General Bayesian theories and the emergence of the exclusivity principle

We construct a general Bayesian framework that can be used to organize beliefs and to update them when new information becomes available. The framework includes classical, quantum, and other alternative models of Bayesian reasoning that may arise in future physical theories. It is only based on the rule of conditional probabilities and the requirement that the agent's beliefs are consistent across time. From these requirements, we show that ideal experiments within every Bayesian theory must satisfy the exclusivity principle, which is a key to explain quantum correlations. As a consequence, the limits to the strength of correlations set by the exclusivity principle can be interpreted as the ultimate limits set by Bayesian consistency. It is an open question whether Bayesian reasoning alone is sufficient to recover all of quantum theory.

We construct a general Bayesian framework that can be used to organize beliefs and to update them when new information becomes available. The framework includes classical, quantum, and other alternative models of Bayesian reasoning that may arise in future physical theories. It is only based on the rule of conditional probabilities and the requirement that the agent's beliefs are consistent across time. From these requirements, we show that ideal experiments within every Bayesian theory must satisfy the exclusivity principle, which is a key to explain quantum correlations. As a consequence, the limits to the strength of correlations set by the exclusivity principle can be interpreted as the ultimate limits set by Bayesian consistency. It is an open question whether Bayesian reasoning alone is sufficient to recover all of quantum theory.
Introduction.-Quantum theory portrays a world where the outcomes of individual measurements cannot be predicted with certainty. And yet, the quantum predictions are strikingly accurate and successfully explain an astonishingly broad range of phenomena. The reasons for this broad applicability still remain controversial. Does the quantum framework capture a bundle of primitive facts about the world? Or it is just a generalpurpose tool for guessing the outcomes of experiments?
Albeit with a variety of nuances, different interpretations of quantum theory tend to favor either one or the other view. For example, Everett's interpretation [1] holds that the quantum framework refers to a multitude of universes. On the other hand, Bohr's interpretation [2] holds that the quantum framework concerns what experimenters can say about nature rather than nature in itself. In a similar way, QBism [3] views the quantum framework as a set of rules that constrain how agents make bets about the outcomes of their experiments.
Different positions about the interpretation are reflected into different attitudes towards the counterintuitive correlations arising in quantum theory. Since Bell [4], we have known that quantum correlations are incompatible with the intuitive worldview known as local realism. But intriguingly, the quantum violations of Bell's inequalities are not maximal: more general theories compatible with relativistic causality could in principle lead to larger violations [5][6][7]. Following up on this observation, various physical principles have been proposed to explain the quantum bounds on correlations [8][9][10][11][12]. Be-hind this approach lies the idea that the quantum bounds are a fact of the world, and, as such, they should be explained in terms of laws the world is subjected to.
However, this is not the only option. Instead of searching for principles constraining how nature behaves, one could search for consistency conditions that an agent should satisfy when assigning probabilities to the outcomes of experiments. Could it be that the quantum bounds on correlations are just consistency constraints on our probability assignments?
In this paper we examine the consequences of adopting a Bayesian point of view [3], referring to an agent who holds prior beliefs about the object of its bets, and who updates those beliefs when a new piece of information becomes available. The beliefs held by the agent are used to make predictions about the outcomes of future experiments. The key assumptions in this framework are that beliefs undergo updates according to the rule of conditional probabilities, and that beliefs at different times are consistent with each other.
Strikingly, we find that such minimalistic assumptions guarantee the existence of a class of ideal experiments that obey the exclusivity principle [13][14][15][16][17], which is a key to unlock many of the puzzling bounds on quantum correlations. Under mild assumptions, the exclusivity principle implies the Tsirelson bound [18] and, more generally, the quantum bounds for any Bell or Kochen-Specker scenario [19]. Therefore, our result suggests that all such bounds could be explained as a consequence of basic Bayesian reasoning. Whether or not all of quantum theory can be derived from Bayesian consistency conditions remains an open problem, but our result provides evidence that an important part of quantum theory can be reconstructed from elementary rules on how Bayesian agents should place their bets.
Beliefs and probabilities.-Consider the situation of an agent who makes bets about a given physical system. The content of the bets is specified by a sample space X, equipped with a σ-algebra of events Σ, namely a collection of subsets of X satisfying the properties (i) X ∈ Σ, In typical cases, X is a finite set and Σ is the power set of X. The simplest bet is to bet on an event E. Operationally, this means that there exists an experiment with two outcomes, corresponding to E and its E = X \ E, and that the agent bets that such experiment will return the outcome corresponding to E. More generally, a bet can involve an experiment with multiple outcomes, corresponding to a partition E = (E i ) i of the sample space X into disjoint events. For brevity, we will identify the experiment and the corresponding partition. These basic experiments will be called principal experiments, in order to distinguish them from more elaborate experiments where the agent performs sequences of operations.
In making a bet, the agent will rely on its beliefs, including beliefs on the laws of physics, or beliefs on the prior history of the physical system involved in the bet. We denote by B the set of all possible beliefs. For a given belief β and for a given experiment E = (E i ) i , the agent assigns a probability distribution E i → p(E i |Eβ) reflecting its expectation that the event E i will occur in the experiment E.
In the following, we will make the standard assumption that the probability assignment p(E i |Eβ) depends only on the event E i , and not on the context E in which the event is considered. In other words, we will assume that every belief β induces a probability distribution p : E → p(E|β) satisfying the usual conditions (i) p(E|β) ≥ 0 for all events E, (ii) p( i E i |β) = i p(E i |β) whenever all E i are mutually disjoint, and (iii) p(X|β) = 1. These conditions are common to most Bayesian approaches to probability [20,21]. In a Dutch book approach, they amount to the idea that a bookie assigns odds to individual events.
It is important to stress that the belief β determines the probability assignment p(E|β), but not vice versa: in general, a belief contains much more than just the outcome probabilities of principal experiments. For example, in quantum theory a belief is described by a density matrix, while a principal experiment corresponds to a projective measurement, with the projectors associated to a given basis. The probability assignment depends only on the diagonal entries of the density matrix in the given basis, and therefore it is not sufficient to identify the density matrix.
Bayesian updates.-The central question in the Bayesian approach is how the agent should update its belief when new information becomes available. Suppose that the agent receives a guarantee that E is the case. The question is how the agent should update its beliefs in order to incorporate this new piece of information. Intuitively, the new belief β ′ should be the old belief β, adjusted to take into account that E is the case. We will denote the new belief as β ′ = Eβ, indicating that the information about the occurrence of E has been incorporated into β. Note that this update makes only sense if the probability p(E|β) is non-zero. Otherwise, the belief β would be incompatible with event E, meaning that the agent would have no way to combine the prior belief β with the occurrence of E. In the following, whenever the notation Eβ is used it is assumed that p(E|β) is non-zero.
The point of updating a belief is to compute conditional probability distributions. We demand that the probability assignment for the updated belief Eβ is given by the rule of conditional probabilities: Axiom 1 (Rule of conditional probabilities). If p(E|β) > 0, then the updated belief Eβ satisfies the rule of conditional probabilities The rule of conditional probabilities implies many properties of the update map β → Eβ. For example, it implies that, once the agent updates its belief based on the event E, the agent becomes certain of the event E. Indeed, one has which follows from letting E = F in Eq. (1). We stress that the update map does not in itself represent a physical process on the observed system, but rather an operation internal to the agent. In the following, we formulate two conditions that the update should satisfy in order for the agent to be consistent with its beliefs at all moments of time.
Forward consistency.-Suppose that the agent believes that the occurrence of the event E is certain, p(E|β) = 1. In this case, the occurrence of E does not add any new information. As a consequence, the update should be trivial, namely Eβ = β.
Axiom 2 (Forward consistency). If the agent is certain that the event E will occur, then the occurrence of E does not change the agents's belief. Mathematically: for every β ∈ B and every E ∈ Σ, p(E|β) = 1 implies Eβ = β.
Since forward consistency is an axiom about the belief of the agent, it is not a law of physics, but just a consistency requirement. It constrains the agent in how it updates the belief forward in time, coherently with the belief held at the present moment. A simple consequence of forward consistency is that the total event E = X does not lead to any update, namely Xβ = β, ∀β ∈ B.
Actions.-So far, we considered the situation where an agent bets directly on the occurrence of a certain event. More generally, the conditions under which a bet is made can be altered before an experiment is performed. We use the word "action" broadly, including situations in which the system evolves under its natural dynamics for a fixed amount of time. In this sense, actions need not be intentional: what matters is only that the agent has a definite belief about what change has occurred to the physical system under consideration.
We denote the set of all actions as Act. It is natural to assume that actions can be composed one after the other, namely that there is an action AB, corresponding to the execution of action A right after action B. The composition is associative, meaning that only the temporal sequence of actions matters, and not how the actions are grouped together. Among the possible actions, we include the trivial action I, which preserves the current state of affairs. Mathematically, this makes Act a monoid [22] (see also [23,24]). An action A is reversible if there exists another action B such that AB = BA = I.
When an action A is performed, the agent will generally change its belief. We will denote the new belief as β ′ = Aβ, meaning that the initial state of affairs described by β has undergone the change induced by the action A. Two actions that induce the same change will be identified: if A ′ β = Aβ for every belief β, then A ′ = A.
Backward consistency.-When actions are included in the picture, additional consistency conditions arise. Suppose that the agent holds the belief β and suppose that, in the context of that belief, the proposition "event F is the case after action A" implies the proposition "event E is the case before action A". Now, consider the following scenarios: Scenario (1) The agent is promised that F will be the case, should the action A be performed. Scenario (2) The agent performs the action A and, only afterwards, it is promised that F is the case. In Scenario (1), if the agent decides to perform the action A, its belief can already be updated to Eβ, even before the action A is performed. After the action A, the belief will be updated to AEβ, and finally to F AEβ, taking into account that F was promised to be the case. In Scenario (2), the agent will update its belief to Aβ, after performing the action A, and then to F Aβ, upon receiving the guarantee that F is the case. Consistency between the two scenarios requires that the final belief of the agent should be the same, namely, It remains to specify what it means for proposition "F is the case after A" to imply proposition "E is the case before A" in the context of the belief β.
To state a relation between the propositions "E is the case" and "F is the case", the agent can assign a joint probability distribution j β,E,A,F (E i , F k ) to the events E i ∈ E and F k ∈ F. Consistency with the agent's beliefs requires the joint probability distribu- In particular, we say that the proposition "F is the case after A" implies the proposition "E is the case before A" in the context of the belief β if the following condition holds Axiom 3 (Backward consistency). If the proposition "event F is the case after action A" implies the proposition "event E is the case before action A" in the context of belief β, then the occurrence of F after A leads to the same updated belief as the occurrence of F after A and after the occurrence of E. Mathematically: In Appendix A we show that quantum theory satisfies forward and backward consistency if the events are represented by orthogonal projectors and the updates follow Lüders' rule.
General Bayesian theories.-A general Bayesian theory is a tuple (B, Act, Σ, U, p), consisting of a set of beliefs B, a monoid of actions Act, acting on the beliefs, a σ-algebra of events Σ, acting on the beliefs through an update map U : (E, β) → Eβ, and a probability assignment p(E|β). The update map is required to satisfy Bayes' rule and the properties of forward and backward consistency.
Note that two beliefs that assign the same probabilities to all possible experiments are equivalent for every practical purpose. Hence, we will regard them as the same belief. In other words, we assume that the condition p(E|Aβ ′ ) = p(E|Aβ), ∀E, ∀A, implies β ′ = β. Mathematically, this assumption identifies the belief β with the function f β (E, A) := p(E|Aβ). We say that the space of beliefs B is finite-dimensional if the vector space generated by the corresponding functions is finite-dimensional (see Appendix C for the details).
Sequential experiments and ideal experiments.-A sequential experiment consists of a sequence of actions interspersed by principal experiments. For example, (A, E, B, F) represents a sequence consisting of an action A, followed by a principal experiment E, followed by another action B, and by another principal experiment F. The joint probability distribution of the outcomes are computed via the rule of conditional probabilities. For example, the probability distribution of the sequential experiment We now focus our attention on a special class of experiments that leave the agent with the option of gathering more refined pieces of information in the future. We say that an experiment (A ′ , F) with partition F = (F i,l ) i,l is a refinement of another experiment (A, E) with partition E = (E i ) i if the following condition holds: The experiment (A, E) is sequentially refinable if there exists an action B such that, for every refinement (A ′ , F) and for every initial belief β, the probability of the event F i,l in the experiment (A ′ , F) is equal to the joint probability of the events (E i , F i,l ) in the sequential experiment (A, E, B, A ′ , F). In formula, for every refinement (A ′ , F), for every event F i,l ∈ F, and for every belief β ∈ B. Operationally, this means that the coarse-grained experiment (A, E) does not alter the probability assignment for the fine-grained experiment (A ′ , F), provided that the agent performs the "correction action" B between them. The proof is provided in the Appendix C. An example of an ideal family is the family of principal experiments (I, * ). For a classical system, this is the only ideal family of experiments, up to relabelling of the outcomes. In quantum theory, instead, there exist infinitely many families of ideal experiments: up to outcome relabelling, there is one ideal family for every maximal algebra of mutually commuting observables.
Recovering the exclusivity principle.-If an experiment (A, E) belongs to an ideal family, we call it ideal. We now show that ideal experiments satisfy the exclusivity principle [13-19, 26, 36], also named global exclusivity [14] or consistent exclusivity [16,17,36]. In the language of general Bayesian theories, the exclusivity principle refers to a collection of ideal experiments, W r = (A r , E r ). Two events E and E ′ that occur with the same probability for every initial belief β are considered equivalent, and the equivalence class is called an effect: the effect corresponding to the event E ∈ E r in the experiment W r will be denoted as eff(E; W r ), and the probability assigned to the effect eff(E; W r ) for the belief β is be denoted by eff(E; W r ), β := p(E|A r β).
Two effects e 1 and e 2 are called mutually exclusive if there exists an ideal experiment W = (A, E) such that eff(E 1 ; W ) = e 1 and eff(E 2 ; W ) = e 2 for two disjoint events E 1 ∩ E 2 = ∅. The exclusivity principle states that, if every pair of effects in the set { e n } n is mutually exclusive, then the corresponding probabilities satisfy the bound n e n , β ≤ 1.
Crucially, the above bound holds in every general Bayesian theory: Theorem 2. Every general Bayesian theory satisfies the exclusivity principle.
The proof is provided in Appendix D.
The exclusivity principle is known to characterize the exact quantum bounds on the set of correlations in a variety of scenarios [13][14][15][16][17][18][19]26]. Combined with this fact, our result suggests that the bounds on quantum correlations could be interpreted as the ultimate limit posed by Bayesian consistency.
Conclusions.-We have built a framework for general Bayesian theories, which can be applied to the classical and quantum domain, or to more general alternatives. In this framework, the agent holds a belief in order to place bets about the outcomes of experiments. The belief is updated according to actions and perceptions of the agent. The constraints that the update has to satisfy are the rule of conditional probabilities and the conditions of forward consistency and backward consistency, which express the consistency of beliefs at different moments of time. These three requirements allow us to define a privileged class of ideal experiments. We showed that the results of the class of ideal experiments satisfy the exclusivity principle which, under mild assumptions, implies tight bounds on the set of quantum correlations for any Bell or Kochen-Specker scenario [19]. Therefore, our result shows that an important part of quantum theory can be reconstructed from elementary rules on how a Bayesian agent should bet about the outcomes of future experiments.
Whether or not all of quantum theory can be derived from Bayesian consistency conditions remains an open problem. If such a derivation turns out to be possible, it would support the view that quantum theory is a consequence of how agents subjectively organize experiences [27]. If such a derivation is not possible, the attempt to find it might point out to the crucial physical ingredient that identifies quantum theory among other physical theories [28]. In contrast to earlier reconstructions of quantum theory [29][30][31][32][33][34][35]  Here we show that forward and backward consistency hold for projective measurements in quantum theory.
We start with forward consistency. Let ρ be a density matrix, E be a projector, and E = I − E be its orthogonal complement, I being the identity matrix. The condition that the event described by E happens with certainty on ρ is tr(Eρ) = 1. Forward consistency is the statement that, under this condition, the updated state EρE/ tr(Eρ) is equal to the original state ρ. The proof is simple. The certainty condition tr(Eρ) = 1 is equivalent to tr(EρE) = 0, which in turn is equivalent to EρE = 0. Since this equation is of the form AA † = 0 with A = E √ ρ, we have A = 0 and thus E √ ρ = 0 and √ ρ E = 0. This leads to proving that quantum theory satisfies forward consistency. We now show that quantum theory satisfies backward consistency, too. Let A be a completely positive and trace preserving map with Kraus representation A : ρ → i A i ρA † i , ρ be a density matrix, E, F be projectors, and E = I − E. Suppose that the occurrence of the event F after A excludes the prior occurrence of the event E, namely Then we have F A i (EρE)A † i F = 0 for all i and by the same argument as above, proving that quantum theory satisfies backward consistency.

Appendix B: Beliefs and actions in general Bayesian theories
One of the standing assumptions in our framework is that two beliefs β and β ′ are different only if they assign different probabilities to at least one experiment (A, E). Equivalently, this means that the belief β can be identified with the function f β (E, A) := p(E|Aβ). In this way, the set of beliefs can be regarded as a subset of the vector space of functions from Σ × Act to the reals. We define as Span R (B) the linear span of the functions f β , with β ∈ B. When Span R (B) is finite dimensional, we say that the space of beliefs is finite dimensional.
In the vector space picture, the actions act as linear maps: Proposition 1. For every action A ∈ Act, the relation T A (f β ) := f Aβ uniquely defines a linear map on Span R (B).
Proof. Every element of Span R (B) is a linear combination of functions of the form f β , say f = i c i f βi for some beliefs {β i } and some real coefficients {c i }. Hence, the requirement that T A be linear implies the relation This relation defines the action of T A on every element of Span R (B). It remains to check that the definition is consistent, meaning that if f admits another . This is equivalent to showing the implication the last equality following from the condition i c i f βi = 0.
Mathematically, the correspondence A → T A defines a representation of the monoid Act on the space of linear maps. In particular, it is easy to check that one has T AB = T A T B for every pair of actions (A, B), and T I = I D , where where D is the dimension of the vector space Span R (B).
The following fact will be used in the proof of Theorem 1 in the main text: Proposition 2. Let B be a finite dimensional belief space, and let A ∈ Act and B ∈ Act be two actions. If AB = I, then BA = I.
Proof. Since AB = I, the corresponding linear maps must satisfy the condition T A T B = I D . Since Span R (B) is finite dimensional, the relation T B T A = I D must also hold. Now, for every belief β, one has Since f BAβ = f β , we conclude that BAβ is equal to β. Moreover, since the equality holds for every β, we conclude that BA is equal to the identity transformation.
Appendix C: Proof of Theorem 1 The proof uses four lemmas, provided in the following: Lemma 3. Let (C, F) be a refinement of (I, E), with F = (F i,l ) i,l and E = (E i ) i , and let F i be the event defined by F i := l F i,l . Then, the equality holds for every outcome i and for every β ∈ B such that p(E i |β) = 0.
Proof. Since (C, F) is a refinement of (I, E), one has the equality the first equality following from the additivity of the probabilities of disjoint events. Applying Eq. (C2) to the initial belief E i β, we obtain the condition the second equality following from Eq. (2) in the main text. Since the agent is sure that the event F i will occur, forward consistency implies that no update should take place on the belief CE i β. Hence, we obtain the relation Now, consider the joint probability distribution Using Equation (C3), we obtain j E,C,F (E i , F j ) = δ i,j p(E i |β), and, therefore j E,C,F (E i |F i ) = 1. Then, backward consistency implies Combining Eqs. (C4) and (C5), we obtain Eq. (C1). Proof. We need to show that there exists an action B such that, for every partition E = (E i ) i , every refinement (C, F) of (I, E), and every belief β, the ideality condition is satisfied. Note that the condition is trivially satisfied when p(E i |β) = 0. Hence, we only need to consider the case of p(E i |β) = 0. Let (C, F) be a refinement of (I, E), with F = (F i,l ) i,l and E = (E i ) i , and let F i be the event defined by F i := l F i,l . Using the rule of conditional probabilities, we obtain the last equality following from Eqs. (C1) and (C2). Eq. (C7) is valid for every refinement (C, F) and for every belief β such that p(E i |β) is non-zero. This condition is nothing but condition (C6), with B = I. Hence, the principal experiment (I, E) is sequentially refinable. Since the partition E is arbitrary, this proves that the family (I, * ) is ideal. Proof. We need to show that there exists an action B such that, for every partition E = (E i ) i , for every refinement (A ′ , F) of (A, E), and for every belief β, the ideality condition is satisfied. Since A is reversible, all elements of B can be obtained as Aβ for some β, namely AB = B. Setting β ′ := Aβ, B = A −1 , and C := A ′ B, the condition (C8) becomes Our goal is to show that this relation holds. Note that, by construction, the condition that (A ′ , F) is a refinement of (A, E) is equivalent to the condition which in turn is equivalent to the fact that (C, F) is a refinement of (I, E). Then, we can use Eq. (C7) to conclude that Eq. (C9) holds, and therefore Eq. (C8) holds. Hence, the experiment (A, F) is sequentially refinable. Since the above argument holds for every partition E, the family (A, * ) is ideal.
Lemma 6. Suppose that the space of beliefs B is finite dimensional. If the family of experiments (A, * ) is ideal, then the action A is reversible.
Proof. Let B be the action such that the ideality condition (C8) holds for every partition E, for every refinement (A ′ , F), and for every belief β ∈ B.
Consider the trivial partition E 0 = (X). By definition, every experiment (C, F) is a refinement of this experiment. Hence, condition (C8) becomes (C11) Since the beliefs β and BAβ predict the same probabilities for any possible experiment (A ′ , F), they are identified. Hence, one has β = BAβ. Since the belief β is arbitrary, one also has BA = I. Since the belief space is finite dimensional, Proposition 2 implies BA = AB = I.
With the above lemmas at hand, the proof of Theorem 1 is immediate: Proof of Theorem 1 in the main text. The first statement in the theorem follows from Lemma 5. The second statement follows from Lemma 6.

Appendix D: Proof of Theorem 2
The proof of Theorem 3 follows the same line of argument of Ref. [36]. The key idea is to show that, if any pair of effects in a set { e n } n are mutually exclusive, then all the effects { e n } n can coexist in a single experiment, meaning that there exists an experiment and a subset of outcomes whose probabilities are given by the effects { e n } n . Lemma 7. In every general Bayesian theory, every collection of pairwise exclusive effects can coexist in the same experiment.
Since the probabilities of all outcomes of a single experiment sum to at most unity, the validity of Lemma 7 implies the validity of Theorem 3.
We start by formalizing the idea that a set of effects { e n } n can coexist in the same experiment. In general, the experiment can consist of a sequence of actions and perceptions, generated by the following procedure: 2. If the outcome is i 1 , then perform experiment 3. If the outcome is i 2 , then perform experiment W (i1i2) , and so on, with the experiment at the jth step depending on the outcomes of the previous j − 1 measurements.
The whole experiment will be denoted by the list L = , . . . , W (i1,i2,...,i k−1 ) k , indexed by the possible outcomes (i 1 , i 2 , . . . , i k−1 ). For a given sequence of events i2 , . . . , E (i1,i2,...,i k−1 ) i k ; L) the corresponding effect. We say that a set of effects { e n } n coexists in the same experiment if there exists a sequential experiment that contains all the e n among its effects.
Now, note that the experiment Z is a refinement of the experiment W . Indeed, one has and for every initial belief β. Equations (D3) and (D4) imply that Z is more refined than W . Then, the sequential refinability of the experiment W implies Combining Eq. (D2) with Eq. (D5) we obtain Since the equality holds for every belief β, we conclude eff(E, E ′ ; L) = e ′ .
We are now ready to prove Lemma 7.
Proof of Lemma 7. Let (e n ) k n=1 be a collection of pairwise mutually exclusive effects. For the pair (e m , e n ), let W m,n = (A m,n , E m,n ) be an ideal experiment such that e m = eff(E Here, i can take any value between 1 and k − 1. We can also allow i = k if we define e k+1 := e 1 , because then e k and e k+1 are mutually exclusive.
Consider the sequential experiment defined by the following procedure: (1) Set i = 1.
(ii) Perform the experiment W (i) . If the outcome is E (i,i+1) i , then record the outcome m = i and stop. Else, perform the action B i,i+1 associated to experiment W (i) in the ideality condition (C2), and move to the next step.
(iii) Increment i by one (i.e. i → i + 1). If this gives i = k + 1, then record the outcome m = k + 1 and stop. Otherwise go back to step (ii).
The above procedure defines a sequential experiment with k + 1 outcomes m ∈ {1, . . . , k + 1}. For i ≤ k, the i-th outcome is the outcome corresponding to the event E (i,i+1) i in the experiment W (i) . We denote by q β (m) the probability assigned to the outcome m when the initial belief is β.
We now show that q β (m) = e m , β for every m ∈ {1, . . . , k}. Formulating the proof in full generality for all possible k is somewhat tedious (it can be done using induction); we will thus present the proof for the first non-trivial case of k = 3 only (the case k = 2 is trivial, since a single pair of exclusive effects does by definition coexist in the same experiment). However, it will become obvious how to generalize the proof to k ≥ 4.