Foundations for conditional probability

TL;DR

This paper presents a foundational perspective on probability, characterizing it as a subset of conditional expectation derived from a plausible preorder on random variables, extending traditional definitions to undefined or zero-probability cases.

Contribution

It introduces a novel characterization of probability through plausible preorders, broadening the scope of conditional probability beyond classical rules.

Findings

Probability is characterized as a subset of conditional expectation.

The approach extends conditional probability to cases with zero or undefined probabilities.

Provides a new perspective on the foundations of probability theory.

Abstract

The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is coherent as confirmed by its common formalizations, and by our result that a function is coherent if and only if it is a subset of conditional expectation induced by a plausible preorder on random quantities. In addition to offering a different perspective on conditional probability, our use of a plausible preorder in the role of a fundamental notion extends conditional probability to cases in which the calculation of conditional probability using the P(A|C)=\frac{P(A\wedge C)}{P(C)} rule fails: if P is a coherent function, then it can be extended so that for every event A and nonzero event C holds that P(A|C)=0 if A\wedge C=0 and P(A|C)=1 if A\wedge C=C, no matter whether the unconditional probability P(C) is zero or whether it is defined.

Equations

$$\underset{i=1}{\overset{n}{\bigvee }}{A}_i\lesssim \sum _{i=1}^n{A}_i$$

$$E(X|C)=E(X.C|C)$$

$$E(X+r|C)=E(X|C)+r$$

$$E(X+Y|C)=E(X|C)+E(Y|C)$$

$$E(rX|C)=rE(X|C)$$

$$P(C|D)=E(C|D)$$

$$P(B|D)\le P(C|D)$$

$$0=P(0|D)\le P(C|D)\le P(D|D)=P(1|D)=1$$

$$P(C|D)=0$$

$$P(C|D)=1$$

$$P(\underset{i=1}{\overset{n}{\bigvee }}{A}_i|D)\le \sum _{i=1}^{n}P(A_i|D)$$

$$E(X.C|D)=E(X|C.D).P(C|D)$$

$$E(X.C|D)=0$$

$$E\left(X.C|D\right)=0$$

$$E\left(X.C|D\right)=E\left(X|C.D\right)$$

$$E(X|C.D)=\frac{E(X.C|D)}{P(C|D)}$$

$$E\left(X|C.D\right)=0$$

$$E(X|C.D)=E(X.C|D)$$

$$E\left(X|C.D\right)=\frac{+\infty }{p}$$

$$P(C|D)=\frac{E(X.C|D)}{E(X|C.D)}$$

$$P\left(C|D\right)=0$$

$$0\ne \sum _{i=1}^n{q}_i{C}_i+\sum _{j=1}^m{r}_j(X_j+s_j).D_j$$

$$PV(A+B|C)=PV(A|C)+PV(B|C)$$

$$PV(\underset{i=1}{\overset{n}{\bigvee }}{A}_i|C)\le \sum _{i=1}^{n}PV(A_i|C)$$

$$s=\{\begin{array}{ll}1& \mathrm{if}E(X|C.D)=+\infty \\ -1& \mathrm{if}E(X|C.D)=-\infty \end{array}$$

$$Y=sX$$

$$s=\{\begin{array}{ll}1& \mathrm{if}E(X.C|D)=+\text{∞}\\ -1& \mathrm{if}E(X.C|D)=-\text{∞}\end{array}$$

$$Y=sX$$

Full text

According to supplementary results, probability is invariably described as a coherent function and every coherent function can be extended to a plausibly complete function. Thus, probability can be, without loss of generality, formalized as a plausibly complete function.

As an illustration, consider a plausibly complete probability $P$. Then for every event $A$ and nonzero event $C$ holds that $P(A|C)=0$ if $A\wedge C=0$ and $P(A|C)=1$ if $A\wedge C=C$, no matter whether the unconditional probability $P(C)$ is zero or whether it is defined. In contrast to that, the common approach using the $\frac{P(A\wedge C)}{P(C)}$ ratio to define the conditional probability $P(A|C)$ leaves the probability plausibly incomplete in general, since it leaves the conditional probability undefined whenever $P(C)$ is zero or undefined.

1 Introduction

The probability foundations provided by A. N. Kolmogorov Kolmogorov (1933) define conditional probability as a ratio of unconditional probabilities. A. Hájek Hájek (2003) brings several reasons why a more adequate formalization of conditional probability is needed.

R.T. Cox Cox (1961) contributed a theorem deriving the laws of conditional probability from a set of postulates. According to J. Halpern Halpern (1999), M. J. Dupré and F. J. Tipler Dupré, Tipler (2009), J. B. Paris Paris (2006) as well as other authors, Cox’s approach is non-rigorous. To be valid, it needs additional assumptions which are complicated and nontrivial.

B. de Finetti de Finetti (1975) developed the foundations of conditional probability around the idea of a partially ordered algebra of random quantities, on which existence of a real-valued function having the fundamental properties of conditional expectation is postulated.

We take a more general approach. Instead of postulating the existence of a real-valued function having the fundamental properties of conditional expectation, we examine a relation on random quantities called a plausible preorder. We show that a plausible preorder on random quantities naturally induces a set of conditional preorders. The conditional preorders naturally induce a conditional expectation that, in general, is a partial function assigning elements of the extended real line to pairs consisting of a random quantity and a nonzero event. The conditional expectation is demonstrated to satisfy a generalized form of probability rules. In the final section, we provide a formal description of the notion of coherence and prove that all formalizations of probability discussed in this article are coherent. Finally, we demonstrate that a function is coherent if and only if it can be extended to a conditional expectation naturally induced by a regular plausible preorder.

2 Random quantities

We define the notion of a random quantity using the axiomatic approach proposed by B. de Finetti de Finetti (1975).

2.1 Definition

Let $\mathcal{T}$ denote the set of random quantities. We postulate that $\mathcal{T}$ is a unital associative commutative algebra over real numbers, i.e. $\mathcal{T}$ is a set equipped with addition, multiplication by real numbers and multiplication, such that

• •

if $X,Y,Z\in\mathcal{T}$, then $\left(X+Y\right)+Z=X+\left(Y+Z\right)$ (associativity of addition)

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if $X,Y\in\mathcal{T}$, then $X+Y=Y+X$ (commutativity of addition)

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there exists an element $\mathbf{0}\in\mathcal{T}$, such that $X+\mathbf{0}=X$ for every $X\in\mathcal{T}$ (identity element of addition)

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if $X\in\mathcal{T}$, then there exists an element $-X\in\mathcal{T}$ such that $\mathbf{0}=X+\left(-X\right)$ (inverse elements of addition)

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if $r,s$ are real numbers and $X\in\mathcal{T}$, then $\left(rs\right)X=r\left(sX\right)$ (compatibility of multiplication by real numbers with real multiplication)

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if $X\in\mathcal{T}$, then $1X=X$ (identity element of multiplication by real numbers)

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if $r$ is a real number and $X,Y\in\mathcal{T}$, then $r\left(X+Y\right)=rX+rY$ (distributivity of multiplication by real numbers with respect to addition)

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if $r,s$ are real numbers and $X\in\mathcal{T}$, then $\left(r+s\right)X=rX+sX$ (distributivity of multiplication by real numbers with respect to real addition)

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if $X,Y,Z\in\mathcal{T}$, then $\left(X.Y\right).Z=X.\left(Y.Z\right)$ (associativity of multiplication)

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if $X,Y\in\mathcal{T}$, then $X.Y=Y.X$ (commutativity of multiplication)

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if $X,Y,Z\in\mathcal{T}$, then $\left(X+Y\right).Z=X.Z+Y.Z$ (distributivity)

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if $r,s$ are real numbers and $X,Y\in\mathcal{T}$, then $\left(rX\right).\left(sY\right)=\left(rs\right)\left(X.Y\right)$ (compatibility with multiplication by real numbers)

• •

there exists an element $\mathbf{1}\in\mathcal{T}$, such that $X.\mathbf{1}=X$ for every $X\in\mathcal{T}$ (identity element of multiplication)

2.2 Canonical embedding of real numbers

Per 2.1, $\mathcal{T}$ has got an identity element of multiplication that we can denote $\mathbf{1}$. We define a function $F$ from the set of real numbers $\mathbb{R}$ to $\mathcal{T}$ such that $F(r)=r\mathbf{1}$ for every real number $r$. Defined this way, $F$ is a map embedding the set of real numbers in $\mathcal{T}$. We call this embedding the canonical embedding of real numbers in $\mathcal{T}$. Using the canonical embedding of real numbers, instead of writing $r\mathbf{1}\in\mathcal{T}$ for a real $r$, we simply write $r\in\mathcal{T}$ from now on.

2.3 Motivational example

Alice is going to throw a coin in the presence of a notary. Bob knows that Carol shall pay him a specific amount $S_{H}$ if Alice throws heads, and a specific amount $S_{T}$ if Alice throws tails.

Bob conceives a set $\mathcal{T}$ containing pairs of real numbers $(X_{H},X_{T})$ and defines

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addition: $(X_{H},X_{T})+(Y_{H},Y_{T})=(X_{H}+Y_{H},X_{T}+Y_{T})$

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multiplication by real numbers: $r(X_{H},X_{T})=(rX_{H},rX_{T})$

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multiplication: $(X_{H},X_{T}).(Y_{H},Y_{T})=(X_{H}Y_{H},X_{T}Y_{T})$

Bob’s $\mathcal{T}$ with these operations is unital, since $(1,1)$ is its identity element with respect to multiplication, associative and commutative algebra over reals. Per 2.1, the elements of $\mathcal{T}$ are random quantities. Denoting $H=(1,0)$ and $T=(0,1)$ and using the canonical embedding of real numbers, $H+T=1$ and $H.T=0$. Carol’s payment is represented by random quantity $S=(S_{H},S_{T})=S_{H}H+S_{T}T$.

3 Events

3.1 Definition

Let $A$ be a random quantity. We say that $A$ is an event if it is idempotent, i.e. if $A.A=A$. We denote the set of events $\mathcal{E}(\mathcal{T})$ and the set of nonzero events $\mathcal{E}_{0}(\mathcal{T})$. On $\mathcal{E}(\mathcal{T})$ we define

• •

negation, for event $A$ its negation $\neg A$ is defined as $1-A$

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conjunction, for events $A,B$ their conjunction $A\wedge B$ is defined as $A.B$

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disjunction, for events $A,B$ their disjunction $A\vee B$ is defined as $A+B-A.B$

With these operations,

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$1$ is the unit element of conjunction, i.e. if $A$ is an event, then $A\wedge 1=1\wedge A=A$

• •

[math] is the unit element of disjunction, i.e. if $A$ is an event, then $A\vee 0=0\vee A=A$ and

• •

$\mathcal{E}(\mathcal{T})$ is a Boolean algebra.

3.2 Natural order

Let $\mathcal{A}$ be a Boolean algebra and $A,B$ be its elements. The natural order on $\mathcal{A}$ is defined so that $A\leq B$ if $A\wedge B=A$.

Then

• •

the natural order is a partial order,

• •

the minimal element of $\mathcal{A}$ in the natural order is 0 and

• •

the maximal element of $\mathcal{A}$ in the natural order is 1.

Since $\mathcal{E}(\mathcal{T})$ is a Boolean algebra, there is a natural order on $\mathcal{E}(\mathcal{T})$.

3.3 Atoms

Let $\mathcal{A}$ be a Boolean algebra. We say that $D$ is an atom of $\mathcal{A}$, if $D\in\mathcal{A}$ and $D$ is a minimal nonzero element of $\mathcal{A}$ in the natural order.

We say that $\mathcal{A}$ is atomic if for every nonzero element $A\in\mathcal{A}$ there is an atom $D\in\mathcal{A}$ such that $D\leq A$ in the natural order.

3.4 Motivational example

Consider the algebra $\mathcal{T}$ defined in 2.3. Then

• •

$\mathcal{E}\left(\mathcal{T}\right)=\{0,H,T,1\}$

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$\mathcal{E}_{0}\left(\mathcal{T}\right)=\{H,T,1\}$

• •

the atoms of $\mathcal{E}\left(\mathcal{T}\right)$ are $H$ and $T$ and

• •

$\mathcal{E}\left(\mathcal{T}\right)$ is atomic.

3.5 Positive combinations of nonzero events

Let $n\geq 1$, $p_{1},\ldots,p_{n}$ be positive real numbers and $C_{1},\ldots,C_{n}$ be nonzero events. Then $0\neq\sum_{i=1}^{n}p_{i}C_{i}$.

4 Plausible preorder

4.1 Definition

We say that a relation $\lesssim$ on $\mathcal{T}$ is a plausible preorder if it has these properties:

4.1.1 Plausible property

If $A$ is an event, then $0\lesssim A$.

4.1.2 Additive property

If $0\lesssim X$ and $0\lesssim Y$, then $0\lesssim X+Y$.

4.1.3 Multiplicative property

If $0\lesssim X$ and $q$ is a nonnegative real number, then $0\lesssim qX$.

4.1.4 Extension property

$X\lesssim Y$ if and only if $0\lesssim Y-X$.

4.2 Motivational example

In the algebra $\mathcal{T}$ described in 2.3, Bob defines

• •

$0\lesssim X=(X_{H},X_{T})$ if $0\leq X_{H}+X_{T}$

• •

$X\lesssim Y$ if $0\lesssim Y-X$

It is easy to verify that $\lesssim$ is a plausible preorder.

4.3 The greatest plausible preorder

Relation $\lesssim=\mathcal{T}\times\mathcal{T}$ is a plausible preorder and the greatest relation* *on $\mathcal{T}$ with respect to inclusion.

4.4 Properties

4.4.1 Reflexivity

A plausible preorder is reflexive.

4.4.2 Transitivity

A plausible preorder is transitive.

4.4.3 Relation to the natural order of events.

A plausible preorder contains the natural order of events as its subset.

4.4.4 Relation to the order of real numbers.

A plausible preorder contains the order of real numbers as its subset.

4.4.5 Intersection of a set of plausible preorders

If $\mathcal{X}$ is a nonempty set containing plausible preorders, then $\bigcap\mathcal{X}$ is a plausible preorder.

4.4.6 The smallest plausible preorder containing a relation

If $R$ is a relation on $\mathcal{T}$, then there is a relation $\lesssim$ that is the smallest plausible preorder with respect to inclusion containing $R$.

4.4.7 Subadditivity

If $A_{1},\ldots,A_{n}$ are events and $\lesssim$ is a plausible preorder, then

[TABLE]

5 Plausible equivalence

5.1 Definition

Let $\lesssim$ be a plausible preorder. We say that a relation $\sim$ is the equivalence part of $\lesssim$ if for any $X,Y$ holds that $X\sim Y$ if and only if $(X\lesssim Y)\wedge(Y\lesssim X)$. We say that a relation $\sim$ on $\mathcal{T}$ is a plausible equivalence if there is a plausible preorder $\lesssim$ such that $\sim$ is its equivalence part.

5.2 Motivational example

Let $\lesssim$ be the plausible preorder defined in 4.2. Then $0\sim X=\left(X_{H},X_{T}\right)$ if and only if $0=X_{H}+X_{T}$.

5.3 Fundamental properties

5.3.1 Plausible property

Let $p_{1},\ldots,p_{n}$ be positive real numbers, $A_{1},\ldots,A_{n}$ be events and $0\sim\sum_{i=1}^{n}p_{i}A_{i}$. Then $0\sim A_{i}$ for every $i\in\left\{1,\ldots,n\right\}$.

5.3.2 Reflexivity

$0\sim 0$.

5.3.3 Additive property

If $0\sim X$ and $0\sim Y$, then $0\sim X+Y$.

5.3.4 Multiplicative property

If $0\sim X$ and $r$ is a real number, then $0\sim rX$.

5.3.5 Extension property

$X\sim Y$ if and only if $0\sim Y-X$.

5.4 Sufficiency of the fundamental properties

Every relation $\sim$ having the fundamental properties of a plausible equivalence is a plausible equivalence.

6 Plausible strict partial order

6.1 Definition

Let $\lesssim$ be a plausible preorder. We say that a relation $\lnsim$ is the strict part of $\lesssim$ if for any $X,Y$ holds that $X\lnsim Y$ if and only if $\left(X\lesssim Y\right)\land\lnot\left(Y\lesssim X\right)$. We also say that a relation $\lnsim$ is a plausible strict partial order if there is a plausible preorder $\lesssim$ such that $\lnsim$ is its strict part.

6.2 Motivational example

Let $\lesssim$ be the plausible preorder defined in 4.2. Then $0\lnsim X=\left(X_{H},X_{T}\right)$ if and only if $0<X_{H}+X_{T}$.

6.3 Fundamental properties

6.3.1 Plausible property

If $A$ is an event, $\neg(0\lnsim A)$ and $0\lnsim X$, then $0\lnsim X+A$ and $0\lnsim X-A$.

6.3.2 Antireflexivity

$\neg(0\lnsim 0)$.

6.3.3 Additive property

If $0\lnsim X$ and $0\lnsim Y$, then $0\lnsim X+Y$.

6.3.4 Multiplicative property

If $0\lnsim X$ and $p$ is a positive real number, then $0\lnsim pX$.

6.3.5 Extension property

$X\lnsim Y$ if and only if $0\lnsim Y-X$.

6.4 Sufficiency of the fundamental properties

Every relation $\lnsim$ having the fundamental properties of a plausible strict partial order is a plausible strict partial order.

7 Conditional preorder

7.1 Definition

Let $\lesssim$ be a plausible preorder and $C$ be an event. We define the conditional preorder $\lesssim_{C}$ so that $X\lesssim_{C}Y$ if $X.C\lesssim Y.C$.

7.2 Properties

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a conditional preorder is a plausible preorder

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$\lesssim_{1}$ is identical with $\lesssim$

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$\lesssim_{0}$ is the greatest plausible preorder

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$\lnsim_{0}$ is empty, i.e. there are no random quantities $X,Y$ such that $X\lnsim_{0}Y$

8 Regularity of a plausible preorder

8.1 Definition

We say that a plausible preorder $\lesssim$ is

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degenerate if $0\sim 1$

• •

regular if for every nonzero event $C$ holds $0\lnsim C$

8.2 Motivational example

The plausible preorder $\lesssim$ defined in 4.2 is regular.

8.3 Properties

• •

A conditional preorder $\lesssim_{C}$ is degenerate if and only if $0\sim C$.

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The greatest plausible preorder is degenerate.

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If a plausible preorder is degenerate, then for every pair of random quantities $X,Y$ in the linear span of $\mathcal{E}(\mathcal{T})$ holds $X\sim Y$. In particular, for every pair of real numbers $r,s$ holds $r\sim s$, and for every pair of events $A,B$ holds $A\sim B$.

• •

A plausible preorder is nondegenerate if and only if it coincides with the order of real numbers on $\mathbb{R}$.

9 Extended real line

9.1 Definition

We define the extended real line as the set $\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,+\infty\}$, where $\mathbb{R}$ is the set of real numbers.

9.2 Order

We extend the order of real numbers to $\overline{\mathbb{R}}$ so that $-\infty\leq x\leq+\infty$ for every $x\in\overline{\mathbb{R}}$, turning $\overline{\mathbb{R}}$ into a linearly ordered set. In this order, every subset $U$ of $\overline{\mathbb{R}}$ has both the least upper bound (supremum) denoted $\sup U$ and the greatest lower bound (infimum) denoted $\inf U$. In particular,

• •

$\sup\emptyset=-\infty$

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$\inf\emptyset=+\infty$

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$\sup\mathbb{R}=\sup\overline{\mathbb{R}}=+\infty$

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$\inf\mathbb{R}=\inf\overline{\mathbb{R}}=-\infty$

9.3 Arithmetic

We extend the artithmetic operations on real numbers to $\overline{\mathbb{R}}$ so that

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if $x\neq-\infty$, then $(+\infty)+x=x+(+\infty)=+\infty$

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if $x\neq+\infty$, then $(-\infty)+x=x+(-\infty)=-\infty$

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if $x>0$, then $x.(+\infty)=(+\infty).x=+\infty$

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if $x>0$, then $x.(-\infty)=(-\infty).x=-\infty$

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if $x<0$, then $x.(+\infty)=(+\infty).x=-\infty$

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if $x<0$, then $x.(-\infty)=(-\infty).x=+\infty$

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if $x$ is a real number, then $\frac{x}{+\infty}=\frac{x}{-\infty}=0$

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if $x$ is a positive real number, then $\frac{+\infty}{x}=+\infty$

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if $x$ is a positive real number, then $\frac{-\infty}{x}=-\infty$

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if $x$ is a negative real number, then $\frac{+\infty}{x}=-\infty$

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if $x$ is a negative real number, then $\frac{-\infty}{x}=+\infty$

Other expressions than the above are undefined. For example, the expressions

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$(+\infty)+(-\infty)$

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$(-\infty)+(+\infty)$

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$0.(+\infty)$

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$0.(-\infty)$

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$(+\infty).0$

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$(-\infty).0$

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$\frac{x}{0}$

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$\frac{+\infty}{+\infty}$

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$\frac{+\infty}{-\infty}$

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$\frac{-\infty}{+\infty}$

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$\frac{-\infty}{-\infty}$

are all undefined.

10 Expectation naturally induced by a plausible preorder

10.1 Definition

Let $\lesssim$ be a plausible preorder and $X$ be a random quantity. Denoted $E\left(X\right)$, the expectation of $X$ (more precisely, the expectation of $X$ naturally induced by $\lesssim$) is

• •

a real number $x$, if for every positive real number $\epsilon$ holds $-\epsilon\lnsim X-x\lnsim\epsilon$

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$+\infty$, if for every real number $y$ holds $y\lnsim X$

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$-\infty$, if for every real number $y$ holds $X\lnsim y$

• •

not defined, if none of the above holds

10.2 Motivational example

Consider the plausible preorder defined in example 4.2. For every random quantity $X=(X_{H},X_{T})$ holds that $E(X)=\frac{1}{2}X_{H}+\frac{1}{2}X_{T}$. In particular, $E(H)=E(T)=\frac{1}{2}$.

10.3 Relation to regularity of plausible preorder

Let $\lesssim$ be a plausible preorder and $r$ be a real number.

• •

if $0\lnsim 1$, then $E(r)=r$

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if $0\sim 1$, then $E(r)$ is not defined

• •

if $\lesssim$ is the maximal plausible preorder and $X$ is a random quantity, then $E\left(X\right)$ is not defined

10.4 Preorder consistency

Let $r$ be a real number and $X,Y$ be random quantities. 10.1 implies that

• •

if $E(X)$ exists and $r<E(X)$, then $r\lnsim X$

• •

if $E(X)$ exists and $E(X)<r$, then $X\lnsim r$

• •

if both $E(X)$ and $E(Y)$ exist and $E(X)<E(Y)$, then $X\lnsim Y$

• •

if both $E(X)$ and $E(Y)$ exist and $X\lesssim Y$, then $E(X)\leq E(Y)$

10.5 Existence and uniqueness

Let $\lesssim$ be a plausible preorder and $X$ be a random quantity. Then $X$ has expectation if and only if in the extended real line $\overline{\mathbb{R}}$ holds that $\sup\{r\in\mathbb{R}|r\lnsim X\}=\inf\{r\in\mathbb{R}|X\lnsim r\}$. In such case, $E(X)=\sup\{r\in\mathbb{R}|r\lnsim X\}=\inf\{r\in\mathbb{R}|X\lnsim r\}$.

11 Conditional expectation

11.1 Definition

Let $\lesssim$ be a plausible preorder, $X$ be a random quantity and $C$ be an event. Denoted $E(X|C)$, the conditional expectation of $X$ given $C$ is defined using 10.1 as the expectation of $X$ naturally induced by the conditional preorder $\lesssim_{C}$.

11.2 Motivational example

Consider the plausible preorder defined in 4.2. For every random quantity $X=(X_{H},X_{T})$ holds that $E(X|H)=X_{H}$ and $E(X|T)=X_{T}$.

11.2.1 Relation to regularity of plausible preorder

Let $A,C$ be events and $r$ be a real number. Then

• •

if $0\lnsim C$ and $r$ is a real number, then $E\left(r|C\right)=E(rC|C)=r$

• •

if $0\sim C$ then neither $E(r|C)$ nor $E(rC|C)$ is defined

11.3 As a function

Per 10.3 and 10.5, conditional expectation is a partial function from $\mathcal{T}\times\mathcal{E}_{0}(\mathcal{T})$ to $\overline{\mathbb{R}}$.

12 Rules

Let $\lesssim$ be a plausible preorder, $E(X|C)$ be the conditional expectation naturally induced by $\lesssim$, $X,Y$ be random quantities, $B,C,D$ be events and $r$ be a real number. The rules the conditional expectation follows are:

12.1 Consistency

$E(X|C)$ exists if and only if $E(X.C|C)$ exists. In case it exists,

[TABLE]

12.2 Real additivity

If $E(X|C)$ exists, then

[TABLE]

12.3 General additivity

If the expression $E(X|C)+E(Y|C)$ makes sense, then

[TABLE]

12.4 Homogeneity

If the expression $rE(X|C)$ makes sense, then

[TABLE]

12.5 Conditional probability

If $E(C|D)$ exists, we, compatibly with Thomas Bayes Bayes (1763), denote

[TABLE]

and say that it is the conditional probability of $C$ given $D$. For $P(C|1)$ we also use a simpler notation $P(C)$.

12.6 Monotonicity

If both $P(B|D)$ and $P(C|D)$ exist and $B\leq C$ in the natural order of events, then

[TABLE]

12.7 Minimal and maximal probability

If $P(C|D)$ exists, then

[TABLE]

12.8 Completeness

• •

If $P(B|D)=0$ and $C\leq B$ in the natural order of events, then

[TABLE]

• •

If $P(B|D)=1$ and $B\leq C$ in the natural order of events, then

[TABLE]

12.9 Subadditivity

If $A_{1},\ldots,A_{n}$ are events and all of $P(A_{1}|D),\ldots,P(A_{n}|D),P(\bigvee_{i=1}^{n}A_{i}|D)$ exist, then

[TABLE]

12.10 Bayes’ rule

12.10.1 Chain form

If the expression $E(X|C.D).P(C|D)$ makes sense, then

[TABLE]

12.10.2 Chain form, zero E(X|C.D)

If $E(X|C.D)=0$, then

[TABLE]

12.10.3 Chain form, zero P(C|D)

If $P\left(C|D\right)=0$ and there is a real number $p$ such that $-p\lesssim_{C.D}X\lesssim_{C.D}p$, then

[TABLE]

12.10.4 Chain form, infinite E(X|C.D)

If $E\left(X|C.D\right)\in\{-\infty,+\infty\}$ and there is a positive real number $p$ such that $p\lesssim_{D}C$, then

[TABLE]

12.10.5 Conditional form

If the expression $\frac{E(X.C|D)}{P(C|D)}$ makes sense, then

[TABLE]

12.10.6 Conditional form, zero E(X.C|D)

If $E\left(X.C|D\right)=0$ and there is a positive real number $p$ such that $p\lesssim_{D}C$, then

[TABLE]

12.10.7 Conditional form, infinite E(X.C|D)

If $E(X.C|D)\in\{-\infty,+\infty\}$, then

[TABLE]

12.10.8 Conditional form, zero P(C|D)

If $P(C|D)=0$ and there is a real number $p$ such that $1\lesssim_{D}pX.C$, then $p\neq 0$ and

[TABLE]

12.10.9 P form

If the expression $\frac{E(X.C|D)}{E(X|C.D)}$ makes sense, then

[TABLE]

12.10.10 P form, zero E(X.C|D)

If $E\left(X.C|D\right)=0$ and there is a real number $p$ such that $1\lesssim_{C.D}pX$, then

[TABLE]

12.10.11 P form, infinite E(X|C.D)

If $E\left(X|C.D\right)\in\left\{-\infty,+\infty\right\}$ and there is a real number $p$ such that $-p\lesssim_{D}X.C\lesssim_{D}p$, then

[TABLE]

13 Coherence

13.1 Definition

We say that $PV$ is a coherent function if it is a partial function from $\mathcal{T}\times\mathcal{E}\left(\mathcal{T}\right)$ to $\overline{\mathbb{R}}$ such that if

• •

$n\geq 0,m\geq 1$ are integers

• •

$q_{1},\ldots,q_{n}$ are nonnegative real numbers

• •

$r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$ are real numbers

• •

$C_{1},\ldots,C_{n}$, $D_{1},\ldots,D_{m}$ are events

• •

$X_{1},\ldots,X_{m}$ are random quantities

• •

$r_{j}(PV(X_{j}|D_{j})+s_{j})>0$ for every $j\in\left\{1,\ldots,m\right\}$

then

[TABLE]

13.2 Kolmogorovian plausible values

Let $\mathbb{R}_{\geq 0}$ denote the set of nonnegative real numbers. We say that $PV$ is a Kolmogorovian plausible value if

• •

$PV$ is a function from $\mathcal{F}$ to $\mathbb{R}_{\geq 0}$, where $\mathcal{F}$ is a nonempty subset of $\mathcal{E}(\mathcal{T})$ closed under negation and conjunction

• •

$PV(1)=1$ (unitarity)

• •

if $A,B\in\mathcal{F}$ and $A.B=0$, then $PV(A+B)=PV(A)+PV(B)$ (additivity)

Using the notation $PV(A|1)=PV(A)$, we can handle every Kolmogorovian plausible value as a function from $\mathcal{F}\times\{1\}$ to $\mathbb{R}_{\geq 0}$.

13.2.1 Coherence

Every Kolmogorovian plausible value is coherent.

13.3 Coxian plausible values

We say that $PV$ is a Coxian plausible value if

• •

$PV$ is a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$, where $\mathcal{F}$ is a nonempty subset of $\mathcal{E}(\mathcal{T})$ closed under negation and conjunction and $\mathcal{F}_{0}$ is the set containing all elements of $\mathcal{F}$ except for [math]

• •

if $C\in\mathcal{F}_{0}$, then $PV(C|C)>0$ (positivity)

• •

if $A\in\mathcal{F}$ and $C\in\mathcal{F}_{0}$, then $PV(1-A|C)=1-PV(A|C)$ (negation formula)

• •

if $A,C,D$ are elements of $\mathcal{F}$ and $C.D\neq 0$, then $PV(A.C|D)=PV(A|C.D).PV(C|D)$ (Bayes’ rule)

13.3.1 Basic properties

Let $PV$ be a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$ that satisfies definition 13.3, $A\in\mathcal{F}$ and $C\in\mathcal{F}_{0}$. Then

• •

$PV(C|C)=1$

• •

$PV(1|C)=1$

• •

$PV(0|C)=0$

• •

$PV(A.C|C)=PV(A|C)$

13.3.2 Sum rule

Let $PV$ be a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$ that satisfies definition 13.3. Let $A,B\in\mathcal{F}$ such that $A.B=0$ and let $C\in\mathcal{F}_{0}$. Then

[TABLE]

13.3.3 Subadditivity

Let $PV$ be a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$ that satisfies definition 13.3. Let $A_{1},\ldots,A_{n}\in\mathcal{F}$ and let $C\in\mathcal{F}_{0}$. Then

[TABLE]

13.3.4 Coherence

Every Coxian plausible value is coherent.

13.4 Dupré-Tiplerian plausible values

We say that $PV$ is a Dupré-Tiplerian plausible value if

• •

$PV$ is a partial function from $\mathcal{T}\times\mathcal{C}$ to $\mathbb{R}$, where $\mathcal{C}$ is a subset of $\mathcal{E}_{0}(\mathcal{T})$ closed under disjunction

• •

if $A$ is an event and $C\in\mathcal{C}$, then $PV(A|C)$ exists and $PV(A|C)\geq 0$ (nonnegativity)

• •

if $C\in\mathcal{C}$, then $PV(C|C)>0$ (positivity)

• •

if $r\in\mathbb{R}$, $C\in\mathcal{C}$, $X$ is a random quantity and $PV(X|C)$ exists, then $PV(rX|C)=r.PV(X|C)$ (homogeneity)

• •

if $C\in\mathcal{C}$, $X$, $Y$ are random quantities and both $PV(X|C)$ and $PV(Y|C)$ exist, then $PV(X+Y|C)=PV(X|C)+PV(Y|C)$ (additivity)

• •

if $C.D\in\mathcal{C}$, $D\in\mathcal{C}$, $X$ is a random quantity and $PV(X|C.D)$ exists, then $PV(X.C|D)=PV(X|C.D).PV(C|D)$ (Bayes’ rule)

13.4.1 Subadditivity

Let $PV$ be a function from $\mathcal{T}\times\mathcal{C}$ to $\mathbb{R}$ that satisfies definition 13.4. Let $A_{1},\ldots,A_{n}$ be events and let $C\in\mathcal{C}$. Then

[TABLE]

13.4.2 Coherence

Every Dupré-Tiplerian plausible value is coherent.

13.5 Characterizations

Let $PV$ be a partial function from $\mathcal{T}\times\mathcal{E}(\mathcal{T})$ to $\overline{\mathbb{R}}$. Then the following characterizations are equivalent:

$PV$ is coherent 2. 2.

$PV$ can be extended to conditional expectation naturally induced by a regular plausible preorder 3. 3.

$PV$ can be extended to conditional expectation naturally induced by a plausible preorder

13.6 Probability as a plausibly complete function

We say that a function is plausibly complete, if it is naturally induced by a regular plausible preorder. According to 13.2.1, 13.3.4, 13.4.2 and 13.5, we can, without loss of generality, characterize probability as a plausibly complete function.

14 Conclusion

Assigning the role of a primitive notion to the notion of a plausible preorder, our formalization offers a different perspective on the foundations of probability than the formalizations discussed in the introduction. Our approach neither forces us to define conditional probability by a ratio of unconditional probabilities which is criticized as inadequate, nor does it force us to postulate conditional probability to have other properties open to doubt. The formalization is supported by theorem 13.5, confirming that it encompasses all coherent instances of probability. We supplement it by verifying that according to all formalizations of the probability notion discussed in the introduction, probability is coherent. To illustrate that our formalization satisfies the main Hájek’s Hájek (2003) requirements, consider a nonzero event $C$ such that $P\left(C\right)$ is either zero or undefined. Because of that, the ratio $\frac{P\left(A\wedge C\right)}{P\left(C\right)}$ leaves the conditional probability $P\left(A|C\right)$ undefined. On the other hand, once probability is coherent, theorem 13.5 confirms that it can be extended to a plausibly complete function, i.e. to a conditional expectation naturally induced by a regular plausible preorder. Definition 11.1 applied to a regular plausible preorder yields that $P\left(A|C\right)=0$ if $A\wedge C=0$ and $P\left(A|C\right)=1$ if $A\wedge C=C$, no matter whether $P\left(C\right)$ is zero or whether it is defined.

15 Appendix

15.1 Proof of 3.5

Let $n\geq 1$, $p_{1},\ldots,p_{n}$ be positive real numbers and $C_{1},\ldots,C_{n}$ be nonzero events. Let $\mathcal{A}$ be the Boolean subalgebra of $\mathcal{E}\left(\mathcal{T}\right)$ generated by $C_{1},\ldots,C_{n}$. Since $\mathcal{A}$ is finitely generated, it is finite and atomic Givant, Halmos (2008). Since $C_{1}$ is nonzero, there is an atom $D$ of $\mathcal{A}$ such that $D\leq C_{1}$ in the natural order, i.e. $C_{1}.D=D$. Since $D$ is an atom, for every $i\in\{2,\ldots,n\}$ either $C_{i}.D=D$ or $C_{i}.D=0$. Therefore, $\left(\sum_{i=1}^{n}p_{i}C_{i}\right).D=p_{1}D+\sum_{i=2}^{n}p_{i}C_{i}.D=pD$, where $p\geq p_{1}$ is a real number. Since $D$ is nonzero and $p$ is a positive real number, $pD$ is nonzero. Since $\left(\sum_{i=1}^{n}p_{i}C_{i}\right).D$ is nonzero, $\sum_{i=1}^{n}p_{i}C_{i}$ is nonzero.

15.2 Proof of 4.4.7

Let $A_{1},\ldots,A_{n}$ be events and $\lesssim$ be a plausible preorder. Due to reflexivity of $\lesssim$, the inequality $\bigvee_{i=1}^{n}A_{i}\lesssim\sum_{i=1}^{n}A_{i}$ holds for $n=0$ and $n=1$. Let the inequality $\bigvee_{i=1}^{n}A_{i}\lesssim\sum_{i=1}^{n}A_{i}$ hold for some integer $n\geq 1$ and arbitrary events $A_{1},\ldots,A_{n}$. Let $A_{1},\ldots,A_{n+1}$ be events. Define events $B_{1}=A_{1},\ldots,B_{n-1}=A_{n-1},B_{n}=A_{n}\vee A_{n+1}$. Then $\bigvee_{i=1}^{n+1}A_{i}=\bigvee_{i=1}^{n}B_{i}\lesssim\sum_{i=1}^{n}B_{i}=\sum_{i=1}^{n+1}A_{i}-A_{n}.A_{n+1}\lesssim\sum_{i=1}^{n+1}A_{i}$. By mathematical induction the inequality holds for every integer $n$.

15.3 Proof of 5.3.1

Let $\sim$ be the equivalence part of a plausible equivalence $\lesssim$, let $p_{1},\ldots,p_{n}$ be positive real numbers, $A_{1},\ldots,A_{n}$ be events and $0\sim\sum_{i=1}^{n}p_{i}A_{i}$. Without loss of generality, we assume that $n\geq 1$ and prove that $0\sim A_{1}$. Per the assumption, $\sum_{i=1}^{n}p_{i}A_{i}\lesssim 0$. Per 4.1, $-\sum_{i=2}^{n}p_{i}A_{i}\lesssim 0$. Therefore, $A_{1}=\frac{1}{p_{1}}(\sum_{i=1}^{n}p_{i}A_{i}-\sum_{i=2}^{n}p_{i}A_{i})\lesssim 0$. Together with the plausible property of $\lesssim$ guaranteeing that $0\lesssim A_{1}$, we obtain that $0\sim A_{1}$.

15.4 Proof of 5.4

Let $\sim$ be a relation having the fundamental properties of a plausible equivalence. We define a relation $\lesssim$ so that $0\lesssim X$ if $X=U+\sum_{i=1}^{n}q_{i}A_{i}$ for some random quantity $U\sim 0$, integer $n\geq 0$, nonnegative real numbers $q_{1},\ldots,q_{n}$ and events $A_{1},\ldots,A_{n}$. We also define that $X\lesssim Y$ if $0\lesssim Y-X$.

The task to verify that the relation $\lesssim$ defined this way is a plausible preorder and that $\sim$ is its equivalence part, is left as an exercise to the reader. Note also that the relation $\lesssim$ defined this way is the smallest plausible preorder with respect to inclusion such that $\sim$ is its equivalence part.

15.5 Proof of 6.4

Let $\lnsim$ be a relation having the fundamental properties of a plausible strict partial order. Let $X$ be a random quantity. We define that $0\lesssim X$ if either $0\lnsim X$ or if there are real numbers $r_{1},r_{2},\ldots,r_{n}$ and events $A_{1},A_{2},\ldots,A_{n}$ such that $X=\sum_{i=1}^{n}r_{i}A_{i}$ and for every $i\in{1,2,\ldots,n}$ holds $\neg(0\lnsim A_{i})$. We also define that $X\lesssim Y$ if $0\lesssim Y-X$.

The task to verify that the relation $\lesssim$ defined this way is a plausible preorder and that $\lnsim$ is its strict part, is left as an exercise to the reader. Note also that the relation $\lesssim$ defined this way is the smallest plausible preorder with respect to inclusion such that $\lnsim$ is its strict part.

15.6 Proof of 10.2

Let $X=\left(X_{H},X_{T}\right)$. Then $X-\left(\frac{1}{2}X_{H}+\frac{1}{2}X_{T}\right)=\left(\frac{1}{2}X_{H}-\frac{1}{2}X_{T}\right)H+\left(\frac{1}{2}X_{T}-\frac{1}{2}X_{H}\right)T$. Since $\left(\frac{1}{2}X_{H}-\frac{1}{2}X_{T}\right)+\left(\frac{1}{2}X_{T}-\frac{1}{2}X_{H}\right)=0$, 4.2 implies that $X-\left(\frac{1}{2}X_{H}+\frac{1}{2}X_{T}\right)\sim 0$. Let $\epsilon$ be a positive real number. Per 4.2, $0\lnsim\epsilon$. Therefore, $-\epsilon\lnsim 0\lesssim X-\left(\frac{1}{2}X_{H}+\frac{1}{2}X_{T}\right)\lesssim 0\lnsim\epsilon$.

15.7 Proof of 10.5

Let $\lesssim$ be a plausible preorder and $X$ be a random quantity.

If $E(X)=+\infty$, then in the extended real line, $\sup\{r\in\mathbb{R}|r\lnsim X\}=\sup\mathbb{R}=+\infty=\inf\emptyset=\inf\{r\in\mathbb{R}|X\lnsim r\}$.

If $E(X)=-\infty$, then $\sup\{r\in\mathbb{R}|r\lnsim X\}=\sup\emptyset=-\infty=\inf\mathbb{R}=\inf\{r\in\mathbb{R}|X\lnsim r\}$.

If $E(X)=x\in\mathbb{R}$, then per 10.4, for $r<x$ holds that $r\lnsim X$ and for $x<r$ holds that $X\lnsim r$. Due to antireflexivity and transitivity of $\lnsim$, there is no real number $r$ such that both $r\lnsim X$ and $X\lnsim r$. Therefore, if $r\lnsim X$ then also $r\leq x$. Similarly, if $X\lnsim r$, then also $x\leq r$. This means that $x$ is an upper bound of $\{r\in\mathbb{R}|r\lnsim X\}$. $x$ is also the least upper bound of $\{r\in\mathbb{R}|r\lnsim X\}$, since for every real number $r$ smaller than $x$ there is a greater real number $s$ that is still smaller than $x$, which means that $r<x$ is not an upper bound of $\{r\in\mathbb{R}|r\lnsim X\}$. Similarly, $x$ is the greatest lower bound of $\{r\in\mathbb{R}|X\lnsim r\}$.

Vice versa, let in the extended real line $\sup\{r\in\mathbb{R}|r\lnsim X\}=x=\inf\{r\in\mathbb{R}|X\lnsim r\}$.

If $x=+\infty$, then $\sup\{r\in\mathbb{R}|r\lnsim X\}=+\infty$, which means that if $y$ is a real number, then there is a real number $r$ such that $y<r$ and $r\lnsim X$. Per 4.4.4 also $y\lnsim X$, which proves that $E(X)=+\infty=x$.

If $x=-\infty$, then $\inf\{r\in\mathbb{R}|X\lnsim r\}=-\infty$, which means that if $y$ is a real number, then there is a real number $r$ such that $r<y$ and $X\lnsim r$. Per 4.4.4 also $X\lnsim y$, which proves that $E(X)=-\infty=x$.

If $x\in\mathbb{R}$ and $\epsilon$ is a positive real number, then since $\sup\{r\in\mathbb{R}|r\lnsim X\}=x$, there is a real number $r$ such that $x-\epsilon<r\lnsim X$, and due to 4.4.4 also $x-\epsilon\lnsim X$. Since $x=\inf\{r\in\mathbb{R}|X\lnsim r\}$, there is a real number $r$ such that $X\lnsim r<x+\epsilon$ , and due to 4.4.4 also $X\lnsim x+\epsilon$. Per 10.1, $E(X)=x$.

15.8 Proof of 12.10.4

Let $E\left(X|C.D\right)\in\left\{+\infty,-\infty\right\}$ and $p$ be a positive real number such that $p\lesssim_{D}C$. Per 7.1, $pD\lesssim C.D$. Define

[TABLE]

and

[TABLE]

Per 12.4, $E\left(Y|C.D\right)=sE\left(X|C.D\right)=+\infty$. Let $y$ be a real number. Define $z=\frac{\max\left(0,y\right)}{p}$. Then $z\geq 0$. Since $E\left(Y|C.D\right)=+\infty$ and $z$ is a real number, $zC.D\lnsim Y.C.D$. Therefore, $yD\lesssim\max\left(0,y\right)D=zpD\lesssim zC.D\lnsim Y.C.D$. This proves that $E\left(Y.C|D\right)=+\infty$. Since $X=sY$ and per 12.4, $E\left(X.C|D\right)=sE\left(Y.C|D\right)=E\left(X|C.D\right)$.

15.9 Proof of 12.10.1

Let $x=E\left(X|C.D\right)$ be a real number, $c=P\left(C|D\right)$ be a real number and let $\epsilon$ be a positive real number. Define $\delta=\frac{\epsilon}{1+\left|x\right|}$. Then $\delta$ is a positive real number and since $x=E\left(X|C.D\right)$, $-\delta C.D\lnsim X.C.D-xC.D\lnsim\delta C.D$. In the natural order of events $C.D\leq D$ and per 4.4.3, $C.D\lesssim D$. Therefore, $-\delta D\lnsim X.C.D-xC.D\lnsim\delta D$. Since $c=P\left(C|D\right)$, $-\delta D\lnsim C.D-cD\lnsim\delta D$, implying that $-\left|x\right|\delta D\lesssim xC.D-xcD\lesssim\left|x\right|\delta D$. Summing inequalities, we get that $-\left(1+\left|x\right|\right)\delta D\lnsim X.C.D-xcD\lnsim\left(1+\left|x\right|\right)\delta D$. Since $\left(1+\left|x\right|\right)\delta D=\epsilon D$, also $-\epsilon D\lnsim X.C.D-xcD\lnsim\epsilon D$, which proves that $E\left(X.C|D\right)=xc$.

The only case when the formula $E\left(X|C.D\right).P\left(C|D\right)$ makes sense and at least one of $E\left(X|C.D\right)$, $P\left(C|D\right)$ is not a real number, is the case when $E\left(X|C.D\right)\in\{-\infty,+\infty\}$ and $c=P\left(C|D\right)$ is a positive real number. In this case, note that $0<\frac{c}{2}<c$, use 10.4 to obtain that $\frac{c}{2}\lnsim_{D}C$ and use 12.10.4 proven above to get that $E\left(X.C|D\right)=E\left(X|C.D\right)=E\left(X|C.D\right).P\left(C|D\right)$.

15.10 Proof of 12.10.2

Let $\epsilon$ be a positive real number. If $E\left(X|C.D\right)=0$, then per 11.1, $-\epsilon C.D\lnsim X.C.D\lnsim\epsilon C.D$. In the natural order of events $C.D\leq D$ and per 4.4.3, $C.D\lesssim D$. Therefore, $-\epsilon D\lesssim-\epsilon C.D\lnsim X.C.D\lnsim\epsilon C.D\lesssim\epsilon D$, which proves that $E\left(X.C|D\right)=0$.

15.11 Proof of 12.10.3

Let $P\left(C|D\right)=0$ and $p$ be a real number such that $-pC.D\lesssim X.C.D\lesssim pC.D$. Define $q=\max\left(1,p\right)$. Then both $q>0$ and $q\geq p$. Therefore, $pC.D\lesssim qC.D$. Let $\epsilon$ be a positive real number. Define $\delta=\frac{\epsilon}{q}$. Since $P\left(C|D\right)=0$, $-\delta D\lnsim C.D\lnsim\delta D$, i.e. also $qC.D\lnsim q\delta D$. Therefore, $-\epsilon D=-q\delta D\lnsim-qC.D\lesssim-pC.D\lesssim X.C.D\lesssim pC.D\lesssim qC.D\lnsim q\delta D=\epsilon D$, which proves that $E\left(X.C|D\right)=0$.

15.12 Proof of 12.10.7

Let $E\left(X.C|D\right)\in\{-\infty,+\infty\}$. Define

[TABLE]

and

[TABLE]

Per 12.4, $E\left(Y.C|D\right)=s.E\left(X.C|D\right)=+\infty$. Let $y$ be a real number. Define $z=\max\left(0,y\right)$. Then both $z\geq 0$ and $z\geq y$. Since $E\left(Y.C|D\right)=+\infty$, $zD\lnsim Y.C.D$. In the natural order of events $C.D\leq D$ and per 4.4.3, $C.D\lesssim D$. Therefore, $yC.D\lesssim zC.D\lesssim zD\lnsim Y.C.D$, which proves that $E\left(Y|C.D\right)=+\infty$. Since $X=sY$ and per 12.4, $E\left(X|C.D\right)=sE\left(Y|C.D\right)=E\left(X.C|D\right)$.

15.13 Proof of 12.10.5

Let $x=E\left(X.C|D\right)$ be a real number, $c=P\left(C|D\right)$ be a nonzero real number. Per 12.7, $c>0$. Let $\epsilon$ be a positive real number. Define $\delta=\frac{c}{2.\left(1+\frac{\left|x\right|}{c}\right)}.\min(1,\epsilon)$. Then $0<\delta\leq\frac{c}{2}$. Equality $c=P\left(C|D\right)$, the fact that $\delta$ is positive and 10.1 give $-\delta D\lnsim C.D\text{\textendash}cD\lnsim\delta D$. Therefore, $\frac{c}{2}D=cD-\frac{c}{2}D\lesssim cD-\delta D\lnsim C.D$, i.e. $D\lnsim\frac{2}{c}C.D$. By 4.1.3, $-\delta\frac{\left|x\right|}{c}D\lesssim-\frac{x}{c}C.D+xD\lesssim\delta\frac{\left|x\right|}{c}D$. Equality $x=E\left(X.C|D\right)$ and 10.1 give $-\delta D\lnsim X.C.D-xD\lnsim\delta D$. Summing the inequalities, we get that $-\delta\left(1+\frac{\left|x\right|}{c}\right)D\lnsim X.C.D-\frac{x}{c}C.D\lnsim\delta\left(1+\frac{\left|x\right|}{c}\right)D$. Also, $\delta\left(1+\frac{\left|x\right|}{c}\right)D\lnsim\delta\left(1+\frac{\left|x\right|}{c}\right)\frac{2}{c}C.D=\min\left(1,\epsilon\right)C.D\lesssim\epsilon C.D$. Combining these inequalities, we get that $-\epsilon C.D\lnsim-\delta\left(1+\frac{\left|x\right|}{c}\right)D\lnsim X.C.D-\frac{x}{c}C.D\lnsim\delta\left(1+\frac{\left|x\right|}{c}\right)D\lnsim\epsilon D$, which proves that $E\left(X|C.D\right)=\frac{x}{c}$.

The only remaining case when the expression $\frac{E\left(X.C|D\right)}{P(C|D)}$ makes sense is the case when $E\left(X.C|D\right)\in\left\{-\infty,+\infty\right\}$ and $P(C|D)$ is a positive real number. In this case, use the equality $E\left(X|C.D\right)=E\left(X.C|D\right)$ from 12.10.7 proven above and the fact that $E\left(X.C|D\right)=\frac{E\left(X.C|D\right)}{P(C|D)}$ to finally obtain $E\left(X|C.D\right)=\frac{E\left(X.C|D\right)}{P(C|D)}$.

15.14 Proof of 12.10.6

Let $E\left(X.C|D\right)=0$ and let $p$ be a positive real number such that $pD\lesssim C.D$. Let $\epsilon$ be a positive real number. Define $\delta=\epsilon p$. Since $E\left(X.C|D\right)=0$ and $\delta>0$, per 10.1 $-\delta D\lnsim X.C.D\lnsim\delta D$, implying that $-\epsilon C.D\lesssim-\epsilon pD=-\delta D\lnsim X.C.D\lnsim\delta D=\epsilon pD\lesssim\epsilon C.D$, which proves that $E\left(X|C.D\right)=0$.

15.15 Proof of 12.10.8

Let $P\left(C|D\right)=0$ and let $p$ be a real number such that $1\lesssim_{D}pX.C$. Defining $Y=pX$ we get $1\lesssim_{D}Y.C$ and per 11.1, $D\lesssim Y.C.D$. Let $y$ be a real number. Define $\epsilon=\frac{1}{\max\left(1,y\right)}$. Then $\epsilon$ is a positive real number and since $P\left(C|D\right)=0$, $-\epsilon D\lnsim C.D\lnsim\epsilon D$. Therefore, $\frac{1}{\epsilon}C.D\lnsim D$ and $yC.D\lesssim\max\left(1,y\right)C.D=\frac{1}{\epsilon}C.D\lnsim D\lesssim Y.C.D$. This proves that $E\left(Y|C.D\right)=+\infty$. It also proves that $0\lnsim Y$, implying that $Y\neq 0$ and since $Y=pX$ also $p\neq 0$ and $X=\frac{1}{p}Y$. Per 12.4, $E\left(X|C.D\right)=\frac{1}{p}E\left(Y|C.D\right)=\frac{+\infty}{p}$.

15.16 Proof of 12.10.11

Let $E\left(X|C.D\right)\in\left\{-\infty,+\infty\right\}$ and let $p$ be a real number such that $-p\lesssim_{D}X.C\lesssim_{D}p$. Per 7.1, $-pD\lesssim X.C.D\lesssim pD$. Define

[TABLE]

[TABLE]

and

[TABLE]

Then $q>0$ and $q\geq p$. Therefore, $-qD\lesssim-pD\lesssim X.C.D\lesssim pD\lesssim qD$. Since $Y=sX$ and $s\in\left\{-1,1\right\}$, also $-qD\lesssim Y.C.D\lesssim qD$. Per 12.4, $E\left(Y|C.D\right)=s.E\left(X|C.D\right)=+\infty$. Let $\epsilon$ be a positive real number. If we define $y=\frac{q}{\epsilon}$, then $y>0$. Since $E\left(Y|C.D\right)=+\infty$, $yC.D\lnsim Y.C.D$. Combining inequalities, we get that $yC.D\lnsim qD$. Therefore, $C.D\lnsim\frac{q}{y}D=\epsilon D$. Per 4.4.3, $0\lesssim C.D$. Combining inequalities, we get that $-\epsilon D\lnsim-C.D\lesssim 0\lesssim C.D\lnsim\epsilon D$, proving that $P\left(C|D\right)=0$.

15.17 Proof of 12.10.9

Let $v=E\left(X.C|D\right)$ be a real number and let $x=E\left(X|C.D\right)$ be a nonzero real number. Let $\epsilon$ be a positive real number. Define $\delta=\frac{\left|x\right|}{2}\epsilon$. Then $\delta$ is a positive real number and since $v=E\left(X.C|D\right)$, $-\delta D\lnsim X.C.D-vD\lnsim\delta D$. Since $x=E\left(X|C.D\right)$, $-\delta C.D\lnsim X.C.D-xC.D\lnsim\delta C.D$. In the natural order of events, $C.D\leq D$ and per 4.4.3 $C.D\lesssim D$. Therefore, $-\delta D\lesssim-\delta C.D\lnsim-X.C.D+xC.D\lnsim\delta C.D\lesssim\delta D$. Summing inequalities, we get $-2\delta D\lnsim xC.D-vD\lnsim 2\delta D$, i.e. $-\epsilon D=-\frac{2}{\left|x\right|}\delta D\lnsim C.D-\frac{v}{x}D\lnsim\frac{2}{\left|x\right|}\delta D=\epsilon D$. This proves that $E\left(C|D\right)=\frac{v}{x}$.

If $E\left(X.C|D\right)\in\left\{-\infty,+\infty\right\}$, then per 12.10.7, $E\left(X|C.D\right)=E\left(X.C|D\right)$, i.e. the expression $\frac{E\left(X.C|D\right)}{E\left(X|C.D\right)}$ does not make sense.

Let $v=E\left(X.C|D\right)$ be a real number and let $E\left(X|C.D\right)\in\left\{-\infty,+\infty\right\}$. Define $p=1+\left|v\right|$. Per 11, $-1D\lnsim X.C.D-vD\lnsim 1D$, i.e. $-pD=-\left(1+\left|v\right|\right)D\lesssim-\left(1-v\right)D\lnsim X.C.D\lnsim\left(1+v\right)D\lesssim\left(1+\left|v\right|\right)D=pD$. This demonstrates that the assumptions of 12.10.11 hold. Therefore, $P\left(C|D\right)=0=\frac{v}{E\left(X|C.D\right)}$.

15.18 Proof of 12.10.10

Let $E\left(X.C|D\right)=0$ and let $p$ be a real number such that $1\lesssim_{C.D}pX$. Define $Y=pX$. Per 12.4, $E\left(Y.C|D\right)=pE\left(X.C|D\right)=0$. Also, $1\lesssim_{C.D}Y$. Per 7.1, $C.D\lesssim Y.C.D$. Let $\epsilon$ be a positive real number. Since $E\left(Y.C|D\right)=0$, $-\epsilon D\lnsim Y.C.D\lnsim\epsilon D$. Combining inequalities, $C.D\lnsim\epsilon D$. Per 4.1.1, $0\lesssim C.D$. Combining inequalities, we get $-\epsilon D\lnsim-C.D\lesssim 0\lesssim C.D\lnsim\epsilon D$, proving that $P\left(C|D\right)=0$.

15.19 Proof of 13.2.1

Let $PV$ be a function from $\mathcal{F}$ to $\mathbb{R}_{\geq 0}$ that satisfies 13.2. Let $n\geq 0$, $m\geq 1$ be integers, $q_{1},\ldots,q_{n}$ be nonnegative real numbers, $r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$ be real numbers, $C_{1},\ldots,C_{n}$ be events, $A_{1},\ldots,A_{m}$ be elements of $\mathcal{F}$ and let $r_{j}\left(PV\left(A_{j}\right)+s_{j}\right)>0$ for every $j\in\left\{1,\ldots,m\right\}$.

Let $\mathcal{A}$ be a Boolean algebra generated by $A_{1},\ldots,A_{m}$ and $\mathcal{B}$ be a Boolean algebra generated by $A_{1},\ldots,A_{m},C_{1},\ldots,C_{n}$. Then $\mathcal{A}\subseteq\mathcal{B}\subseteq\mathcal{F}$. Since both $\mathcal{A}$ and $\mathcal{B}$ are finitely generated, they are finite and atomic Givant, Halmos (2008). Let $\mathrm{at}\left(\mathcal{A}\right)$ be the set of atoms of $\mathcal{A}$ and $\mathrm{at}\left(\mathcal{B}\right)$ be the set of atoms of $\mathcal{B}$. Let $\mathrm{span}\left(\mathcal{B}\right)$ be the linear span of $\mathcal{B}$. Let $G\in\mathrm{at}\left(\mathcal{B}\right)$. If $X\in\mathrm{span}\left(\mathcal{B}\right)$, then there is a unique real number $r$ such that $X.G=rG$. This allows us to define a function $\varphi$ from $\mathrm{span}\left(\mathcal{B}\right)\times\mathrm{at}\left(\mathcal{B}\right)$ to $\mathbb{R}$ such that $X.G=\varphi\left(X,G\right)G$. The properties of the function $\varphi$ are:

• •

If $B\in\mathcal{B}$, then $\varphi\left(B,G\right)\in\{0,1\}$, implying that $\varphi\left(B,G\right).\varphi\left(B,G\right)=\varphi\left(B,G\right)$. (idempotence)

• •

If $X,Y\in\mathrm{span}\left(\mathcal{B}\right)$ and $X.Y=0$, then $\varphi\left(X,G\right).\varphi\left(Y,G\right)=\varphi\left(X.Y,G\right)=0$. (orthogonality)

• •

If $X,Y\in\mathrm{span}\left(\mathcal{B}\right)$, then $\varphi\left(X+Y,G\right)=\varphi\left(X,G\right)+\varphi\left(Y,G\right)$. (additivity)

• •

If $r$ is a real number and $X\in\mathrm{span}\left(\mathcal{B}\right)$, then $\varphi\left(rX,G\right)=r\varphi\left(X,G\right)$. (homogeneity)

Define a function $\nu$ from $\mathcal{C}$ to $\mathbb{R}$ so that for $B\in\mathcal{B}$, $\nu\left(B\right)=\sum_{G\in\mathrm{at}\left(\mathcal{B}\right)}\varphi\left(B,G\right)$. The properties of the function $\nu$ are

• •

If $0\neq B\in\mathcal{B}$, then $\nu\left(B\right)>0$.

Define a function $F$ from $\mathrm{span}\left(\mathcal{B}\right)$ to $\mathbb{R}$ so that for $X\in\mathrm{span}\left(\mathcal{B}\right)$,

[TABLE]

The properties of the function $F$ are

• •

If $B\in\mathcal{B}$, then $F\left(B\right)\geq 0$. (nonnegativity)

• •

$F$ is additive.

• •

$F$ is homogeneous.

• •

$F$ coincides with $PV$ on $\mathcal{A}$.

Therefore, $\sum_{j=1}^{m}r_{j}\left(F\left(A_{j}\right)+s_{j}F\left(1\right)\right)=\sum_{j=1}^{m}r_{j}\left(PV\left(A_{j}\right)+s_{j}PV\left(1\right)\right)>0$. Also, $F\left(\sum_{i=1}^{n}q_{i}C_{i}\right)\geq 0$ and $F\left(\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}\left(A_{j}+s_{j}\right)\right)>0$. Since $F\left(0\right)=0$, this proves that $\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}\left(A_{j}+s_{j}\right)\neq 0$.

15.20 Proof of 13.3.2

Let $PV$ be a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$ that satisfies definition 13.3. Let $A,B$ be elements of $\mathcal{F}$ such that $A.B=0$ and $C\in\mathcal{F}_{0}$. Then $A+B=1-(1-A).(1-B)$ is an element of $\mathcal{F}$.

If $(1-B).C=0$, then $B.C=C$, i.e. $PV(B|C)=PV(B.C|C)=PV(C|C)=1$. Also $A.C=A.B.C=0$ implying $PV(A|C)=PV(A.C|C)=0$. Finally, $PV(A+B|C)=PV((A+B).C|C)=PV(A.C+B.C|C)=PV(C|C)=1$. Therefore, $PV(A+B|C)=PV(A|C)+PV(B|C)$.

If $(1-B).C\neq 0$, then $PV(A+B|C)=1-PV((1-A).(1-B)|C)=1-PV((1-A)|(1-B).C).PV(1-B|C)=1-[1-PV(A|(1-B).C)].PV(1-B|C)=1-PV(1-B|C)+PV(A|(1-B).C).PV(1-B|C)=PV(B|C)+PV(A.(1-B)|C)=PV(B|C)+PV(A-A.B|C)=PV(B|C)+PV(A|C)$.

15.21 Proof of 13.3.4

Let $PV$ be a function from $\mathcal{F}\times\mathcal{F}_{0}$ to $\mathbb{R}_{\geq 0}$ that satisfies definition 13.3. Let $n\geq 0,m\geq 1$ be integers, $q_{1},\ldots,q_{n}$ be nonnegative real numbers, $r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$ be real numbers, $C_{1},\ldots,C_{n}$ be events, $A_{1},\ldots,A_{m}$ be elements of $\mathcal{F}$, $D_{1},\ldots,D_{m}$ be elements of $\mathcal{F}_{0}$ and for every $j\in{1,\ldots,m}$, $r_{j}(PV(A_{j}|D_{j})+s_{j})>0$.

Let $D=\bigvee_{j=1}^{m}D_{j}$. Since $PV$ is subadditive, $PV(\bigvee_{j=1}^{m}D_{j}|D)\leq\sum_{j=1}^{m}PV(D_{j}|D)$. Since $0<PV(D|D)$, there is a $k\in\{1,\ldots,m\}$ such that $0<PV(D_{k}|D)$.

Let $j\in{1,\ldots,m}$. Due to Bayes’ rule and nonnegativity,

[TABLE]

If $k\in\{1,\ldots,m\}$ is such that $0<PV(D_{k}|D)$, then $r_{k}(PV(A_{k}D_{k}|D)+s_{k}PV(D_{k}|D))>0$. Since such $k\in\{1,\ldots,m\}$ exists, $\sum_{j=1}^{m}r_{j}(PV(A_{j}D_{j}|D)+s_{j}PV(D_{j}|D))>0$.

Let $\mathcal{A}$ be the Boolean algebra generated by $A_{1},\ldots,A_{m},D_{1},\ldots,D_{m}$. Since $\mathcal{A}$ is finitely generated, it is finite and atomic Givant, Halmos (2008). Let $\mathrm{at}\left(\mathcal{A}\right)$ be the set of atoms of $\mathcal{A}$.

Let $\mathcal{B}$ be the Boolean algebra generated by $A_{1},\ldots,A_{m},D_{1},\ldots,D_{m},C_{1},\ldots,C_{n}$. Since $\mathcal{B}$ is finitely generated, it is finite and atomic Givant, Halmos (2008). Let $\mathrm{at}\left(\mathcal{B}\right)$ be the set of atoms of $\mathcal{B}$ and $\mathrm{span}\left(\mathcal{B}\right)$ be the linear span of $\mathcal{B}$.

Define a function $\varphi$ from $\mathrm{span}\left(\mathcal{B}\right)\times\mathrm{at}\left(\mathcal{B}\right)$ to $\mathbb{R}$ so that if $X\in\mathrm{span}\left(\mathcal{B}\right)$ and $G\in\mathrm{at}\left(\mathcal{B}\right)$, then $X.G=\varphi(X,G).G$. Also define a function $\nu$ from $\mathcal{B}$ to $\mathbb{R}$ so that if $H\in\mathcal{B}$, then $\nu(H)=\sum_{G\in\mathrm{at}\left(\mathcal{B}\right)}\varphi(H,G)$. Finally, define a function $F$ from $\mathrm{span}\left(\mathcal{B}\right)\times\{D\}$ to $\mathbb{R}$ so that if $X\in\mathrm{span}\left(\mathcal{B}\right)$, then

[TABLE]

The reader can verify that $F$ is homogeneous and additive on $\mathrm{span}\left(\mathcal{B}\right)\times\{D\}$, nonnegative on $\mathcal{B}\times\{D\}$ and that it coincides with $PV$ on $\mathcal{A}\times\{D\}$. Therefore,

[TABLE]

Also, $F(\sum_{i=1}^{n}q_{i}C_{i}|D)\geq 0$. Therefore, $F\left(\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}(A_{j}+s_{j}).D_{j}|D\right)>0$. Since $F(0|D)=0$, this proves that $\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}(A_{j}+s_{j}).D_{j}\neq 0$.

15.22 Proof of 13.4.2

Let $PV$ be a partial function from $\mathcal{T}\times\mathcal{C}$ to $\mathbb{R}$ that satisfies definition 13.4. Let $n\geq 0,m\geq 1$ be integers, $q_{1},\ldots,q_{n}$ be nonnegative real numbers, $r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$ be real numbers, $C_{1},\ldots,C_{n}$, $D_{1},\ldots,D_{m}$ be events, $X_{1},\ldots,X_{m}$ be random quantities and for every $j\in{1,\ldots,m}$, $r_{j}(PV(X_{j}|D_{j})+s_{j})>0$.

Due to additivity of $PV$, if $C\in\mathcal{C}$, then $PV(0|C)=PV(0|C)+PV(0|C)$, which implies that $PV(0|C)=0$.

Let $D=\bigvee_{j=1}^{m}D_{j}$. Due to subadditivity, $PV(D|D)\leq\sum_{j=1}^{m}PV(D_{j}|D)$. Since $0<PV(D|D)$, there is a $k\in\{1,\ldots,m\}$ such that $0<PV(D_{k}|D)$.

Let $j\in{1,\ldots,m}$. Due to Bayes’ rule and nonnegativity,

[TABLE]

If $k\in\{1,\ldots,m\}$ is such that $0<PV(D_{k}|D)$, then $r_{k}(PV(X_{k}.D_{k}|D)+s_{k}PV(D_{k}|D))>0$. Since such $k\in\{1,\ldots,m\}$ exists, $\sum_{j=1}^{m}r_{j}(PV(X_{j}.D_{j}|D)+s_{j}PV(D_{j}|D))>0$. Additivity and homogeneity of $PV$ imply that $PV\left(\sum_{j=1}^{m}r_{j}(X_{j}+s_{j}).D_{j}|D\right)>0$. Moreover, $PV(\sum_{i=1}^{n}q_{i}C_{i}|D)\geq 0$, i.e. also $PV\left(\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}(X_{j}+s_{j}).D_{j}|D\right)>0$. Since $PV(0|D)=0$, this proves that $\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}(X_{j}+s_{j}).D_{j}\neq 0$.

15.23 Proof of 13.5

We start by proving that $1\Rightarrow 2$.

Let $PV$ be a coherent partial function from $\mathcal{T}\times\mathcal{E}\left(\mathcal{T}\right)$ to $\overline{\mathbb{R}}$.

Define the relation $\lnsim$ so that $0\lnsim X$ if $X=\sum_{i=1}^{n}p_{i}C_{i}+\sum_{j=1}^{m}r_{j}\left(X_{j}+s_{j}\right).D_{j}$ for some nonnegative integers $n$, $m$, positive real numbers $p_{1},\ldots,p_{n}$, real numbers $r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$, nonzero events $C_{1},\ldots,C_{n}$, events $D_{1},\ldots,D_{m}$ and random quantities $X_{1},\ldots,X_{m}$, such that at least one of $n$, $m$ is nonzero and $r_{j}\left(PV\left(X_{j}|D_{j}\right)+s_{j}\right)>0$ for every $j\in\left\{1,\ldots,m\right\}$. Define $X\lnsim Y$ if $0\lnsim Y-X$. According to this definition, for every event $C$ holds that $0\lnsim C$. This guarantees both regularity and plausible property 6.3.1. The antireflexivity of $\lnsim$ is a consequence of the coherence of $PV$ and 3.5. We leave the task to verify that $\lnsim$ has the remaining fundamental properties of a plausible strict partial order listed in 6.3 as an exercise to the reader.

Take $\lesssim$ as a plausible preorder having $\lnsim$ as its strict part. By 6.4, such a plausible preorder exists.

If $PV\left(X|C\right)=+\infty$ and $y$ is a real number, then $PV\left(X|C\right)-y=+\infty>0$, i.e. $yC\lnsim X.C$, proving that $E\left(X|C\right)=+\infty$.

If $PV\left(X|C\right)=-\infty$ and $y$ is a real number, then $\left(-1\right).\left(PV\left(X|C\right)-y\right)=+\infty>0$, i.e. $X.C\lnsim yC$, proving that $E\left(X|C\right)=-\infty$.

If $r=PV\left(X|C\right)$ is a real number and $\epsilon$ is a positive real number, then $PV\left(X|C\right)-r+\epsilon>0$, i.e., $-\epsilon C\lnsim X.C-rC$, and $\left(-1\right).\left(PV\left(X|C\right)-r-\epsilon\right)>0$, i.e. $X.C-rC\lnsim\epsilon C$, proving that $E\left(X|C\right)=r$. This completes the proof that the expectation naturally induced by $\lesssim$ extends the function $PV$.

The implication $2\Rightarrow 3$ is trivial.

We finish by proving that $3\Rightarrow 1$.

Let $\lesssim$ be a plausible preorder and $E$ be the conditional expectation induced by $\lesssim$. Let $n\geq 0$, $m\geq 1$ be integers, $q_{1},\ldots,q_{n}$ be nonnegative real numbers, $r_{1},\ldots,r_{m}$, $s_{1},\ldots,s_{m}$ be real numbers, $C_{1},\ldots,C_{n}$, $D_{1},\ldots,D_{m}$ be events, $X_{1},\ldots,X_{m}$ be random quantities and for every $j\in\{1,\ldots,m\}$, $r_{j}\left(E\left(X_{j}|D_{j}\right)+s_{j}\right)>0$. By 4.1, $0\lesssim\sum_{i=1}^{n}q_{i}C_{i}$. By real additivity and homogeneity, $E\left(r_{j}\left(X_{j}+s_{j}\right)|D_{j}\right)=r_{j}\left(E\left(X_{j}|D_{j}\right)+s_{j}\right)>0$. By preorder consistency, if $E\left(r_{j}\left(X_{j}+s_{j}\right)|D_{j}\right)>0$, then $0\lnsim r_{j}\left(X_{j}+s_{j}\right).D_{j}$, yielding $0\lnsim\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}\left(X_{j}+s_{j}\right).D_{j}$. Due to 6.3.2, $0\neq\sum_{i=1}^{n}q_{i}C_{i}+\sum_{j=1}^{m}r_{j}\left(X_{j}+s_{j}\right).D_{j}$.