingeneral.Forquasi-Borelspaces(X,MX) C. Equivalent measures on quasi-Borel spaces
and (Y,M ), we always have Σ ⊗Σ ⊆Σ , but
Y MX MY MX×Y Recall (Def. 10) that a probability measure (α,µ) on a
notalways⊇.Indeed,ΣMRR⊗ΣR 6=ΣM(RR×R),byProp.6. quasi-Borel space (X,MX) is a pair (α,µ) of a function
V. A monad of probability measures α ∈ M and a probability measure µ on R. Random
X
variables are often equated when they describe the same
In this section we will show that the probability mea-
distribution. Every probability measure (α,µ) determines
sures on a quasi-Borel space form a quasi-Borel space
a push-forward measure α µ on the corresponding mea-
again. This gives a commutative monad that generalizes ∗
surable space (X,Σ ), that assigns to U ⊆ X the real
the Giry monad for measurable spaces [15]. MX
number µ(α−1(U)). We will identify two probability mea-
A. Monads sures when they define the same push-forward measure,
and write ∼ for this equivalence relation.
We use the Kleisli triple formulation of monads (see
This is a reasonable notion of equality even if we
e.g. [26]). Recall that a monad on a category C comprises
put aside the notion of measurable space, because two
• for any object X, an object T(X); probabilitymeasureshavethesamepush-forwardmeasure
• for any object X, a morphism η: X →T(X); precisely when they have the same integration operator:
• for any objects X,Y, a function (α,µ) ∼ (α0,µ0) if and only if R fd(α,µ) = R fd(α0,µ0)
(>>=): C(X,T(Y))→C(T(X),T(Y)). for all morphisms f: (X,MX)→R.
We write (t>>=f) for (>>=)(f)(t). D. A probability monad
This is subject to the conditions (t>>=η)= t, (η(x)>>=f)= We now explain how to build a monad of probability
f(x), and t>>=(λx.(f(x)>>=g))=(t>>=f)>>=g. measures on the category of quasi-Borel spaces, modulo
The intuition is that T(X) is an object of computations this notion of equivalence. This monad P will inherit
returning X, that η is the computation that returns properties from the Giry monad. Technically, the functor
immediately, and that t>>=f sequences computations, first L: QBS → Meas (Prop. 14) is a ‘monad opfunctor’
running computation t and then calling f with the result. taking P to the Giry monad G, which means that it
WhenC iscartesianclosed,amonadisstrong if(>>=)in- extends to a functor from the Kleisli category of P to the
ternalizes to an operation (>>=): (T(Y))X → (T(Y))T(X), Kleisli category of G [36].
a) On objects: For a quasi-Borel space (X,M ), let
X
P(X)={(α,µ) probability measure on (X,M )}/∼,
X
M ={β: R→P(X)|∃α∈M .∃g ∈Meas(R,G(R)).
P(X) X
∀r ∈R.β(r)=[α,g(r)]}.
Note that
P(X)∼={α µ∈G(X,Σ ) | α∈M , µ∈G(R)} (3)
∗ MX X
so l ([α,µ]) = α µ defines a measurable injection
X ∗
l : L(P(X))(cid:26)G(X,Σ ).
X MX
b) Monad unit (return): Recall that the constant
Fig.2. Illustrationof1000sampledfunctionsfromtheprioronRR
functions (λr.x) are all in MX. For any measure µ on R, forBayesianlinearregression(5).
the push-forward measure (λr.x) µ on (X,Σ ) is the
∗ MX
Dirac measure on x, with ((λr.x) µ)(U)=1 if x∈U and
∗
0 otherwise. Thus (λr.x,µ) ∼ (λr.x,µ0) for all measures d) Prior: Lines 2–4 define a prior measure on RR:
µ,µ0 on R. The unit of P at (X,M ) is the morphism
X def
prior = (let [s (sample (normal 0.0 3.0))
η: X →P(X) given by
b (sample (normal 0.0 3.0))]
η (x)=[λr.x, µ] (4) (fn [x] (+ (* s x) b)))
(X,MX)
for an arbitrary measure µ on R. To describe this semantically, observe the following.
c) Bind: Todefine(>>=): P(Y)X →(P(Y))P(X),sup-
Proposition 22. Let (Ω,Σ ) be a standard Borel space,
Ω
pose f: X → P(Y) is a morphism and [α,µ] in P(X).
and (X,M ) a quasi-Borel space. Let α: R(Ω,Σ ) → X
Sincef isamorphism,thereisameasurableg: R→G(R) X Ω
be a morphism and µ a probability measure on (Ω,Σ ).
Ω
and a function β ∈ MY such that (f ◦α)(r) = [β,g(r)]. Any section-retraction pair (Ω →−ς R −→ρ Ω) = id has a
Set ([α,µ]>>=f)=[β,µ>>= g], where µ>>= g is the bind Ω
G G probabilitymeasure[α◦ρ,ς µ]∈P(X),thatisindependent
of the Giry monad. This matches the bind of the Giry ∗
of the choice of ς and ρ.
monad, since ((α µ)>>= (l ◦f))=β (µ>>= g).
∗ G Y ∗ G
Write [α,µ] for the probability measure in this case.
Theorem 20. Thedata(P,η,(>>=))abovedefinesastrong
Now,theprogramfragmentprior describesthedistribu-
monad on the category QBS of quasi-Borel spaces. tion[α,ν⊗ν]inP(RR)whereν isthenormaldistribution
Proof notes. The monad laws can be reduced to the laws on R with mean 0 and standard deviation 3, and where
for the monad G on Meas [15]. The monad on QBS α: R×R→RR isgivenbyα(s,b)d=efλr.s·r+b.Informally,
is strong because (>>=): P(Y)X → (P(Y))P(X) is a mor-
prior =[α,ν⊗ν]∈P(RR). (5)
phism, which is shown by expanding the definitions.
(cid:74) (cid:75)
Figure 2 illustrates this measure [α,ν ⊗ ν]. This deno-
Proposition 21. The monad P satisfies these properties:
tational semantics can be made compositional, by using
1) For f: (X,M ) → (Y,M ), the functorial action
X Y the commutative monad structure of P and the cartesian
P(f): P(X)→P(Y) is [α,µ]7→[f ◦α,µ].
closed structure of the category QBS (following e.g. [26],
2) Itisacommutativemonad,i.e.theorderofsequenc-
[34]), but in this paper we focus on this example rather
ing doesn’t matter: if p∈P(X), q ∈P(Y), and f is
than spelling out the general case once again.
a morphism X × Y → P(Z), then p >>= λx.q >>=
e) Likelihood: Lines 5–9 define the likelihood of the
λy.f(x,y) equals q >>=λy.p>>=λx.f(x,y).
observations:
3) The faithful functor L: QBS→Meas with
L(X,MX)=(X,ΣMX) extends to a faithful obs d=ef (observe (normal (f 1.0) 0.5) 2.5)
functor Kleisli(P)→Kleisli(G), i.e. (L,l) is a monad
...(observe (normal (f 5.0) 0.5) 8.0)
opfunctor [36].
4) When(X,Σ )isastandardBorelspace,themapl This program fragment has a free variable f of type RR.
X X
of Eq. (3) is a measurable isomorphism. Let us focus on line 5 for a moment:
VI. Example: Bayesian regression obs d=ef(observe (normal (f 1.0) 0.5) 2.5)
1
We are now in a position to explain the semantics of Given a function f: R→R, the likelihood of drawing 2.5
the Anglican program in Figure 1. The program can be
fromanormaldistributionwithmeanf(1.0)andstandard
splitintothreeparts:aprior,alikelihood,andaposterior.
deviation 0.5 is
Recall that Bayes’ law says that the posterior is propor-
tionate to the product of the prior and the posterior. f: RR ‘obs =d(f(1.0),2.5),
1
(cid:74) (cid:75)
where d: R2 → [0,∞) is the density of the normal level of standard Borel spaces, and so a defunctionalized
distribution function with standard deviation 0.5: version would be in some sense equivalent, but from the
q programming perspective it helps to abstract away from
d(µ,x) = 2 e−2(x−µ)2.
π this. The regression program in Fig. 1 is quickly adapted
to fit other kinds of functions, e.g. polynomials, or even
Noticethatweuseanormaldistributiontoallowforsome
programs from a small domain-specific language, simply
noiseinthemeasurement.Informally,wearenotrecording
by changing the prior in Lines 2–4.
an observation that f(1.0) is exactly 2.5, since this would
make regression impossible; rather, f(1.0) is roughly 2.5.
VII. Random functions
Overall, lines 5–9 describe a likelihood weight which is
theproductofthelikelihoodsofthefivedatapoints,given
We discuss random variables and random functions,
f: RR.
starting from the traditional setting. Let (Ω,Σ ) be a
Ω
f: RR ‘obs =d(f(1),2.5)·d(f(2),3.8)·d(f(3),4.5) measurable space with a probability measure. A random
(cid:74) (cid:75) variable is a measurable function (Ω,Σ ) → (X,Σ ). A
·d(f(4),6.2)·d(f(5),8.0). Ω X
random function between measurable spaces (X,Σ ) and
X
f) Posterior: We follow the recipe for a semantic (Y,Σ ) is a measurable function (Ω × X,Σ ⊗ Σ ) →
Y Ω X
posteriorgivenin[34].Puttingthepriorandlikelihoodto- (Y,Σ ).
Y
gethergivesaprobabilitymeasureinP(RR×[0,∞))which We can push forward a probability measure on Ω along
is found by pushing forward the measure prior ∈ P(R) arandomvariable(Ω,Σ )→(X,Σ )togetaprobability
Ω X
along the function (cid:74) (cid:75) measure on X, but in the traditional setting we cannot
push forward a measure along a random function. Mea-
(id, obs ): RR →RR×[0,∞).
surable spaces are not cartesian closed (Prop. 6), and so
(cid:74) (cid:75)
This push-forward measure we cannot form a measurable space YX and we cannot
curry a random function in general.
P(id, obs )( prior )∈P(RR×[0,∞))
Now, if we revisit these definitions in the setting of
(cid:74) (cid:75) (cid:74) (cid:75)
quasi-Borel spaces, we do have function spaces, and so we
is a measure over pairs (f,w) of functions together with
can push forward along random functions. In fact, this
their likelihood weight. We now find the posterior by
is somewhat tautologous because a probability measure
multiplying the prior and the likelihood, and dividing by
(Def. 10) on a function space is essentially the same thing
a normalizing constant. To do this we define a morphism
as random function: a probability measure on a function
norm: P(X×[0,∞))→P(X)]{error} by
( space (Y,ΣY)(X,ΣX) is defined to be a pair (f,µ) of a
[α,ν /(ν (R))] if 06=ν (R)6=∞ probability measure µ on R, our sample space, and a
norm([(α,β),ν])d=ef β β β
error otherwise morphism f: R→YX; but to give a morphism R→YX
is to give a morphism R×X → Y (Prop. 17) as in the
where νβ: ΣR → [0,∞] d=ef λU. Rr∈U(β(r))dν. The idea is traditional definition of random function.
that if β: R → [0,∞) and ν is a probability measure on We have already encountered an example of a random
R then ν is always a posterior measure on R, but it is function in Section VI: the prior for linear regression is a
β
typically not normalized, i.e. ν (R) 6= 1. We normalize it random function from R to R over the measurable space
β
by dividing by the normalizing constant, as long as this (R×R,ΣR⊗ΣR) with the measure ν⊗ν. Random func-
division is well-defined. tionsaboundthroughoutprobabilitytheoryandstochastic
Now, the semantics of the entire program in Figure 1 processes. The following section explores their use in the
is norm(P(id, obs )( prior )), which is a measure in so-called randomization lemma, which is used throughout
P(RR). Calculat(cid:74)ing(cid:75)thi(cid:74)s post(cid:75)erior using Anglican’s infer- probability theory. By moving to quasi-Borel spaces, we
ence algorithm lmh gives the plot in the lower half of can state this lemma succinctly (Theorem 25).
Figure 1.
g) Defunctionalized regression and non-linear regres- A. Randomization
sion: Of course, one can do regression without explicitly
An elementary but useful trick in probability theory is
considering distributions over the space of all measurable
that every probability distribution on R arises as a push-
functions, by instead directly calculating posterior distri-
forward of the uniform distribution on [0,1]. Even more
butions for the slope s and the intercept b. For example,
useful is that this can be done in a parameterized way.
one could defunctionalize the program in Fig. 1 in the
style of Reynolds [29]. But defunctionalization is a whole- Proposition 23 ([23], Prop. 3.22). Let (X,Σ ) be
X
program transformation. By structuring the semantics a measurable space. For any kernel k: X×ΣR →[0,1]
usingquasi-Borelspaces,weareabletoworkcomposition- there is a measurable function f: R×X →R such that
ally, without mentioning s and b explicitly on lines 5–10. k(x,U)=υ{r | f(r,x)∈U}, where υ is the uniform dis-
The internal posterior calculations actually happen at the tribution on [0,1].
For quasi-Borel spaces we can phrase this more suc- We have (⊇) since q is a morphism, so it remains to
cinctly: it is a result about a quotient of the space of show (⊆). Consider β ∈ M . We must show that
(P(R))X
random functions. We first define quotient spaces. β =q◦αforsomeα∈M .ByProp.17,β ∈M
P(RX) (P(R))X
means that the uncurried function (uncurryβ): R×X →
Proposition 24. Let (X,M ) be a quasi-Borel space, let
X P(R) is a morphism. As above, this morphism corre-
Y beaset,andletq: X →Y beasurjection.Then(Y,M )
Y sponds to a kernel (uncurryβ)]: (R×X)×ΣR → [0,1].
is a quasi-Borel space with M ={q◦α | α∈M }.
Y X We use the randomization lemma (Prop. 23) to find a
We call such a space a quotient space. measurable function f : R × (R × X) → R such that
β
(uncurryβ)]((r,x),U) = υ{s | f (s,(r,x)) ∈ U}. By
Theorem 25. Let (X,M ) be a quasi-Borel space. The β
X Prop. 14(1) and the fact that the σ-algebra of a prod-
space (P(R))X of kernels is a quotient of the space P(RX)
uct quasi-Borel space R × (R × X) includes the prod-
of random functions.
uct σ-algebras ΣR ⊗ ΣMR×X, this function fβ is also
Before proving this theorem, we use Prop. 23 to give an a morphism. Let γ: R → (RX)R be given by γ d=ef
alternative characterization of our probability monad. λr.λs.λx.f (s,(r,x)). This is a morphism since we can
β
interpret λ-calculus in a cartesian closed category. Let
Lemma 26. Let (X,M ) be a quasi-Borel space. The
function q: XR → P(X)Xgiven by q(α) d=ef [α,υ] is a sur- α: R→P(RX)begivenbyα(r)d=ef[γ(r),υ];thisfunction
is in M by Lemma 26. Moreover a direct calculation
jection, with corresponding quotient space (P(X),M ): P(RX)
P(X) gives β =q◦α, as required.
MP(X) ={λr ∈R.[γ(r),υ] | γ ∈MXR}, (6) VIII. De Finetti’s theorem
where υ is the uniform distribution on [0,1]. De Finetti’s theorem [8] is one of the foundational re-
sultsinBayesianstatistics.Itsaysthateveryexchangeable
Proof notes. The direction (⊆) follows immediately from sequence of random observations on R or another well-
Prop.23.Forthedirection(⊇)wemustconsiderγ ∈M
XR behaved measurable space can be modeled accurately by
andshowthat(λr ∈R.[γ(r),υ])isinM .Thisfollows
P(X) the following two-step process: first choose a probability
by considering the kernel k: R → G(R×R) with k(r) = measure on R randomly (according to some distribution
υ⊗δ , so that [γ(r),υ] = [uncurry(γ),k(r)]. Here we are
r on probability measures) and then generate a sequence
using Prop. 22.
with independent samples from this measure. Limiting
observations to values in a well-behaved space like R in
Proof of Theorem 25. We consider the evident morphism
q: P(RX) → (P(R))X that comes from the monadic the theorem is important: Dubins and Freedman proved
def that the theorem fails for a general measurable space [9].
strength. That is, (q([α,µ]))(x) = [λr.α(r)(x),µ]. We
In this section, we show that a version of de Finetti’s
show that q is a quotient morphism.
theorem holds for all quasi-Borel spaces, not just R. Our
We first show that q is a surjection. Note that to give
resultdoesnotcontradictDubinsandFreedman’sobstruc-
a morphism k: (X,M ) → P(R) is to give a measur-
X tion;probabilitymeasuresonquasi-Borelspacesmayonly
able function (X,ΣMX) → G(R), since (P(R),MP(R)) ∼= use R as their source of randomness, whereas those on
(G(R),M ) (Prop. 4(4)) and by using the adjunc-
ΣG(R) measurablespacesareallowedtouseanymeasurablespace
tion between measurable spaces and quasi-Borel spaces
forthesamepurpose.Aswewillshowshortly,thiscareful
(Prop. 14(1)). Directly, we understand a morphism
choice of random source makes it possible to generalize
k: (X,MX)→P(R)asthekernelk]: X×ΣR →[0,1]with key arguments behind a proof of de Finetti’s theorem [2]
k](x,U) d=ef µ (α -1(U)) whenever k(x) = [α ,µ ]. The
x x x x to quasi-Borel spaces.
definition of k] does not depend on the choice of αx,µx. Let(X,M )beaquasi-Borelspaceand(Xn,M )the
X Xn
Now we can use the randomization lemma (Prop. 23) productquasi-BorelspaceQn X foreachpositiveinteger
to find a measurable function f : R × X → R such i=1
k] n. Recall that P(X) consists of equivalence classes [β,ν]
that k](x,U) = υ{r | f (r,x) ∈ U}. In general, if a
k] of probability measures (β,ν) on X. For n ≥ 1, define a
function Y ×X →Z is jointly measurable then it is also morphism iid : P(X)→P(Xn) by
n
a morphism from the product quasi-Borel space (but the
converse is unknown). So fk] is a morphism, and we can iidn([β,ν])=(cid:2)(Qni=1β◦ιn), (cid:0)(cid:0)ι-n1(cid:1)∗Nni=1ν(cid:1)(cid:3)
form (curryfk]): R→RX. So, where ιn is a measurable isomorphism R → Qni=1R, and
Nn ν is the product measure formed by n copies of
q([curryf , υ])(x)=[λr.curryf (r)(x), υ] i=1
k] k] ν. The name iid represents ‘independent and identically
n
=[λr.f (r,x), υ]=k(x),
k] distributed’.Indeed,iid transformsaprobabilitymeasure
n
(β,ν) on X to the measure of the random sequence in Xn
and q is surjective, as required.
that independently samples from (β,ν). The function iid
Finally we must show that n
is a morphism P(X) → P(Xn) because it can also be
M ={q◦α | α∈M }. written in terms of the strength of the monad P.
(P(R))X P(RX)
Write (Xω,M ) for the countable product Q∞ X. of Theorem 28. This semantics-preserving rewriting and
Xω i=1
some post-processing will give us the equivalence claimed
Definition 27. A probability measure (α,µ) on Xω is ex-
in the lemma.
changeable if for all permutations π on positive integers,
Thefirstdefinitiontounpackisthenotionofprobability
[α,µ]=[α ,µ], measureinP(P(X)).Herearethecrucialfactsthatenable
π
this unpacking. First, for every probability measure (β,ν)
where α (r) d=efα(r) for all r ∈R and i≥1. on P(X), there exist a function γ: R → X in M and a
π i π(i) X
measurable k: R→G(R) such that for all r ∈R,
Theorem 28 (WeakdeFinettiforquasi-Borelspaces). If
(α,µ) is an exchangeable probability measure on Xω, then β(r)=[γ,k(r)].
there exists a probability measure (β,ν) in P(P(X)) such
that for all n ≥ 1, the measure ([β,ν] >>= iidn) on P(Xn) Second, conversely, for a function γ: R → X in MX, a
equals P((−) )(α,µ) when considered as a measure measurable k: R → G(R), and a probability measure ν
1...n
on the product measurable space (Xn,Nn Σ ). (Here on R,
i=1 MX
(−) : Xω →Xn is (x) d=ef(x ,...,x ).) (λr.[γ,k(r)],ν)
1...n 1...n 1 n
Inthetheorem,(β,ν)representsarandomvariablethat is a probability measure in P(P(X)). Thus, we can look
has a probability measure on X as its value. The theorem for (γ,k,ν) in the conclusion of the theorem instead of
says that (every finite prefix of) a sample sequence from (β,ν).
(α,µ) can be generated by first sampling a probability The second is the definition of [β,ν] >>= iid . If we do
n
measure on X according to (β,ν), then generating inde- this and use (γ,k,ν) instead of (β,ν), we have
pendent X-valued samples from the measure, and finally
forming a sequence with these samples. ([β,ν]>>=iid )
n
We call the theorem weak for two reasons. First, the σ- =[(Qn γ)◦ι , (ι-1) (ν >>=λr. Nn k(r))]
algebra Σ includes the product σ-algebra Nn Σ , i=1 n n ∗ i=1
MXn i=1 MX
butwedonotknowthattheyareequal;twodifferentprob- Now, recall that two measures p and q on the product
ability measures in P(Xn) may induce the same measure space (Xn,Nn X) are equivalent if and only if for all
on (Xn,Nn Σ ), although they always induce differ- U ,...,U ∈Σi=1 ,
i=1 MX 1 n MX
ent measures on (Xn,Σ ). In the theorem, we equate
such measures, which letMsXuns use a standard technique for p(U1×...×Un)=q(U1×...×Un).
proving the equality of measures on product σ-algebras.
Thus we must show that
Second, we are unable to construct a version of iid for
n
infinitesequences,i.e.amorphismP(X)→P(Xω)imple-
((−) ◦α) µ)(U ×...×U )
menting the independent identically-distributed random 1...n ∗ 1 n
=(cid:0)(Qn γ) (ν >>=λr. Nn k(r))(cid:1)(U ×...×U )
sequence. The theorem is stated only for finite prefixes. i=1 ∗ i=1 1 n
Therestofthissectionprovidesanoverviewofourproof
This equation is equivalent to the one in the statement of
ofTheorem28.Thestartingpointistounpackdefinitions
def
the lemma, where ξ = ν.
in the theorem, especially those related to quasi-Borel
spaces,andtorewritethestatementofthetheorempurely
Thuswejustneedtoshowhowtoconstructξ,kandγin
in terms of standard measure-theoretic notions.
Lemma29fromagivenexchangeableprobabilitymeasure
Lemma 29. Let (α,µ) be an exchangeable probability (α,µ) on Xω. Constructing ξ and γ is easy:
measureonXω.Then,theconclusionofTheorem28holds
def def
if and only if there exist a probability measure ξ ∈G(R), a ξ = µ, γ = λr.α(r)1.
measurable function k: R→G(R), and γ ∈M such that
X Note that these definitions type-check: ξ =µ∈G(R), and
for all n≥1 and all U ,...,U ∈Σ ,
1 n MX γ ∈ MX because α ∈ MXω and the first projection (−)1
Z n ! is a morphism Xω →X.
Y
[α(r) ∈U ] dµ Constructing k is not that easy. We need to use the
i i
r∈R i=1 factthatµisdefinedoverR,astandardBorelspace.This
Z n (cid:18)Z (cid:19) factitselfholdsbecauseallprobabilitymeasuresonquasi-
Y
= [γ(s)∈Ui]d(k(r)) dξ. Borel spaces use R as their source of randomness. Define
r∈Ri=1 s∈R measurable functions αe,αo: (R,ΣR)→(Xω,ΣMXω) by
Here we express the domain of integration and the inte-
def def
grated variable explicitly to avoid confusion. αe(r)i = α(r)2i (even), αo(r)i = α(r)2i−1 (odd).
Proof. Let (α,µ) be an exchangeable probability measure Since µ is a probability measure on R, there exists a
on Xω. We unpack various definitions in the conclusion measurable function k0: (Xω,ΣMXω) → (G(R),ΣG(R)),
called a conditional probability kernel, such that for all is a conditional expectation of λs.[α (s) ∈ U] with
o j
measurable f: R→R and U ∈(α )-1(Σ ), respect to µ and α . The second equation follows from
e MXω e
Z Z (cid:18)Z (cid:19) the exchangeability of (α,µ). We have just shown that
f(r)dµ= fd((k0◦α )(r)) dµ. (7) λr. R [α (s) ∈U]d(k(r)) is a conditional expectation
e s∈R o j
r∈U r∈U R of λs.[α (s) ∈U] with respect to µ and α .
o i e
DeOfinuer kξ,d=ekfka0n◦dαγe.satisfy the requirement in Lemma 29 Lemma 32. For all n≥1 and all U1,...,Un ∈ΣMX,
because of the following three properties, which follow Z n !
Y
from exchangeability of (α,µ). [αo(s)i ∈Ui] d(k(r))
s∈R i=1
Lemma 30. For all n≥1 and all U1,...,Un ∈ΣMX, Yn Z
Z n ! Z n ! = [αo(s)i ∈Ui]d(k(r))
Y[α(r) ∈U ] dµ= Y[α (r) ∈U ] dµ. i=1 s∈R
i i o i i
r∈R i=1 r∈R i=1 holds for µ-almost all r ∈R.
Proof. Consider n ≥ 1 and U ,...,U ∈ Σ . Pick a
1 n MX Proof. We prove the lemma by induction on n≥1. There
permutation π on positive integers such that π(i)=2i−1
is nothing to prove for the base case n = 1. To handle
for all integers 1 ≤ i ≤ n. Then, [α,µ] = [α ,µ] by the
π the inductive case, assume that n > 1. Let U ,...,U be
1 n
exchangeability of (α,µ). Thus subsetsinΣ .Defineafunctionα0: R→Xω asfollows:
MX
Z n ! Z n ! (cid:26)
Y Y α (r) if 1≤i≤n−1
[α(r) ∈U ] dµ= [α (r) ∈U ] dµ, α0(r) = o i
i i π i i i α (r) otherwise.
r∈R i=1 r∈R i=1 e i−n+1
from which the statement follows. Then, α0 is in MXω, so that α0 is a measurable function
Lemma 31. For all U ∈ΣMX and all i,j ≥1, a(Rbl,eΣkR00): (→Xω(,XΣMω,XΣωM)X→ω)(.GT(Rhu),sΣ, Gth(Re)r)e, ceaxlilsetdscaonmdietaiosnuar-l
Z Z probabilitykernel,suchthatforallmeasurablef: R→R,
[α (s) ∈U]d(k(r))= [α (s) ∈U]d(k(r))
o i o j Z
s∈R s∈R λr. fd((k0 ◦α0)(r))
holds for µ-almost all r ∈R. R 0
Proof. Consider a measurable subset U ∈Σ and i,j ≥ isaconditionalexpectationoff withrespecttoµandthe
1. We remind the reader that when viewedMaXs a function σ-algebra generated by α0. Let
on r ∈R,
Z k0: R→G(R)
[αo(s)i ∈U]d(k(r)) k0 =k0 ◦α0.
s∈R 0
is a conditional expectation of the indicator function The function k0 is measurable because so are k0 and α0.
0
λs.[α (s) ∈ U] with respect to the measure µ and the At the end of this proof, we will show that for µ-almost
o i
σ-algebra generated by the measurable function α : R→ all r ∈R,
e
(Xω,Σ ).Bythealmost-sureuniquenessofconditional
MXω Z
expectation, it suffices to show that
[α (s) ∈U ]d(k(r))
o n n
Z s∈R
Z
λr. [α (s) ∈U]d(k(r))
o j = [α (s) ∈U ]d(k0(r)). (8)
s∈R o n n
s∈R
is also a conditional expectation of λs.[α (s) ∈ U] with
o i Fornow,justassumethatthisequationholdsandseehow
respecttoµandα .PickameasurablesubsetV ∈Σ .
e MXω this assumption lets us complete the proof.
Then,
Recall that k = k ◦α and k0 = k0 ◦α0 are defined
Z (cid:18)Z (cid:19) 0 e 0
in terms of conditional expectation. Thus, they inherit
[α (r)∈V]· [α (s) ∈U]d(k(r)) dµ
e o j all the properties of conditional expectation after minor
r∈R s∈R
Z adjustment. In particular, since the σ-algebra generated
= [α (r) ∈U ∧α (r)∈V]dµ
o j e by α0 is larger than the one generated by α and it makes
r∈R e
Z α measurableforall1≤i≤(n−1),wehavethefollowing
i
= [αo(r)i ∈U ∧αe(r)∈V]dµ. equalities: for µ-almost all r ∈ R and all measurable
r∈R functions h: R→R,
The first equation holds because the function
Z n
Z Y
[α (s) ∈U ]d(k(r))
λr. [α (s) ∈U]d(k(r)) o i i
o j
s∈R s∈Ri=1