Order Effects of Measurements in Multi-Agent Hypothesis Testing

03/26/2020 ∙ by Aneesh Raghavan, et al. ∙ University of Maryland 0

All propositions from the set of events for an agent in a multi-agent system might not be simultaneously verifiable. In this paper, we revisit the concepts of event-state-operation structure and relationship of incompatibility from literature and use them as a tool to study the algebraic structure of the set of events. We present an example from multi-agent hypothesis testing where the set of events does not form a Boolean algebra but forms an ortholattice. A possible construction of a 'noncommutative probability space', accounting for incompatible events (events which cannot be simultaneously verified) is discussed. As a possible decision-making problem in such a probability space, we consider the binary hypothesis testing problem. We present two approaches to this decision-making problem. In the first approach, we represent the available data as coming from measurements modeled via projection valued measures (PVM) and retrieve the results of the underlying detection problem solved using classical probability models. In the second approach, we represent the measurements using positive operator valued measures (POVM). We prove that the minimum probability of error achieved in the second approach is the same as in the first approach.

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I Introduction

In the study of stochastic multi-agent decision making problems it is often assumed that the joint distribution of the observations collected by the agents is known. In our prvious work,

[13], we have discussed as to why such an assumption will not lead to truly decentralized policies. Hence the joint distribution of measurements collected by agents in a multi-agent system might not be always available. When a probability space is to be constructed for an agent in the multi-agent system, the first step would be to enumerate the list of events / propositions that the agent can verify. We recall that in Kolomogorov’s axioms for classical probability, it is assumed the set of events ( assocciated with subsets of sets ) form a Boolean algebra, a very specific algebraic structure. Hence, the existence of a classical probability space for formulating and solving decision-making problems imposes restrictions on the set of events, i.e., the set verifiable propositions. By assuming that we can construct a classical probability space we assume that the set of events is a Boolean algebra. This assumption implies that all subsets of events are simultaneously verifiable. In multi-agent systems, agents collect observations and exchange information. In asynchronous multi-agent systems, the agents might not have a common notion of time. Propositions involving information from different agents might not be simultaneously verifiable as the information might not be simultaneously available, thus violating the structure of a Boolean algebra. Hence before we construct a classical probability space for a agent, we would first have to verify that the set of events indeed form a Boolean algebra and cannot assume apriory that a classical probability space can be constructed for the agent.

Our hypothesis is that the algebraic structure of the set events need not be a Boolean algebra, it can be an orthomodular ortholattice. We present an example from multi-agent decision making supporting our hypothesis. This hypothesis is motivated from the observation that for an agent there could exist propositions which are not “simultaneously verifiable” by the agent. Such events exist in quantum mechanical systems, which leads to the set of events forming an orthomodular ortholattice. The algebraic structure of the set of events in quantum mechanical systems have been well investigated in literature. One of the earliest papers in this direction, is [4]. More recently, in [8] the author argues that quantum logic is a fragment of independent friendly logic. Noncommuting observables are assumed to be mutually dependent variables. Independent friendly logic allows all possible patterns of dependence/ independence to be expressed among variables, which is not possible in first order logic. Independent friendly logic violates the law of excluded middle ( every proposition, either its positive or negative form is true). This violation stems from the fact that truth value for propositions is assigned by finding winning strategy for a player in a suitable game. In [8], the author argues that one can a find a suitably analogy between quantum logic and an extension of independent friendly logic.

Our objective is to study multi-agent decision-making problems given “data sets” or samples of (observation, decision) pairs generated from the multi-agent system. Our objective leads us to first study the algebraic structure of the set of events and then “suitably” construct a probability space where the decision-making problems can be formulated and solved. The problem that we consider is the binary hypothesis testing problem with three observers and a central coordinator. There are two possible states of nature, one of which is the true state of nature. There are three observers collecting measurements (samples) that are statistically related to the true state of nature. The joint distribution of the measurements collected by the observers is unknown. Each observer knows the marginal distribution of the observations it alone collects. Each observer performs sequential hypothesis testing and arrives at a binary decision. The binary decision is then sent to a central coordinator. The objective of the central coordinator is to find its own belief about the true state of nature by treating the decision information that it receives as measurements. At the central coordinator a suitable probability space is to be constructed for formulating and solving the hypothesis testing problem.

Our contributions are as follows. The set of events, i.e, the set of propositions that can be verified by the central coordinator is enumerated. We show that the set along with suitable relation of implication and unary operation of orthocomplmentation is not a Boolean algebra. To prove the same, we adopt the methodology developed in [12]. Hence the construction of a classical probability space is ruled out. We construct an event-state structure (a generalization of measure spaces) for the central coordinator along the lines of von -Neumann Hilbert space model. We associate operations (a generalization of conditional probability) with the event-state structure and construct a noncommutative probability space for the central coordinator. We consider the binary hypothesis testing problem in the non-commutative probability framework. We present two approaches to the decision-making problem. In the first approach, We represent the available data as coming from measurements modeled via projection valued measures (PVM) and retrieve the results of the underlying detection problem solved using classical probability models. In the second approach, we represent the measurements using positive operator valued measures (POVM). We prove that the minimum probability of error achieved in the second approach is the same as in the first approach. For specific empirical local distributions at the three observers, we show that different orders of measurement can lead to different probabilities of error at the central coordinator.

The paper is organized as follows. In the next section, section II, we present the methodology from [12] which we can used to investigate the structure of the set of events. In section III, we discuss a specific example from multi-agent decision making supporting our hypothesis. In section IV, we discuss hypothesis testing problem in a non commutative probability space, the probability space from von Neumann Hilbert space model.

Ii Algebraic structure of the set of events

In the following section we introduce some definitions and identities from propositional calculus that have been mentioned in the literature, for e.g., [4]. We have mentioned them to keep this paper self contained.

Ii-a Introduction to propositional calculus

Let be an experiment. Let be the set of experimentally verifiable propositions, i.e., propositions to which we can assign truth value based on the outcome of the experiment .

Example 4.1 [14]. Let the experiment be ‘observing the environment (surroundings)’. Suppose the set of propositions is =it is raining, it is snowing, it is warm, it is cold, the sun is shining, it is not raining, it is not snowing, it is not warm, it is not cold, the sun is not shining. By performing the experiment(i.e., by observing the surroundings) one can assign truth value to each proposition, i.e., each proposition is either true or false. On the domain of propositions, we are given the the relation of implication() which satisfies the following properties:

  • reflexive: for any proposition , ,

  • transitive: for propositions and belonging to , if and , then .

In example 4.1, ‘it is warm ’ ‘it is not snowing’, ‘it is raining’ ‘it is not shining’ and, ‘it is cold’ ‘it is snowing’( this implication need not be true always). We can define the relation of cotestable on the set of propositions as follows: two propositions are cotestable if and only if they can be assigned truth values simultaneously. This relation is reflexive, symmetric but is not transitive. When we verify the relation of implication between two propositions and , we are simultaneously assigning truth value to both the propositions, i.e., we are assuming that the propositions are cotestable. If we impose the condition that the relation of implication between two propositions can be verified only when the propositions are co-testable, we loose the transitivity property of the relation of implication. The concept of simultaneous testability was introduced in [14].

The domain and the relation implication , form a partially ordered set(POSET). The transitivity property of the relation of implication is essential for the construction of the partially ordered set. Hence we assume all propositions are simultaneously verifiable. We assume that the domain includes the identically true proposition, denoted by , and the identically false proposition, denoted by . Both and are partially ordered sets. Using the relation of implication, we can define operations on the set .

Definition II.1.

Let be a POSET. A proposition is said to be the conjunction (greatest lower bound or “meet”) of propositions and if , , and, for any other proposition such that and , . The conjunction of and is denoted by .

Definition II.2.

Let be a POSET. A proposition is said to be the disjunction (least upper bound or “join”) of propositions and if , , and, for any other proposition such that and , . The disjunction of and is denoted by .

Definition II.3.

Let be a POSET. A proposition is said to be logically equivalent to proposition if and .

In the example, the meet and join of the propositions are not included in . We obtain the set , by taking the closure of the set with respect to the conjunction and disjunction operations. is also a partially ordered set.

Definition II.4.

Let be a POSET with with and . A mapping is an orthocomplementation,(denoted by ) provided it satisfies the following identities: for and ,

  1. ,

  2. and ,

  3. implies .

If is an orthocomplementation, the relation of orthogonality() is defined as if and only if .

The relation of orthogonality is not reflexive or transitive. From identity [3], it follows that the relation is indeed symmetric. From the definitions of the conjunction operator, disjunction operator and the identities, [1], [2], and [3], the following result can be proven,

  1. and .

Definition II.5.

A partially ordered set is said to be lattice if: for every proposition and , and belong to .

From the above definition it follows that neither nor are lattices but is a lattice. The distributive identity of propositional calculus can be stated as follows: for ,

  1. and .

A lattice which satisfies [2] and [5] is a Boolean algebra. In classical probability, the probability space consists of a sample space, a sigma algebra of subsets of the sample space and a probability measure on the sigma algebra. The sigma algebra along with set inclusion as the relation of implication, union of sets as the disjunction operation, and intersection of sets as conjunction operation is a Boolean algebra. Hence in classical probability we are defining measures over a Boolean algebra. The modular identity can be stated as follows:

  1. If , then

The finite dimensional subspaces of a Hilbert space, along with subspace inclusion as the relation of implication, closed linear sum (instead of union of sets) as the disjunction operation, and set products (corresponding to intersection of sets)as conjunction operation satisfy the modular identity, but do not satisfy the distributive identity. Thus, if the propositions from the experiment along with implication relation satisfy the modular identity, but not the distributive identity, they can be represented by the finite dimensional subspaces of Hilbert space with the direct sum operation corresponding to the disjunction operation and set product operation corresponding to conjunction operation. In our study we consider the set of propositions as the propositions which describe the outcomes of experiments on multi-agent systems. They can be assigned truth values based on the outcome of the experiments. For propositions which arise from experiments on multi-agent systems, the relation of implication and unary operation of orthocomplementation are yet to be defined, but the properties and identities that they satisfy were discussed in the section.

Ii-B Event state operation structure

Ii-B1 Event-state structures

We are interested in studying the structure of the set experimentally verifiable propositions. We associate operations with the propositions(events as defined below) and measures on the set of propositions. From the properties of the operations and measures we infer the algebraic structure of the set of propositions. We follow the definitions mentioned in [12]:

Definition II.6.

An event state structure is a triple where:

  1. is a set called the logic of the event state structure and an element of is called an event,

  2. is a set and an element of is called an state,

  3. is a function called the probability function and if and then is the probability of occurrence of event in state ,

  4. if , then the subsets and of are defined by , and if () then the event is said to occur (not occur) with certainty in the state ,

  5. axioms to are satisfied.

Axioms:

  1. If belong to and then .

  2. There exists an event such that .

  3. If belong to and then .

  4. If then there exists an event such that and .

  5. If are a sequence of events such that for then there exists a such that (a) for all i (b) if there exits such that for all i, then , and (c) if then .

  6. If such that for every then .

  7. , and then exists an such that for all .

There are different interpretations that could be associated with the state, [9]. The state could refer to the physical state of the system. The state could be interpreted as a special(probabilistic) representation of information about the results of possible measurements on an ensemble of identically prepared systems. The second interpretation is appropriate given our context. An event may be identified with the occurrence or non-occurrence of a particular phenomenon pertaining to the multi-agent system. The event is associated with an observation procedure which interacts with multi-agent system resulting in a yes or no corresponding to the occurrence or non-occurrence of the phenomenon. The interpretation of for and is as follows: we consider an ensemble of the systems such that the state is . We determine the occurrence or non-occurrence of the event by executing the associated the observation procedure associated with on each system in the ensemble. If the ensemble is large enough then the frequency of occurrence of is close to . Axiom [I.1] states the condition for uniqueness of events. Axiom [I.2] guarantees the existence of the certain event. Axiom [I.4] guarantees the existence of the orthocomplement of any event. Axiom [I.3] ensures that the third part of definition II.4 is satisfied. Axiom [I.5] is equivalent to countable additivity of measures. Axiom [I.6] states the condition for uniqueness of states. Axiom [I.7] leads to convexity of the probability function.

Definition II.7.

If is an event state structure, then the relation of implication, , is defined as follows: for , if and only if .

The relation of implication is defined using the states and the probability function. Thus is said to imply if and only if the set of states for which occurs with certainty is a subset of the set of states for which occurs with certainty. Since the subset relation() is reflexive and transitive, it follows that the implication relation is also reflexive and transitive. The antisymmetry property of the subset () relation and axiom [I.1] imply that the implication relation is also antisymmetric. Hence the relation of implication () is partial ordering of .

Definition II.8.

Let be an event state structure. Then the unique event such that and is the certain event. If , then the unique event such that and is called the complement(negation) of . The unique event such that and is the impossible event.

Axiom [I.2] implies the existence of the certain event and axiom [I.1] implies that the certain event is unique. Further, the certain event is the greatest event corresponding to , as , for all . Axiom [I.4] applied to the certain event yields the unique event such that , and for all .

Theorem II.9.

If is an event state structure, then:

  • is a POSET,

  • and are the greatest and least elements of the POSET, ,

  • is an orthocomplementation on ,

  • if , the following are equivalent: (a) (b) (c) ,

  • if , the following are equivalent: (a) (b) (c) ,

  • if , the following are equivalent; (a) (b) (c) .

For the proof of above theorem we refer to [12].

Example 4.2 [12] We consider the classical probability model, the probability space constructed based on Kolomogorov’s axioms. Let be the sample space and be a sigma algebra of subsets of . The relation of implication is defined as follows: if and only if , where the relation is the set theoretic inclusion. is a probability measure if (a) and (b)if is a sequence of pairwise orthogonal events, then . Let be a collection of convex, strongly order determining set of probability measures on . Let . Then is an event-state structure. The sample space corresponds to the certain event (thus verifying axiom I.2) and the corresponds to the impossible event. Axiom [I.1] follows from the strong order determining property of the set . The orthocomplementation is given by , where demotes the set theoretic complement, satisfies axiom [I.4]. Since is a algebra, the countable union of events in also belongs to . This property of the algebra along with countable additivity of the measures imply that axiom [I.5] is also satisfied. Axiom [I.7] follows from the convex property of the set .

Example 4.3 [12] Let be a separable complex Hilbert space. Let denote the set of bounded linear operators which map from to . Let denote the adjoint of . For , let and . Let denote the set of hermitian, positive semi-definite bounded linear operators. For the following definitions and results we refer to [10]. Let denote the set of operators in which have finite rank. The set of compact operators is closed subspace of . The set is dense in with the operator norm. Let denote an orthonormal basis for (since is separable the orthonormal basis exists). For , the trace norm is defined as , where and corresponds to inner product on the Hilbert space . The trace norm is independent of the choice of orthonormal basis. The set of trace class operators is set of operators in which have finite trace norm, . The set of trace class operators is a subspace of

. The vector space

along with the trace norm is a nonreflexive Banach Space. It can be shown that . is a dense subset of the Banach space with the trace norm. For , there exits such that . Since , . When a sequence of compact operators converge to a bounded operator, that operator is also compact. Thus is compact. Hence , i.e., every trace class operator is compact. Let the closed (in norm topology) convex cone of hermitian, positive semidefinite trace class operators be denoted by . Let . Let denote the set of all orthogonal projections onto , . Let for and be defined as . Then is an event state structure. The identity operator () corresponds to the certain event and null operator() corresponds to the impossible event. but does not belong to . Axioms [I.1], [I.2] and [I.3] can be verified. The orthocomplementation is given by which satisfies axiom [I.4]. Axioms [I.5] and [I.6] can be verified. Since is convex and the trace operator is linear, axiom [I.7] is also satisfied. if and only if which is equivalent to stating that . With this definition for the relation of implication, it can be shown that for , is the projection onto the subspace and is the projection on the subspace .

Ii-B2 Relation of compatability

Definition II.10.

The relation of compatibility () is defined on the set of events, , as follows: for , if and only if there exists such that (a) (b) and and (c) and .

The relation on satisfies following properties,[12]:

  1. if and then ,

  2. if and then (a) , (b) (c) and exist in ,

  3. if , , , , and exists then and .

The relation is determined by the following property, [12]: for , if and only if there is Boolean sublogic such that .

Theorem II.11.

Let be an event state structure. If and there exists an such that then the conjunction of with respect to exists and is equal to .

For the proof of above theorem we refer to [12].

Ii-B3 Operations

The concepts of conditional probability and conditional expectation are very important in classical probability theory. They enhance the utility of the theory and deepen the mathematical structure of the theory. They are extensively used in estimation, detection, filtering and control. Conditional probability is defined as a measure on a restricted sample space, with the ’observed event’ leading to the restriction. Conditional expectation of a random variable given a

algebra is a random variable which is measurable with respect to the algebra and its expectation is equal to the expectation of the original random variable over the sets of the algebra. Our goal is to obtain concepts analogous to conditional probability and conditional expectation for general event-state structures. Conditional probability can be viewed as map from a probability measure to a probability measure restricted to the observed event. Since states in the event-state structure are “analogous” to probability measures in classical probability, we first define maps from the set of states to the set of states and its associated properties.

Definition II.12.

Let be an event state structure.

  1. Let denote the set of all maps with domain and range . If and then denotes the image of under .

  2. For , if and only if and .

  3. is defined by .

  4. is defined by and .

  5. If , then is defined by and .

In order to predict the result when consecutive experiments are performed on a system, it is essential to define the composition of maps. The state obtained up on applying the composition of maps and to a state , denoted by (), is the state obtained by applying the map first to and then applying to . We impose an axiomatic framework on the set of maps resulting in “operations” which can be associated with events from the experiment.

Definition II.13.

An event-state-operation structure is a 4-tuple where is an event-state structure and is mapping which satisfies axioms [II.1] to [II.7].

If , then is called the operation corresponding to event . If and , then is called the state conditioned on the event and state . If , then is the probability of conditioned on the event and state . Let denote the subset of defined by . An element of is called as operation.

  1. If , then the domain of coincides with the set .

  2. If , and then .

  3. If and , then .

  4. If , are subsets of , and then .

  5. If , then there exists a such that .

  6. If , and , then .

  7. If , and then .

Example 4.2 Operations for the classical probability space: The event state structure is . For , the operation is defined as follows:

The domain of is , satisfying axiom [II.1]. Axioms [II.2], and [II.3] can be verified. For axiom [II.4], it is given that for all in domain and . Since the operation is commuting, it follows that for all in domain and . For axiom [II.5], let . Domain of is . The states which do not belong to the domain are: . Let , that is the set theoretic complement of . as is a algebra. . Hence there exists unique event satisfying axiom [II.5]. Axioms [II.6] and [II.7] can be verified.

Example 4.3 Operations for the von Neumann Hilbert space model: given an event , the operation corresponding to event is defined as:

The domain of is , satisfying axiom [II.1]. Axioms [II.2], and [II.3] can be verified. For the verification of axioms [II.4] and [II.5] we refer to sections A-A and A-B. Axioms [II.6] and [II.7] can be verified. We note that in this von Neumann Hilbert space model, the orthocomplementation corresponds to orthogonal complement of subspaces and not the set theoretic complement. This concept has been discussed in [8].

Definition II.14.

Let be an event-state-operation structure. The mapping is defined as: if , there exists such that , then .

Axiom [II.4] ensures that even if there are two sequences of operations which result in the same operation, i.e., for , , subsets of such that , then the involution is unique as .

Theorem II.15.

If be an event-state-operation structure, then is a subsemigroup of . Further,

  1. and ,

  2. if , then , i.e., is a projection and the range of ,

  3. is the unique mapping such that

    1. is an involution on the semigroup ,

    2. for all , and

    3. if , then the following properties are equivalent: (i) , (ii) , (iii) , (iv) , (v) .

For proof we refer to [12]. The theorem asserts that is an involution semigroup such that:

  1. For each , is a projection, that is belongs to the set .

  2. is order preserving map of into where means for .

Definition II.16.

If is event state operation structure then the mapping is defined as follows: for , where is the unique element of such that .

Axiom [II.5] ensures the existence of an element as required by the above definition. Uniqueness of the event follows from axiom [I.1]. Axioms [II.4] and [II.5] were included to ensure that the involution and orthocomplementation operations can be defined on the set of operations. These operations are needed in order to construct a specific kind of semigroup, the Baer-semigroup, on the set of operations. This additional structure helps us find equivalence between compatibility of events and the commutativity of their corresponding operations.

Definition II.17.

A Baer-semigroup is an involution semigroup with a zero and a mapping such that if then . If is Baer-semigroup, then an element of is called as closed projection.

Theorem II.18.

Let be a Baer-semigroup.

  1. .

  2. If , then .

  3. is an orthomodular lattice where is the relation on restricted to and is the restriction of to . If , then .

  4. If then the following are equivalent:(i) there exists such that , , , and (ii) . If then .

For the proof of above theorem we refer to [7]. From the axioms associated with operations, we conclude that is a Baer-semigroup. Let . From the above theorem it follows that, is an orthomodular ortholattice.

Commutative Baer-semigroup for Example 4.2: Let denote the set of all maps from to . Let . Since the axioms associated with involution and orthocomplmentation are satisfied, forms Baer-semigroup. Since the set theoretic intersection operation () is commutative the composition operation is commutative, i.e, . Thus is a commutative Baer-semigroup.

Noncommutative Baer-semigroup for Example 4.3: First we note is a semigroup. The usual operator adjoint, is an involution for . Let . It is clear that is an involutive semigroup. For , the orthocomplementation of is the projection corresponding to the unique event satisfying axiom [II.5]. Hence