Conditional Probability Framework (CPF)

• CPF is a foundational framework that defines probability as a subset of conditional expectation, induced by a plausible preorder on random quantities.
• It develops an algebra of random quantities that unifies the treatment of events and expectations, providing coherence even when classical ratio definitions fail.
• The framework’s coherence criterion and order-theoretic approach ensure that traditional probability laws re-emerge, while accommodating zero or undefined probabilities.

Searching arXiv for the foundational CPF paper and closely related conditional-probability-space work. The Conditional Probability Framework (CPF) is a foundational program in which probability is characterized not as a primitive ratio of unconditional probabilities, but as a subset of conditional expectation induced by a plausible preorder on random quantities. In its 2019 foundational formulation, CPF begins from a very general algebra of random quantities, treats plausibility as the primitive order-theoretic notion, derives conditional expectation from conditionalized preorder structure, and then recovers conditional probability as the restriction of conditional expectation to events. Within this formulation, coherence is the exact admissibility criterion: a partial function is coherent if and only if it can be extended to conditional expectation naturally induced by a plausible preorder, and this remains meaningful in cases where the classical formula $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$ fails because $P(C)$ is zero or undefined (Mečíř, 2019).

1. Foundational orientation

In the foundational CPF, the central claim is that probability is not fundamentally a ratio-based notion. Instead, the framework starts from a primitive ordering relation on random quantities and derives both expectation and conditional expectation from that order. Probability then appears as the event-valued fragment of conditional expectation. The paper summarizes this perspective by stating that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities, and that a function is coherent if and only if it is a subset of conditional expectation induced by a plausible preorder on random quantities (Mečíř, 2019).

This reverses the usual order of exposition. In the classical ratio-based presentation, conditional probability is often introduced through $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$ when $P(C)>0$. In CPF, by contrast, conditioning is defined structurally before such ratios are considered. A plausible implication is that the framework is intended not merely as a reformulation of standard conditional probability, but as a replacement for unconditional-probability-ratio foundations in settings where those foundations are too narrow.

The same 2019 literature situates this move alongside a broader revival of conditional probability spaces and Rényi-style foundations. In Rényi spaces, probability is represented by an equivalence class of $\sigma$-finite measures up to positive scaling, with conditioning defined only on a bunch of admissible finite-mass events (Taraldsen, 2019). A closely related treatment of improper priors formulates statistics directly in terms of conditional probability spaces and proves equivalence between the improper-prior formulation and the Rényi conditional-probability-space formulation (Taraldsen et al., 2020). These parallel developments do not identify CPF with the same primitive objects, but they share the general thesis that conditional laws can be foundational rather than derivative.

2. Algebra of random quantities and events

The foundational CPF axiomatizes a set $T$ of random quantities as a unital associative commutative algebra over $\mathbb{R}$. Real numbers are embedded canonically via $r\mapsto r1$, and from then on the theory identifies $r$ with $r1$. This algebraic starting point allows events to be defined internally rather than introduced as a separate primitive domain (Mečíř, 2019).

An element $P(C)$0 is called an event iff it is idempotent: $P(C)$1 The set of events $P(C)$2 is a Boolean algebra, with

$P(C)$3

The natural order on events is

$P(C)$4

This event construction is significant because it lets the framework treat events and general random quantities inside one algebraic object. The Boolean structure of $P(C)$5 is therefore not external to the theory; it is recovered from idempotents of the ambient algebra. A plausible implication is that CPF is designed to support both event-level probability and quantity-level expectation without changing ontological level.

The later development of the theory depends on strict comparisons between random quantities and real numbers. That dependence explains why the paper first builds a single algebraic universe $P(C)$6, then defines order structure on all of $P(C)$7, and only afterward specializes to events.

3. Plausible preorder as primitive structure

The primitive order-theoretic notion in CPF is the plausible preorder $P(C)$8 on $P(C)$9. It is defined by four axioms:

• Plausible property: if $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$0 is an event, then $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$1.
• Additive property: if $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$2 and $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$3, then $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$4.
• Multiplicative property: if $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$5 and $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$6 is a nonnegative real number, then $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$7.
• Extension property:

$P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$8

From these axioms, the relation is reflexive, transitive, and extends both the natural order on events and the order on real numbers. The paper also notes that the greatest plausible preorder is the total relation $P(A\mid C)=\frac{P(A\wedge C)}{P(C)}$9 (Mečíř, 2019).

Two derived notions are central. The equivalence part is

$P(C)>0$0

and the strict part is

$P(C)>0$1

The paper states that the fundamental properties listed in §5.3 characterize plausible equivalence, and that the corresponding fundamental properties characterize plausible strict partial order.

Conditioning is introduced by restricting the preorder to a nonzero event $P(C)>0$2: $P(C)>0$3 This definition is structurally decisive. Conditioning is not defined by normalization, quotienting, or regular conditional distributions, but by passing to the preorder induced on the “slice” $P(C)>0$4. The paper also defines a plausible preorder to be degenerate when $P(C)>0$5, and regular when for every nonzero event $P(C)>0$6, $P(C)>0$7. Regularity is the nondegeneracy condition required for meaningful conditional expectation.

4. Expectation and conditional probability inside CPF

Expectation is defined from the plausible preorder rather than assumed. For a random quantity $P(C)>0$8, its expectation $P(C)>0$9 naturally induced by $\sigma$0 is a real number $\sigma$1 if for every positive real number $\sigma$2 the preorder expresses

$\sigma$3

is $\sigma$4 if every real $\sigma$5 satisfies $\sigma$6, is $\sigma$7 if every real $\sigma$8 satisfies $\sigma$9, and is undefined otherwise. The paper proves the existence and uniqueness characterization

$T$0

and preorder-consistency: if $T$1 and $T$2 exist and $T$3, then $T$4 (Mečíř, 2019).

Conditional expectation is then defined by passing to the conditional preorder: $T$5 is the expectation of $T$6 naturally induced by $T$7. In a regular plausible preorder, the paper highlights the edge-case behavior that if $T$8, then $T$9, while if $\mathbb{R}$0, then neither $\mathbb{R}$1 nor $\mathbb{R}$2 is defined. It also states that $\mathbb{R}$3 is a partial function from $\mathbb{R}$4 to $\mathbb{R}$5.

The derived rules have the expected linear form. Consistency states: $\mathbb{R}$6 and, when it exists,

$\mathbb{R}$7

Real additivity gives

$\mathbb{R}$8

general additivity gives

$\mathbb{R}$9

and homogeneity gives

$r\mapsto r1$0

whenever the relevant expressions make sense.

Conditional probability is then defined by restricting conditional expectation to events: $r\mapsto r1$1 if $r\mapsto r1$2 exists. Standard properties are derived internally: monotonicity, bounds

$r\mapsto r1$3

and completeness conditions for values $r\mapsto r1$4 and $r\mapsto r1$5. The framework also yields a chain form

$r\mapsto r1$6

when the expression on the right makes sense.

The distinctive point is the treatment of edge cases. For a regular plausible preorder, the paper states that if $r\mapsto r1$7, then $r\mapsto r1$8, and if $r\mapsto r1$9, then $r$0, even if $r$1 is zero or undefined. This is the precise sense in which CPF extends conditional probability beyond the domain of the classical ratio rule.

5. Coherence and the characterization theorem

The final layer of the foundational CPF is coherence. A partial function $r$2 from $r$3 to $r$4 is coherent if it satisfies the paper’s no-sure-loss style positivity condition: whenever $r$5, $r$6, $r$7, $r$8, $r$9 are events, $r1$0 are random quantities, and $r1$1 for every $r1$2, then

$r1$3

The paper explicitly states that every Kolmogorovian plausible value is coherent, every Coxian plausible value is coherent, and every Dupré-Tiplerian plausible value is coherent (Mečíř, 2019).

Its main theorem is the equivalence:

1. $r1$4 is coherent;
2. $r1$5 can be extended to conditional expectation naturally induced by a regular plausible preorder;
3. $r1$6 can be extended to conditional expectation naturally induced by a plausible preorder.

This theorem is the central representation result of the framework. It identifies coherence as exactly the condition for representability inside CPF. The paper then defines a function to be plausibly complete if it is naturally induced by a regular plausible preorder, and concludes that probability can be characterized, without loss of generality, as a plausibly complete function.

This characterization distinguishes the framework from purely axiomatic treatments that begin by stipulating probability laws directly. Here, admissible probability assignments are those that can be embedded in preorder-induced conditional expectation. A plausible implication is that CPF is intended to unify several standard formalizations of plausible value under one representation theorem rather than to replace them with a single competing axiom system.

6. Relation to conditional probability spaces and other usages of “CPF”

The acronym and phrase “conditional probability framework” are used in several nonidentical ways in the literature, and the foundational preorder-based CPF should be distinguished from at least three nearby traditions.

First, Rényi-style conditional probability spaces define probability by a family of conditional probability measures indexed by a bunch of admissible conditioning events. In this setting a Rényi state is an equivalence class $r1$7 of $r1$8-finite measures up to positive scaling, and the consistency relation

$r1$9

is primitive. A later note extends Kolmogorov’s conditional expectation to Rényi spaces and defines conditional Rényi states by Radon–Nikodym and disintegration identities (Taraldsen, 2019). A related statistical treatment states that improper priors are naturally represented as equivalence classes of $P(C)$00-finite measures and proves that maximal Rényi spaces are in one-to-one correspondence with conditional measure spaces (Taraldsen et al., 2020).

Second, recent work in epistemic game theory uses the term conditional probability space (CPS) rather than CPF. On a finite state space $P(C)$01, a CPS is a pair $P(C)$02 where $P(C)$03 is a family of nonempty conditioning events and each $P(C)$04 is a probability measure satisfying concentration and the chain rule

$P(C)$05

Using this formalism, an Agreement Theorem is derived without assuming a common prior, information partitions, positivity of measure, or knowledge operators (Yang et al., 28 May 2026).

Third, the acronym “CPF” is also used for unrelated constructs. Examples in the supplied literature include the conditional particle filter in hidden Markov model smoothing (Lee et al., 2018, Karppinen et al., 2020), the conditional prediction function for knockoff-based false discovery rate control (Shi et al., 2023), and a compositional zero-shot learning model that factorizes

$P(C)$06

and explicitly names this factorization a Conditional Probability Framework (Wu et al., 23 Jul 2025). This suggests that “CPF” functions partly as an acronym of local convenience across fields, whereas the preorder-based theory of “Foundations for conditional probability” gives it a specific foundational meaning.

7. Conceptual significance and recurring points of interpretation

The foundational CPF makes five recurring claims. Plausibility is primitive; conditional expectation is induced by conditioning the preorder; conditional probability is conditional expectation on events; coherence is the exact representation criterion; and the framework remains meaningful when $P(C)$07 or $P(C)$08 is undefined (Mečíř, 2019). Those claims together define its conceptual identity.

A common misconception is that the framework merely restates ordinary probability in unfamiliar notation. The representation theorem and the explicit treatment of cases with zero or undefined unconditional probability show that the paper intends a stronger thesis: conditional probability is not derived from unconditional probability by a quotient rule, but reconstructed from a more primitive plausible order. Another common misconception is that such a reconstruction must abandon standard probabilistic behavior. The derived rules for monotonicity, bounds, additivity, homogeneity, and Bayes-style chain identities are presented precisely to show that familiar laws re-emerge within the new foundation.

The relation to Rényi-space and conditional-probability-space traditions also clarifies what is specific about CPF in the foundational sense. Rényi spaces take equivalence classes of $P(C)$09-finite measures and admissible conditioning events as primitive. CPS formalisms take conditional laws indexed by information events as primitive. The preorder-based CPF instead takes random quantities and a plausible preorder as primitive, then derives conditional expectation, probability, and coherence from that order. The frameworks are therefore adjacent rather than identical.

In that narrower foundational sense, CPF denotes an order-theoretic semantics for conditional expectation and probability in which representability by plausible preorder is the decisive criterion. Its principal significance lies in shifting the foundation of probability from unconditional normalization to structured plausibility, while preserving ordinary event-probability laws and extending them to cases where the classical ratio definition is silent.