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1 Learning Entity Specific Models Stefan Niculescu Carnegie Mellon University November, 2003

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2 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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3 Entities and Examples Today, huge databases track the evolution of users, companies or other entities/objects across time Multiple examples are collected per entity –Hospitals (Entities) treat many patients (Examples) –Users (Entities) are observed when handling various Emails (Examples) –In fMRI experiments, a Subject (Entity) is observed across few tens of Trials (Examples)

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4 Entity Specific and General Knowledge Each Entity has its own particularities Different number of attributes may be available for each entity –Different Hospitals may perform different tests on their patients to diagnose a given disease –A certain User may have some software installed while others may not Even two attributes that are present in two entities may relate in a complete different ways However, there are things that are common across entities –Treatments for a disease are the same across hospitals

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5 Goals and Approaches GOAL: Make inference about new examples –From available entities –From a new entity TWO EXTREME APPROACHES: –Learn a General Model by combining all the examples available May have different attributes for different entities Entity Specific params will be an “Average” across all Entities –BAD for making inference about existing Entities –Learn a separate Model for each Entity May not have enough data to learn some dependencies accurately Cannot be used when a new example comes from a previously unseen Entity

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6 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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7 Entity Specific Models Proposed approach: build a model that is somewhere between the two extremes –Takes advantage of multiple Entities to better learn General Knowledge –Also adapts itself to whatever is specific to each Entity Bayes Nets will be adapted to deal with the two issues

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8 Assumptions Examples independent given the parameters of the distribution There is no uncertainty about the entity –This is the case in our studies Data is fully observable (no missing values) Entities have the same sets of attributes Model – Bayes Net –Structure of the Bayes Net is the same for all entities –Parameters of the Bayes Net may vary from entity to entity It is known which parameters are General and which Entity Specific

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9 Notations

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10 Conditional Probability Tables CPT for Entity e 1 CPT for Entity e 2

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11 Maximum Data Likelihood setting Find the hypothesis that maximizes the likelihood of the data –Assumes that all hypotheses have equal priors Constraints: for all entities, each column of their CP tables should sum up to 1 !!!

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12 Optimizing using Lagrange Multipliers Can split in a set of independent optimization problems: Apply Lagrange Multipliers Theory: Solution of P ik is among solutions of: where

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13 Optimizing using Lagrange Multipliers Sanity Check: If all parameters are general (no entity specific params), then this is equivalent to a normal Bayes Net Sanity Check: If all parameters are entity specific, first fraction cancels and we have a collection of independent Bayes Nets

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14 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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15 Inference in Entity Specific models TWO CASES: 1.For a new (partial) example coming from a previously SEEN Entity: Build the Bayes Network corresponding to that Entity, then use any existing BN inference algorithm 2.For a new (partial) example coming from a previously UNSEEN Entity: Average / Weight Average the entity specific parameters corresponding to all previously seen entities into a General BN OR Train a General BN based on all seen entities – gives some prior over all parameters Apply the General BN to make inference about the new example

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16 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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17 EM for maximizing data likelihood

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18 EM for learning Entity Specific models from incomplete data E STEP: – Compute expected counts under current estimated parameters – Because of incomplete data, the counts are random variables M STEP: Reestimate model parameters – Maximize likelihood under observed expected counts

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19 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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20 Conditional Probability Tables CPT for Entity e Given Not given => Estimate! Given for Entity e – May be unknown for other entities Not given for Entity e => Estimate! – May be given for other entities

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21 Maximum Data Likelihood setting Find the hypothesis that maximizes the likelihood of the data –Assumes that all hypotheses have equal priors Constraints: for all entities, each column of their CP tables should sum up to 1 !!!

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22 Optimizing using Lagrange Multipliers Can split in a set of independent optimization problems: Apply Lagrange Multipliers Theory: Solution of P ik is among solutions of: where

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23 Optimizing using Lagrange Multipliers THEOREM: The ML estimators exist and they are unique. In addition, they can be accurately approximated by a bisection method. Proof (Sketch): Given parameters are treated as constants – Do not differentiate with respect to them ! Differentiating to unknown parameters we obtain: …

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24 Proof sketch

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25 Proof sketch where U is strictly increasing on the domain of admissible values U takes both negative and positive values Therefore A exists, is unique and it can be determined by a bisection method Once A is known, it is trivial to find B e values by substituting A in the constraints OBSERVATION: There is no closed form for the ML estimators, but they can be approximated arbitrarily close! Substituting in the constraints, we easily obtain that A is the solution of a polynomial equation:

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26 Outline Introduction Learning Entity Specific Models from complete data Inference in Entity Specific models EM for learning Entity Specific models from incomplete data Learning in presence of simple Domain Knowledge Summary / Future Work

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27 Summary Derived ML estimators for Entity Specific Bayes Nets Modified EM to deal with learning in presence of multiple entities Proved how simple Domain Knowledge can be incorporated in the learning algorithm

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28 Future Work Test the Entity Specific Bayes Net model on artificially generated data Incorporate uncertainty about the entity in the model Modify the model to be able to have different network topologies for different entities Improve the representation power of the domain knowledge that can be incorporated in learning … maybe probabilistic rules

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