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[EQUATION] The left-hand side is [EQUATION] Hence the assertion holds. We show ( 3.5 ) for [MATH] First assume [MATH] A direct computation shows |
[EQUATION] By swapping [MATH] and [MATH] , and summing up, we obtain [EQUATION] These are nothing but the desired relations. Next consider the other cases. For the case [MATH] , we have |
[EQUATION] For the case [MATH] [MATH] , we have [EQUATION] We need the condition [MATH] for the case [MATH] [MATH] 3.2 Cyclic automorphism |
We consider algebra automorphisms corresponding to the rotation of the Dynkin diagram. It is easy to see that the assignment [EQUATION] |
gives an algebra automorphism [MATH] of [MATH] Guay introduced an analogous automorphism for the affine Yangian. Proposition 3.4 |
G1 , Lemma 3.5) The assignment [EQUATION] gives an algebra automorphism [MATH] of [MATH] Lemma 3.5 We have [MATH] Proof. See K3 , Lemma 4.1] |
Proposition 3.6 We have [MATH] Proof. The identity obviously holds for [MATH] [MATH] Hence it is enough to show that the identity also holds for [MATH] since [MATH] is [MATH] [MATH] [MATH] [MATH] ). We show [MATH] for [MATH] The proof for the case [MATH] is similar. The left-hand side is |
[EQUATION] We have [EQUATION] and [EQUATION] This verifies that [MATH] is equal to [MATH] under the assumption [MATH] 3.3 Another version |
Guay’s evaluation map [MATH] has [MATH] as its target space and hence it matches with lowest weight modules of [MATH] To deal with highest weight modules, we introduce an opposite evaluation map by using the anti-isomorphisms [MATH] and [MATH] Moreover we make it one-parameter family of algebra homomorphisms by using [... |
Definition 3.7 For each [MATH] , define [MATH] Here [MATH] and [MATH] are taken as those for [MATH] Theorem 3.8 Assume [MATH] and let [MATH] be a complex number. Then there exists an algebra homomorphism [MATH] uniquely determined by |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] Evaluation modules In this section, the symbol [MATH] denotes [MATH] introduced in Section 2.2 |
Let [MATH] be the integrable irreducible highest weight module of [MATH] with highest weight [MATH] The dominant integral weight [MATH] is determined by the data [MATH] and [MATH] satisfying [MATH] [MATH] ) and [MATH] The correspondence is given by |
[EQUATION] We denote a fixed highest weight vector of [MATH] by [MATH] Assume [MATH] We define a [MATH] -module [MATH] by the pull-back of [MATH] via [MATH] By abuse of notation, we regard [MATH] as a vector of [MATH] The vector [MATH] satisfies the condition of the highest weight vector for affine Yangian. More precis... |
Theorem 4.1 We have [EQUATION] for all [MATH] and [MATH] , where [EQUATION] The goal of the remaining part is to prove this theorem. |
Remark 4.2 In K3 , the author proves that the image of Guay’s evaluation map [MATH] in [MATH] contains [MATH] under the assumption [MATH] This result implies that the image of [MATH] in [MATH] contains [MATH] under the assumption [MATH] Hence the [MATH] -module [MATH] is irreducible when [MATH] We do not use this fact ... |
We provide a general lemma. Lemma 4.3 Let [MATH] be a nonzero element of a [MATH] -module satisfying [EQUATION] for some [MATH] Further assume that [MATH] satisfies |
[EQUATION] [EQUATION] for some [MATH] Then [MATH] satisfies [EQUATION] for all [MATH] , where [MATH] [MATH] is given by [EQUATION] |
Proof. We prove [MATH] by induction on [MATH] The assertion for [MATH] holds by the assumption ( 4.1 ). Assume it for [MATH] Then we have |
[EQUATION] by the induction assumption, and it is equal to [MATH] by the assumptions ( 4.1 ), ( 4.2 ) and again by the induction assumption. |
We prove that [MATH] is an eigenvector of [MATH] with the eigenvalue ( 4.4 ) by induction on [MATH] The assertion for [MATH] holds by the assumption ( 4.2 ). Assume it for [MATH] We have |
[EQUATION] Here we use [MATH] in the second equality. Then we have [EQUATION] and [EQUATION] Hence we conclude [MATH] , which completes the proof. |
The vector [MATH] satisfies the condition ( 4.1 ). In order to apply Lemma 4.3 to our situation, we compute [MATH] and [MATH] Lemma 4.4 |
We have [MATH] for all [MATH] Proof. We give a proof for [MATH] The case [MATH] is similar. Recall the formula for [MATH] and note that all the terms in [MATH] annihilate [MATH] except for those concerning |
[EQUATION] [EQUATION] [EQUATION] Then we see that [MATH] is an eigenvector of [MATH] with the eigenvalue [EQUATION] Lemma 4.5 We have [MATH] for all [MATH] |
Proof. We give a proof for [MATH] The case [MATH] is similar. Recall the formula for [MATH] Then we have [EQUATION] Proof of Theorem 4.1 |
By Lemma 4.3 4.4 4.5 , we see that [MATH] and [MATH] is an eigenvector of [MATH] with the eigenvalue [EQUATION] Let us consider the generating series |
[EQUATION] of the eigenvalues of [MATH] on [MATH] It is equal to [EQUATION] where we put [MATH] The collection [MATH] is an analog of the Drinfeld polynomials of finite-dimensional irreducible modules for the Yangian associated with a simple Lie algebra. |
In K1 K2 , the author constructed the level-one Fock representation [MATH] of [MATH] The polynomials ( [MATH] ) corresponding to its highest weight is |
[EQUATION] by K1 , Theorem 5.7] (the highest weight vector is denoted by [MATH] ). We see that the level-one Fock representation at [MATH] with [MATH] is an example of evaluation modules. Indeed, take [MATH] as the level-one dominant integral weight corresponding to [MATH] and put [MATH] Then the highest weights of [MA... |
# Source: arxiv 1806.09997 # Title: Probabilistic Inference Using Generators - The Statues Algorithm # Sections: all # Downloaded: 2026-03-03T02:19:50.017765+00:00 |
Probabilistic Inference Using Generators The Statues Algorithm version 2 – Abstract We present here a new probabilistic inference algorithm that gives exact results in the domain of discrete probability distributions. This algorithm, named the Statues algorithm, calculates the marginal probability distribution on proba... |
After introduction of several definitions, primitives and compositional rules, we present in details the Statues algorithm. Then, we briefly discuss the interest of this algorithm compared to others and we present possible extensions. Finally, we introduce Lea and MicroLea, two Python libraries implementing the Statues... |
Keywords: probabilistic inference, probabilistic arithmetic, discrete probability distribution, probabilistic model, Bayesian network, marginalization, generator |
Introduction Probabilistic inference is a field that is receiving renewed attention with latest developments in AI and probabilistic programming. Nowadays, problems characterized by some uncertainty can be modeled using different approaches, formalisms, primitives and levels of expressiveness: joint probability distrib... |
In order to perform actual problem resolution, each modeling approach has its own catalogue of algorithms, characterized by different merits and trade-offs – see Russell & Norvig, ( 2003 or De Raedt & Kimmig, ( 2013 for an overview. Several algorithms produce exact results but may be limited practically by complexity b... |
These well-established algorithms, in their original formulations, are specialized for one single modeling approach. In particular, algorithms for probabilistic arithmetic do not handle Bayes reasoning or even simple conditioning; on the other hand, above-cited inference algorithms for BN do not handle arithmetic (e.g.... |
Following this trend, the present paper introduces a new algorithm for the task of exact marginalization inference in the scope of discrete random variables. It is in essence a variant of the enumeration algorithm that provides important improvements for reducing both the number of paths to explore and the number of ca... |
The paper is organized as follows. Section introduces a probabilistic modeling framework using discrete random variables; this framework defines a set of primitives to build up probabilistic models; some examples provide substantiations that these primitives are rich enough to express (in particular) joint probability ... |
Probabilistic modeling framework We shall model randomness using discrete random variables with a finite domain. We do not put any restriction on the domains provided that these are discrete and finite: these can be numbers, matrices, symbols, booleans, tuples, functions, propositions, etc . Also, it is not required to... |
In the following subsections, we shall introduce the different types of RVs, as primitives to build up probabilistic models. We shall distinguish elementary RVs, which are defined on their own, and derived RVs, which are defined in terms of other RVs. |
2.1 Elementary random variables Definition 2.1 An elementary random variable is a random variable with a given finite domain and characterized by a given prior probability mass function. |
Elementary RVs are the most basic RVs. They require specifying prior probability for each possible value of their domains. Note that, since we constrain the domain of RV to be discrete and finite, an elementary RV can be called also a categorical distribution . The probability mass function (pmf) shall obey Kolmogorov ... |
An example of elementary RV is the result obtained by flipping a fair coin. We can model this by defining RV [MATH] with dom [MATH] and a uniform pmf defined using the following notation – borrowed from Williamson, ( 1990 |
[EQUATION] Another example is the value [MATH] got after throwing a fair die: [EQUATION] For the sake of simplicity, we assume here that all pmf are defined by extension (as above), even if other formulation could be handled without much difficulties. Also, we shall forbid values with a null probability and duplicate v... |
It is important to avoid confusion between a RV and the pmf that characterizes it. In the following, a letter with a hat (e.g. [MATH] ) refers to a given pmf; by prepending a diamond on a given pmf, we designate an elementary RV characterized by this pmf. So, [MATH] is a RV having [MATH] as pmf or, for short, |
[EQUATION] Thanks to this formalism, the elementary RV [MATH] seen above could equivalently be defined as [EQUATION] As a special case, we admit any elementary RV having a domain of one unique element; such RV is then certain and has a probability of 1. For instance, the usual [MATH] number can be represented by the RV... |
Let us stress that any two distinct elementary random variables are independent by definition. Each occurrence of the diamond notation creates a brand new independent RV, even if applied on the same pmf. For example, let us consider the pmf [MATH] and the two boolean RVs defined as [MATH] and [MATH] . Then, [MATH] and ... |
Since we admit any domain for our elementary RVs, two special cases are worth mentioning: boolean RVs and joint probability distributions. |
2.1.1 Boolean random variables Definition 2.2 A RV [MATH] is defined as boolean iff the domain of [MATH] contains no other values than booleans [MATH] |
For convenience, we shall use the notation [MATH] to represent a pmf with a given probability [MATH] to be true: [EQUATION] For instance, the elementary RV defining that a fair die shows the value 4 can be notated [MATH] . Note that, as trivial special cases, we can write the relations [MATH] and [MATH] . The classical... |
[EQUATION] 2.1.2 Joint probability distributions The concept of elementary RV allows defining joint probability distributions (also known as multivariate distributions). The way to proceed consists in defining a RV with a set of tuples for domain; each tuple represents a possible outcome and each element of the tuple r... |
[EQUATION] Joint probability distributions allow modeling interdependence between random phenomena. This is the case in the example above since, in particular, the joint probability [MATH] is not equal to the product of marginal probabilities [MATH] . Such construction is of course not suited when the number of outcome... |
2.2 Derived random variables Beside elementary RV, a random variable may also be defined in terms of other random variables. Definition 2.3 |
derived random variable is a random variable that is defined by a given deterministic dependency on a given finite set of random variables. |
The basic idea of this recursive definition is that elementary RV can be used to define derived RV that, in turn, can be used to define other derived RV and so on up to elementary RVs. Of course, the terms ”deterministic dependency” used in the definition is vague; the precise definition of these terms shall be elabora... |
We shall define the dependency of a derived RV as belonging to one of the four types tuple functional conditional and table . These are detailed in the following subsections, after introducing the concept of referential consistency. |
2.2.1 Referential consistency Before covering the types of derived RVs, it is important to state a general rule that they shall obey: when defining a derived RV [MATH] depending on a given RV [MATH] , the value randomly chosen for each occurrence of [MATH] shall be the same. To give a simple example, consider an elemen... |
Note that this constraint is closely linked to the concept of stochastic memoization found at least in Church (Goodman et al. ,, 2008 and WebPPL (Goodman & Stuhlmüller,, 2014 . The two concepts actually enforce the same consistency constraint. The difference lies, to the best of our knowledge, in the fact that referent... |
2.2.2 Tuple random variables The first type of derived RV is defined by grouping a given set of RVs into one tuple. Definition 2.4 |
Be [MATH] random variables [MATH] with [MATH] . The RV [MATH] defined as the tuple [EQUATION] is called a tuple RV For instance, we can define a 2-tuple RV [MATH] that is made up of two elementary RV having Bernoulli distributions: |
[EQUATION] The pmf of [MATH] can be calculated by enumeration: [EQUATION] Note that a tuple is defined as a sequence of elements, so the order of these elements is significant. Let us note for instance that swapping the inner RV in the tuple definition results in another distribution: |
[EQUATION] [EQUATION] A rather contrived example is given when a RV appears twice in the same tuple: [EQUATION] Then, the referential consistency forces the two elements to be the same; the pmf is then |
[EQUATION] Note that a tuple RV containing elementary RV (as seen here) is not equivalent to an elementary RV containing tuples (as seen in 2.1.2 , for joint probability distributions). In both cases, the domain is a set of tuples; however, a tuple RV is a derived RV and it cannot be used to specify a joint probability... |
For non-empty tuples, we shall adopt the following LISP-like notation [MATH] to represent a tuple with [MATH] as first element (the ”head”) and [MATH] as a tuple with remaining elements (the ”tail”). So, the tuple RV [MATH] defined above could be written as [MATH] |
2.2.3 Functional random variables The second type of derived RV is defined by application of a function on other RVs. Definition 2.5 |
Be a random variables [MATH] and an unary function [MATH] which domain includes the domain of [MATH] . The RV [MATH] defined as [EQUATION] |
is called a functional RV Let us stress that [MATH] is meant here to be a pure function, that is deterministic and without side-effect: once the value of argument RV is defined ( [MATH] ), the value of [MATH] is uniquely defined ( [MATH] ). A functional RV can use any algorithm, provided that it can evaluate the result... |
Functional RVs cover, among others, a large set of basic mathematical operations (we assume in the following that [MATH] [MATH] [MATH] [MATH] have numerical domains, and [MATH] [MATH] have boolean domains): |
arithmetic: [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] , etc comparison: [MATH] [MATH] [MATH] [MATH] , etc logical: [MATH] [MATH] [MATH] [MATH] [MATH] , etc |
and any combinations of these operations, like [EQUATION] which use standard function composition. Note that, to strictly conform to our definition, the value 3 is here considered as an elementary RV giving 3 with a probability 1; also, the infix subexpressions shall be translated using unary functions as explained abo... |
[EQUATION] Let us come back on the rule of referential consistency ( 2.2.1 ). To exemplify the idea, let us consider the following dummy functional RVs: [MATH] is certainly [MATH] [MATH] is certainly true, [MATH] is certainly false, etc; also, assuming that the RV’s domains are natural numbers, the RV [MATH] defined ab... |
Beside the afore-mentioned common mathematical functions, we could add many other useful functions: checking the membership of an element in a given set, taking the minimum/maximum element of a tuple, summing the elements of a tuple, getting the attribute of an object, etc. Among these functions, the indexing of a give... |
[EQUATION] 2.2.4 Conditional random variables The third type of derived RV is defined by filtering the values of one given RV according to a condition expressed in a given boolean RV. |
Definition 2.6 Be the RV [MATH] and the boolean RV [MATH] . The RV [MATH] defined as [MATH] under the condition [MATH] is noted as |
[EQUATION] and is called a conditional RV The idea is here to build a new RV from an existing one [MATH] , with the assurance that the possible values of [MATH] are such that the given condition [MATH] is true. [MATH] could represent an evidence, an assumption or a constraint; [MATH] has its own prior probability to be... |
changes when evidence [MATH] is provided; actually, [MATH] keeps its definition unchanged whatever it may happen; [MATH] is just a new RV, which is meant to capture some evidence absent from [MATH] ’s definition. |
Although not required, the interesting cases happen of course when [MATH] and [MATH] are dependent each from each other; this occurs if the evidence is a functional RV referring to the conditioned variable – i.e. [MATH] – or, more generally, if the evidence and conditioned variables are both functional RVs referring to... |
For example, let us define [MATH] and [MATH] as the respective values of two fair dice and [MATH] as the sum of these two values: |
[EQUATION] and let us assume that we know, by any means, that first die shows 1 and that the dice total is greater than 5: the conditional RV for the dice total is then written |
[EQUATION] which is characterized by the pmf [MATH] . To make a link with the classical concept of conditional probability, we simply need to transform the above-defined RV so that it becomes a boolean conditional RV; this can be obtained (in particular) by using a functional RV with an equality, e.g. |
[EQUATION] As another example, we could now assume to have evidences on derived RV and query the explaining RVs: the following conditional RV |
[EQUATION] is characterized by the pmf [MATH] . For this instance of causal inference, tuple RVs can bring up explanatory values, by revealing details of each atomic case; for instance, the pmf of the previous RV can easily be understood by tuples giving each die value and their sum: |
[EQUATION] which has pmf [MATH] . Here is a last example, which is more involved: [EQUATION] which has pmf [MATH] . As stated before, we see in all these examples the importance of referential consistency for getting the correct pmf. |
A valid conditional RV [MATH] requires that [MATH] can produce at least one value verifying the condition expressed in [MATH] . This may be violated if [MATH] is unfeasible. For instance, [MATH] is invalid since [MATH] is certainly false (the sum of dice values is strictly greater than any die’s). Let us point out that... |
Many probabilistic algorithms constrain evidence conditions to be observations , which are equalities of the form [MATH] or a conjunction of such equalities [MATH] . Conditional RVs, as defined here, subsume this approach: they cover a far broader class of evidence conditions, for which the usual observations are just ... |
2.2.5 Table random variables The fourth type of derived RV is defined by selecting a random variable in a lookup table based on the value taken by another RV. |
Definition 2.7 Be a RV [MATH] such that dom [MATH] with [MATH] and be [MATH] RV [MATH] . The RV [MATH] depending of [MATH] such that, for any [MATH] |
[EQUATION] is noted [EQUATION] and is called a table RV The order of RVs in the table is irrelevant. The table RVs allow defining conditional probability tables (CPT), which are used in Bayesian networks. Consider for example the well-known example of ”Rain-Sprinkler-Grass” BN. We define three boolean RVs: [MATH] repre... |
[MATH] has a prior probability 0.20; the other probabilities and dependencies are quantified using CPTs: [MATH] ’s probability depends of the weather: if it is raining the probability of [MATH] is 0.01, otherwise it is 0.40; [MATH] depends of both the weather and the grass state; the probabilities for [MATH] depending ... |
[EQUATION] Let us examine some queries we could make on this model using conditional RVs (since there are only boolean RVs, the [MATH] notation can be used): according to the definition of the table RV seen above, [MATH] [MATH] and [MATH] . Of course, the above results are just consistency check of the CPT, bringing no... |
The table RVs allow modeling any CPT. Note that the number of entries shall be exactly equal to the cardinal of the domain of [MATH] . This can be cumbersome if this domain is large, e.g. if the condition is a tuple having many inner RVs (the domain of [MATH] is the cartesian product of these RVs, provided that they ar... |
(Pearl,, 1982 ; Poole & Zhang,, 2011 , redundancies can be avoided. To take an example, let us revisit the model above by assuming now that the probability to find the grass wet ( [MATH] ) given that the sprinkler is on ( [MATH] ) equals 0.95, whatever it rained or not ( [MATH] ). Using the approach above naively, this... |
[EQUATION] By avoiding redundancies in CPT, hence limiting the table size, it is easy to extrapolate the dramatic simplification gained on larger models. In the section dedicated to possible extensions ( ), we shall present an extra type of derived RV, called mixture RV , which offers an alternate way to express a CPT ... |
Another application of the table RV is the modeling of discrete-time Markov chains (DTMC). Consider for example the ”Weather” DTMC represented in the following graph: |
Assuming that the initial state [MATH] at [MATH] is sunny with probability 0.2, [EQUATION] the future states can be modeled by table RVs [MATH] , defined by the following recurrence: |
[EQUATION] Exact marginalization inference So far, we have defined different types of random variables, which could be interdependent. We have seen that the probabilistic primitives and compositional rules allow defining a large set of probabilistic models. The pmf of elementary RVs are known by definition. The pmf of ... |
The examples seen so far demonstrate that the evaluation of the pmf of a given RV can generally not be performed by a simple recursive evaluation, as done for example in usual arithmetic. This constraint includes in particular arithmetic expressions like [MATH] and all non-trivial conditional RV like [MATH] . Actually,... |
3.1 Representing RVs as -expressions The Statues algorithm requires as input a p-expression , a structured object that defines the exact dependencies between random variables up to the elementary ones. For any given RV [MATH] , the associated -expression is noted [MATH] . If [MATH] is a derived RV, then [MATH] shall be... |
type random variable [MATH] -expression [MATH] elementary [MATH] [MATH] tuple [MATH] [MATH] functional [MATH] [MATH] conditional |
[MATH] [MATH] table [MATH] [MATH] For the case of an elementary RV [MATH] [MATH] is meant to designate the pmf of [MATH] ; this definition can be captured by the following relationship, for any pmf [MATH] |
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