Probabilistic EntityRelationship Models, PRMs, and Plate Models

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Probabilistic Entity-Relationship Models, PRMs,
and Plate Models

• David Heckerman, Chris Meek, and Daphne Koller
• Slides from SRL 2004 talk

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History/Motivation

• Began with Plates (stats) PRMs (ML)
• Found it to be important to distinguish between
  entities and relationships
• Discovered the ER model (e.g., Ullman and Widom,
  Ch 2)
• Created probabilistic version of ER model PER
  model
• PER Model is more expressive than Plate Model or
  PRM and helps to show their connections
• PER Model provides a strong link to the db
  community by virtue of being built on top of ER
  Model

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Outline

• Entity-Relationship (ER) Model
• Probabilistic Entity-Relationship (PER) Model
• Connections to plate model, PRM
• Modeling issues

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ER Model

• An abstract representation of data
• The creation of an ER model is often the first
  step in the process of constructing a relational
  database.
• Often constructed before any data has arrived
  (much like we construct models before collecting
  data).

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ER Model -- Example

• A university database maintains records on
  students and their IQs, courses and their
  difficulty, and the courses taken by students and
  the grades they receive.

Entity classes
Attribute classes
Course entities CS107, Stats10, Student
entities John, Mary, Takes relations (John,CS
107), Attributes John.IQ, CS107.Diff
Relationship class
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ER Model generates attributes
ER Model
Skeleton

gt
Attributes
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PER Model -- Example

• Continuing the university database example, a
  student's grade in a course depends both on the
  student's IQ and on the difficulty of the course.

Arc classes
Not shown Local distribution class for grade
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PER Model generates Bayes net
PER Model
Skeleton

gt
Attributes
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Constraints on arc classes
ER Model
Skeleton

gt
Attributes
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More on constraints
A database contains diseases and symptoms for a
given patient. Both diseases and symptoms have
labels from a common set of categories (e.g.,
cardiovascular, neuro, urinary). The possible
causes of a symptom are diseases that have at
least one category in common with that symptom.
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More on constraints
A constraint on the arc class from X.A to Y.B in
a PER model is any first-order expression
involving entities and relationship classes in
the PER model such that the expression is bound
when the tail and head entities are taken to be
constants. To determine whether to draw an arc
from x.A to y.B, we evaluate the first-order
expression using the tail and head entities of
the putative arc. (It must evaluate to true or
false.) We draw the arc from x.A to y.B only if
the expression is true.
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Local distribution classes
E.g., Noisy OR
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Caveat

• Typically, a PER model is not based on the ER
  model of a database

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PER model, plate model, PRM
PER model
Plate model
PRM
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Modeling issues

• Restricted relationships
• Self relationships
• Probabilistic relationships

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Restricted relationship Example
Hierarchical model A binary outcome O is
measured on patients in multiple hospitals. Each
patient is treated in exactly one hospital. It
is believed that outcomes in any given hospital h
are i.i.d. given binomial parameter h.q and that
these binomial parameters are themselves i.i.d.
across hospitals given hyperparameters a.
a
a
Hospital
q
Ç

h1.q
hm.q
In


p11.O
pm1.O
Patient
O
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Restricted, Self, and Uncertain
RelationshipExample
Full
F(p,pf)

• A student's grade in a course depends on whether
  an advisor of the student is a friend of a
  teacher of the course.

Friend
Professor
Teaches
Course
Diff
Takes
Grade
Advises
Student
IQ
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In the paper(Google -gt Heckerman -gt Papers)

• Formal definitions and theorems
• Precise differences between PER models, plate
  models, and PRMs
• Undirected PER models
• PER models for asymmetric independence
• Many more examples