Phantom Probability

Phantom Probability

Yehuda Izhakian Faculty of Management, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel  and  Zur Izhakian Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel CNRS et Universit´e Denis Diderot (Paris 7), 175, rue du Chevaleret 75013 Paris, France
October 2008

Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the introduction of a new probability measure, enabling varying probabilities that are recorded by ring elements to be assigned to events; this measure still provides a Bayesian model, resembling the classical probability model.

By introducing two principles for the possible variation of a probability (also known as uncertainty, ambiguity, or imprecise probability), together with the “correct” algebraic structure allowing the framing of these principles, we present the foundations for the theory of phantom probability, generalizing classical probability theory in a natural way. This generalization preserves many of the well-known properties, as well as familiar distribution functions, of classical probability theory: moments, covariance, moment generating functions, the law of large numbers, and the central limit theorem are just a few of the instances demonstrating the concept of phantom probability theory.

Key words and phrases:
Weak ordered ring, Imprecise probability, Phantom probability measure, Phantom probability space, Random variables, Distribution and mass functions, Moments, Variance, Covariance, Moment generating function, Central limit theorem, Laws of large numbers.
2000 Mathematics Subject Classification:
Primary: 60A, 60B05, 06F25 ; Secondary: 13B
The second author acknowledges the support of the Chateaubriand scientific post-doctorate fellowship, Ministry of Science, French, 2007-2008.


Over the years much effort has been invested in trying human beings have tried to understand aspects of probability in which the evaluations of occurrences, as well as their likelihoods of happening, are uncertain. Although the terminology for this type of phenomena is varied (uncertainty for physicists, ambiguity for economists, imprecise probability for mathematicians, and phantom for us), fundamentally, the absence of theory enabling the formulation of such phenomena is a common problem for many fields of study. In this paper we introduce a new approach, supported by a novel probability measure, allowing a natural mathematical framing of this type of problems.

Two main principles underlie our approach to treating probability measures associated with varied evaluations:

  • For each event, the sum of its probability and its possible distortion lies in the real interval ;

  • The overall distortions always sum up to .

Having the right algebraic structure, termed here the ring of phantom numbers that naturally records probabilities and their oriented variations, these principles lead to the introduction of our new phantom probability measure, on which much of the theory of classical probability can be generalized. This generalization captures both the uncertainty of outcomes and ambiguous likelihoods, and it is still Bayesian.

The ring of phantom numbers consists of elements of the form , each of which is a compound of the real term  and the phantom term  (notated, like the complex numbers, by instead of ), and whose operations, addition and multiplication respectively, are

This arithmetic makes suitable for the purpose of carrying a theory of probability. In many ways this ring resembles the field of complex numbers, but its arithmetic is different; here is idempotent, i.e. , while for the complexes. Similar structures, though sometimes using different terminology, have been studied in the literature, mainly from the abstract point of view of algebra; the innovation of this paper is the utilization in probability theory, which requires some special setting like phantom conjugate, reduced elements, absolute value, and norm. With these notions suitably defined, the way toward the development of a phantom probability theory is prepared.

One of the main advantages of phantom functions , mainly polynomial-like functions, is that they can be rewritten as

where and are real functions . We call this property, which plays a main role in our exposition, the realization property of phantoms functions.

With this realization property satisfied, most of the phantom calculations are reduced simply to the real familiar calculations. Moreover, for , the real term  of is the only argument involved in ; this shows that when is a pantomization of a real function , the real component  of is just . Surprisingly, the pantomizations of all classical probability functions (moments, variances, covariances, etc.) admit the realization property.

Using the phantom ring structure, together with our measure principles, we keep track of the evolution of the classical theory of probability. The leading motif throughout our exposition is that restricting the theory to the real terms of all the arguments involved always leaves ones with the well-known classical theory. Given this foundation, as well as the appropriate definitions, the probability insights are much clearer and their proofs become more transparent.

The main topics covered by this paper include:

  • Conditional probability, independence, and Bayes’ rule;

  • Random variables (discrete, continuous, and multiple);

  • Attributes of random variables: moments, variances, covariances, moment generating functions;

  • Inequalities (appropriately defined);

  • Limit theorems.

Along our exposition we also provide many examples demonstrating how classical results naturally carry over to the phantom framework. further results and applications will be appear in our future papers.

The fact that the phantom probability space provides a Bayesian probability model paves the way for developing a theory of phantom stochastic processes and phantom Markov chains [9] with a view towards applications in dynamical systems.

We use the notion of imprecise probability as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities [16]. The known results of past efforts to find a theory that frames imprecise probability give only partial or complicated answers. For example, fuzzy probability [17] only treats uncertain outcomes but not varying probabilities; conversely, complex probability provides a partial answer for deformed probabilities but only for fixed outcomes [1, 18]. On the other hand, the operator measure theory [14] is very complicated and not intuitive, while the min-max model [7] is not Bayesian and [13] sometimes becomes non-additive.

These probability theories have a tremendous range of applications, like quantum mechanics, statistics, stochastic processes, dynamical systems, game theory, economics, mathematical finance, or decision making theory; to name just a few. Our development, together with the attendant examples, which smoothly extend the known theories that have already proven to be significant, lead one to believe that phantom theory could contribute to these applications, and make for a better understanding of phenomenons that arise in the real life.

1. The phantom ground ring

1.1. Ground ring structure

The central idea of our new approach is a generalization of the field of real numbers to a ring structure whose binary operations are induced by the familiar addition and multiplication of . Focusing on application to probability theory, to make our exposition clearer, we give the explicit description for the certain extension of of order , which is suitable enough for the scope of this paper. For the sake of completeness, and in an effort to attract audiences from various fields of study, we recall some of the standard algebraic definitions (see [15]) and present the full proofs related to the basics of the algebraic structure for the extension of order . The more general phantom framework is outlined in the next subsection.

Set theoretically, our ground ring , called a ring with phantoms or phantom ring, for short, is the Cartesian product ; for simplicity, we write for a pair . We say that is a phantom ring of order over the reals, and denote it as , for short. (The general case of order is spelled out later in Subsection 1.2.) The elements of are called phantom numbers, usually denoted .

In what follows we use the generic notation that for reals and write for a phantom number ; we call the real term of while is termed the phantom term of . We use the notation

for the real term and the phantom term of , respectively. (The reason for calling the second argument “phantom” arises from the meaning assigned to this value in the extension of the probability measure, as explained in Section 2.)

The set is then equipped with the two binary operations, addition and multiplication, respectively,

to establish the phantom ring (to be proved next), , with unit and zero . We write for , for , and for .


In general, similar algebraic structures (with different terminologies) are known in the literature, mainly for graded algebras or -algebras in semiring theory, usually applied to a tensor , where is a module over semiring , c.f., [10]. However, as will be seen immediately, in this paper we push the algebraic theory much further for the special case where is a field; then, in this case, has a much richer structure. Moreover, some of our definitions are unique with the aim of serving applications in probability and measure theory.

Although the multiplication of is somehow reminiscent of the multiplication of the complex numbers , it is different: for the phantoms is multiplicative idempotent, while for the complexes is not idempotent.


is an Abelian semigroup.


Given , where , we have

which proves associativity. Commutativity is obtained by

This shows that is a (multiplicative) Abelian semigroup. ∎


is a commutative ring.


Since is defined coordinate-wise, and is an (additive) commutative group, it is clear that is also a commutative group. The unique additive inverse of is

The pair is a (multiplicative) Abelian semigroup, by Proposition 1.1, so we need to prove the distributivity of over :

All together we have proved that has the structure of a commutative ring. ∎

Note that is not a group, and thus is not a field, since there are non-zero numbers without an inverse; for example .

Recalling that a nonzero ring element is a zero divisor if there exists a nonzero element such that , one observes that the phantom ring is not an integral domain, i.e. it has zero divisors; for example

and thus and are zero divisors.


All the zero divisors of are of the form


for some .


Assume , and are nonzero elements such that , that is

Suppose , then by the real term of the product . So, by the phantom term, we should have . But , since , and thus . This means that and m as required. ∎

A nonzero element which is not of the form (1.1) is called a nonzero divisor; the collection of all zero divisors in is denoted

We sometimes write for the union .


The phantom conjugate of is defined to be

The real number

is called the (real) reduction of .

Having the notion of (real) reduction, we can write the product of two phantom numbers as :


By Proposition 1.1, one sees that in is a zero divisor iff or ; when both of them are zero then . Moreover, in this view, given a suitable topology on , the complement of in is dense, so we can omit the zero divisor without detracting form the abstract theory.

One can easily verify the following properties for phantom conjugates and (real) reductions: {properties} For any the following properties are satisfied:

  1. ,

  2. ,

  3. ,

  4. ,

  5. ,

  6. ,

  7. ,

  8. ,

  9. ,


The elements of can be understood as intervals in . This means that is an element stands for the interval that starts at and ends at , i.e. the reduction of . Thus, can be realized as a ring of intervals, given by:


In order to get a canonical interval representation, is assigned to the half-open interval .

In this view, zero divisors are intervals with as one of their endpoints. This view also provides the motivation for the definition of the conjugate: and represent the same interval but with switched endpoints.

In fact, has a much richer structure than a standard ring; the division is well defined for all nonzero divisors in , and each has an inverse. Given a nonzero divisor , we define the multiplicative inverse of to be


indeed is an inverse of ,

One can easily verify that is unique, and that the reduction of an inverse number has the form:

Since the multiplicative inverse, defined only for all , and the additive inverse are unique, we define the division and the substraction, respectively, for as

where is defined only for a nonzero divisor . Accordingly, we write


which leads to the following useful form:


A phantom number is said to be positive if and . When and we say that is pseudo positive. If and , then is said to be negative and when and we say that is pseudo negative. When is pseudo positive or it is termed pseudo nonnegative, and if is pseudo negative or is called pseudo nonpositive.

Clearly, any positive (negative) phantom number is also pseudo positive (negative). In particular, if is pseudo positive, or pseudo negative, then , cf. Remark 1.1, and is multiplicatively invertible.


Given two pseudo nonnegatives then:

  1. Their sum is pseudo nonnegative,

  2. Their product is pseudo nonnegative,

  3. is pseudo nonnegative for each ,

  4. When is pseudo positive, the fraction is pseudo nonnegative.


  1. Write , since is positive, and and , the proof is clear.

  2. By Equation (1.2) . Then, by the hypothesis, and .

  3. Use (ii) with , or write directly , so and then .

  4. Writing as in Equation (1.6), and are (real) positives, and thus

Next, we outline the view of our structure in the category of rings. Categorically, we have the trivial embedding

given by sending . On the other hand, we also have the onto projection

given by sending for some real numbers and . If , the projection is phantom forgetful, i.e. , while is real forgetful when .


Viewing as an -module, we define the scalar multiplication as

for any , which is written as , for simplicity. Similarly, we write for .

It is easy to check that the set of real numbers forms a subfield in the ring of phantom numbers , and the phantom numbers whose real term is zero establish an ideal in .

1.2. Generalization

In the previous subsection we described the extension of order of the field of real numbers. For completeness, we present the general definition of a phantom ring of arbitrary order.

Given a field of characteristic , usually the field of real numbers, the phantom ring of order , or -phantom ring, for short, is built over the product of copies of indexed . Accordingly, the elements of are just -tuples and denoted, respectively, as and . is then equipped with the following binary operations, addition and multiplication, respectively:


where and .

(Note that the notation here is different from that used in the previous subsection, in particular the and the , stand for the phantom terms for the respective level.)

Numbering the copies of sequentially, the first copy is considered as the real part of while , , is said to be the phantom of level of . Note that is just , which is a subfield of .

Having the operations rigorously defined for any , using the arithmetic defined in (1.7), we can push to infinity and also define the -phantom ring .

In the sequel, for simplicity, we apply our development only to , which as we have said is denoted , though we note that extends smoothly to any , with , defined over a suitable field . Generalizing the future definitions suitably to , the -phantom ring carres also the same properties as , to be described in the next sections.


Notations: For the rest of this paper, assuming that the reader is familiar with the arithmetical nuances, we write for , for , for the product , for the division , and for repeated times. The phantom ring is denoted , for short.

1.3. Relations and orders

In the sequel, mainly for the development of phantom probability theory, we need some relations that help to utilize the structure of .

To make our paper reasonably self-contained, let us recall the property of a binary relation on a set for being an order:


A binary relation is a weak order on a set if the following properties hold:

  1. Reflexivity: for all ;

  2. Transitivity: and implies ;

  3. Comparability (trichotomy law): for any , either or .

(When and we write .) A weakly ordered set is a pair where is a set and is a weak order on . When consists of phantom numbers we say that is a phantom weakly ordered set.

Adding the extra axiom:

  1. Antisymmetry: and implies ;

the order is then a total order, or order, for short, and is denoted as . Clearly, induces an equivalent relation on , the classes of which are , and when is a total order is replaced by full equality . We use the notation to distinguish this order, mainly when writing , from the other orders used in the sequel. Therefore, the symbol and always denote the usual order of the real numbers.

Although, in general, many relations may serve as weak order on , in this paper we require the weak order to have the following properties: {properties}

  1. Compatibility with the standard order of the reals, that is

    for any and .

  2. Compatibility with the arithmetic operations of :

    1. if then , for any ;

    2. if then , for any pseudo positive ;

    3. if then , for any pseudo positive .

Viewing as an Euclidian space, we usually assume that all the elements that are form a connectable set.


Viewing as , our main example for a total order on is the lexicographic order defined as


which is a total order satisfying the above conditions.

In the continuation, when writing , we assume the weak order is provided with the set structure. The reader should keep in mind that one interpretation for an order which is also total is the lexicographic order .


Note that our definition of pseudo positivity, cf. Definition 1.1, is independent of the given order on .

In the sequel, mainly for probability theory, we also use the notation for the (real) relation


the other real relations , , , and are defined similarly.

We define the real-valued function , with a real positive parameter , given by


and write for the image of in . Then, determines the equivalence relation on given by

and written . (Note that in the special case when , by this definition, we always have .) The quotient ring of , taken with respect to , is denoted as ; clearly .

In the same way, induces a weak order on , provided as


and satisfying Properties 1.3; the relations , , and are determined similarly.

These relations are very important for advanced topics in phantom probability theory and their applications, mainly discussed in the sequel papers [8, 9].

1.4. Powers and exponents

Writing , with , for the product with repeated times, for any we have

as usual, is identified with the unit . This form leads to the following friendly formula:


Following accepted standards, we write for , and therefore get the extension to integral powers of phantom numbers.

Equation (1.12) plays a main role throughout our development and, together with Equation (1.2), leads to the next important formula, which is used frequently in the sequel:


(To verify this equality, combine Equation (1.12) and Equation 1.2.)


Given a phantom number , then:

  1. ,

  2. ,

  3. ^{j}=^{i}=z^{ij}\),

for any .

Of course, one can take an arbitrary finite number of multiplicands, , and get recursively


for any . {definition} When a phantom equation can be written in terms of two real equations, and , as

we say that has a realization form, or equivalently, that it admits the realization property. For that matter an equation might be an arithmetic expression or a function, where and stand respectively for the real and the reduction of each argument involved in .

For example, Equations (1.12), (1.13), and (1.14) above admit the realization property. In the sequel, we will see that many other familiar equations admit this nice property. Having this property, as spelled out later for probability theory, phantom results are induced by known results for reals, which makes the development much easier.

Since the realization property us satisfied for each and any natural power , cf. Equation (1.12), it easy is to determine the n’th root, if it exists, of a phantom number as:


where and are, respectively, the real ’th roots of and , and is a real positive number. Clearly, when is even, both and must be nonnegative.

In the usual way, we sometimes write for , and have the properties: {properties} Given pseudo nonnegative phantom numbers , , and then:

  1. ,

  2. , for pseudo positive ,

  3. ^{m}=^{m}=z^{\frac{m}{n}}\),

for any positive .

In the specific case when , clearly, each pseudo nonnegative phantom number has a square root


Actually, in the equation stands for ; therefore there always exists a nonnegative square root of , i.e. a root whose real and phantom terms are both nonnegative.

In the standard way, we define the exponent of an element to be the infinite phantom sum


Given , and in then:

  1. = 1,

  2. \),

  3. ,

  4. .


(1) is by definition. (2) Expand and use Equation (1.12), i.e.

(3) Using the identity in (2), write