Alternatively, the CPD can be represented as a tree [3], where the interior vertices represent splits on the value of some parent of R.
A, and the leaves contain distributions over the values of R. In this representation, we find the conditional distribution over R.
There are a number of other possible compact functional forms for the distribution such as noisy-or and noisy-max. In this section, we define two distinct semantics for the models. One, which we call the possible worlds approach, is based on defining a distribution over possible database instances.
We will see that this approach is useful when we are attempting to generalize from observed data. This is useful for example, if we are trying to infer values for missing attributes in an incomplete database. The other semantics, which we call the domain frequency approach, is based on the defining a distribution over a randomly chosen tuple or a collection of randomly chosen tuples. We will see that this approach is useful when we are attempting to represent a compact statistical summary of a particular database instance.
The PRM syntax specifies a template for a probability distribution over a database. The template includes the relational component, that describes the relational schema for our 1 Most often we assume that the attribute domains are discrete and finite; continuous domains can also be supported in which case we must specify conditional density functions.
The possible worlds semantics for PRMs, which we will denote as PRM pw , provides a coherent formal se- mantics in terms of probability distributions over sets of relational logic interpretations.
Given a set of ground objects, a PRMpw specifies a probability distribution over a set of interpretations involving these objects. A probabilistic schema, together with a partial database describing tuples and relations, defines a probability dis- tribution over the unspecified attributes of the tuples.
As such, it can be used effectively for representing degree of belief, which for example can be used when reasoning over missing values is required. We will refer to a database instance D with no missing or unknown values as a complete instantiation of a schema.
Each of these complete instantiations is considered a possible world, and a PRM pw defines a probability distribution over database instances. In order to make this probability space well-defined, we need to constrain the space in some manner. Several different ways of specifying the probability space have been studied, with varying representational power. Attribute uncertainty is the simplest way of defining the probability space.
Intuitively, we assume that the set of objects and the relations between them are fixed, i. We specify the tuples and relations using a relational skeleton. Then, the PRM pw defines a probability distribution over the assignments to the attributes of the objects in the model. Definition 4: A relational skeleton D s of a relational schema is a partial specification of a database instance.
It specifies the value for the primary key R. A PRMpw defines a distribution over the possible worlds consistent with the relational skeleton. The re- lational skeleton implicitly defines the random variables in our domain; we have a random variable for each attribute of each object in the skeleton. A PRM pw then specifies a probability distribution over completions D of the skeleton. The probabilistic schema defines the qualitative dependency structure, S.
As we saw, the dependency struc- ture associates with each attribute R. A a set of parents Pa R. These correspond to formal parents; for different objects, the relational skeleton D s determines the actual parents. In other words, the formal parents are defined by the probabilistic schema, however the relational skeleton provides the interpretation.
The interpreta- tion tells us which tuples join; this in turn tells us which attributes depend on each other. We will typically use capital letters to denote the generic attributes, R.
A, and use lowercase letters to denote tuple variables, so that r. A refers to the attribute of a specific tuple. A depends probabilistically on its parents according to the probabilistic schema and the relational skeleton, PaDs r.
A , as specified by the probabilistic schema. A PaDs r. There are three primary differences. First, our random variables correspond to the attributes of the tuples defined by the relational skeleton.
Thus, a different relational skeleton will result in a distribution defined over a different set of random variables. Second, the set of parents of a random variable can vary according to the relational context of the object — the set of objects to which it is related.
These are also determined by the relational skeleton. Third, the parameters are shared; the parameters of the local probability models for attributes of objects in the same class are identical. As in any definition of this type, we must take care that the resulting function from instances to numbers defines a coherent probability distribution, i. In Bayesian networks, where the joint probability is also a product of CPDs, this requirement is satisfied if the dependency graph is acyclic: a variable is not an ancestor of itself.
A similar condition is sufficient to ensure coherence in PRM pw s as well. Additional details are given in [4]. In the discussion so far, we have assumed that the relational skeleton is external to the probabilistic model. The PRMpw framework can be extended to accommodate uncertainty about the structural relationships between objects as well as about their properties [5].
One approach to structural uncertainty we have developed is called existence uncertainty. Existence uncertainty is a simple approach to modeling the probability that a relationship exists between any two entities. We add a Boolean attribute, the Exists attribute, and build a probabilistic model for it, just as any other attribute in our domain.
We can model the probability that a person purchases a particular item. This can depend on both properties of the person and properties of the item. The approach is described more fully in [5]. We have also investigated the representation and learning of PRMs with class hierarchies. A discussion of these issues can be found in [4]. These semantics describe a statistical model of a particular database instantiation. The model captures the tuple frequencies in the database, and in particular it captures the frequencies with which tuples join.
We will refer to this flavor of probabilistic relational models as a PRM df. These semantics are useful for describing a compact statistical model of a database. This compact model can then be used to efficiently answer question about the expected number of tuples that will satisfy a query.
What makes PRM df s unusual is they model correlations across tuples and they can flexibly answer a wide collection of queries. This has many possible uses, including for use by a query optimizer in choosing the appropriate query plan, and for use in computing approximate aggregate query answers. These semantics were first introduced in [8] and their utility for selectivity estimation was shown. Here, we give an introduction to the semantics, but for full details see [4]. As a simple illustration of the domain frequency semantics that we hope to capture, consider two tables R and S such that R has a foreign key, R.
F , that points to S. We define a joint probability space over R and S using an imaginary sampling process that randomly samples a tuple r from R and independently samples a tuple s from S.
The two tuples may or may not join with each other. We introduce a new join indicator variable to model this event. This variable, J F , is binary valued; it is true when r. K and false otherwise. Now, consider any query Q over R and S of the form: r.
K where we abbreviate a multidimensional select using vector notation. Generalizing from this example to more general select-join queries, let Q be a query over tuple variables r1 ,. We can write Q as follows:. We introduce an indicator variable I Q indicating when the equalities in Q hold. The probability that the query is satisfied is:. While the expression in Definition 6 is computable for any select-join query, we will find it more useful if we can define a unique distribution induced by our database.
To achieve this, we restrict our attention to a finite, acyclic collection of foreign-key joins. F with V R. Now we can define a new relation U, which is the universal foreign-key closure of a database D with respect to a table stratification. This relation is never actually materialized, we merely use it as a tool in defining a unique distribution induced by our database. Intuitively, the query that we construct introduces a tuple variable for each table in our schema, and has a unique tuple variable for each foreign-key in the schema.
The universal foreign-key closure of D is defined by the query U we construct below. T U will be the set of tuples variables in our query. Each tuple variable is initially marked unprocessed. We will construct the full set of tuple variables in the query as follows. F refers to S, add a new unique tuple variable s to T U. This tuple variable is marked unprocessed. We say that s is the tuple variable associated with r. We also add the join r.
U is simply a query over the cross product of the relations with a new copy of the relation introduced for each tuple variable that we add. Given this query U, we can define the probability distribution P U. It is the distribution induced by the occurrence frequency of different value combinations for the attributes of the tuple variables in U and join events among the tuple variables. PRMdf s allow us to compactly model PU. A probabilistic schema PS describes a set of independence assumptions and the local distributions of attributes given their parents.
This is made more precise in [4]. Further, PRMdf s allow us to efficiently answer certain frequency queries by constructing a small query- specific Bayesian network that can be used to compute the desired frequencies.
We begin by restricting attention to a form of select-join queries over multiple tables we call inverted-tree foreign-key-join queries. These queries are over a subset of the tuple variables of the universal foreign-key closure. Intuitively, they are over an upwardly closed fragment of the forest defined via the foreign-key joins in U, and may themselves form a forest. We refer to these as legal queries.
Let Q be a legal query. V ars Q is a set of random variables which includes, for each attribute r. A, and for each r. K, it has a random variable r. For every node r. V introduced, r. V has parents Pa r. V defined by S and the CPD of r. It turns out that although the semantics for the models are very different, the learning algorithms are largely the same and are closely related to the work in learning Bayesian networks [10].
The input to the construction algorithm is the relational schema, including the possible foreign-key joins between tuples; and a database instance. We can set this problem up as an optimization problem. Hypothesis Space. A hypothesis PS specifies a set of parents for each attribute R.
We must restrict attention to probabilistic schemas which will generate a consistent probability model for any skeleton or query we are likely to see. We can do this by constructing a dependency graph for the candidate structure and ensuring that the class graph is acyclic. We maintain the graph during learning, and consider only models whose dependency structure passes the appropriate test; see [4] for more details.
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