Generalized Loopy 2U: A New Algorithm for Approximate Inference in Credal Networks Alessandro Antonucci, Marco Zaffalon, Sun Yi, Cassio de Campos IDSIA Lugano, Switzerland PGM ’08 - Hirtshals September 19th, 2008
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) • More generally, a closed convex set of probability mass functions K ( X ) This is a credal set (Levi, 1980) • Complete ignorance? A vacuous credal set K 0 ( X ) • Lower (and upper) expectation E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) • More generally, a closed convex set of probability mass functions K ( X ) This is a credal set (Levi, 1980) • Complete ignorance? A vacuous credal set K 0 ( X ) • Lower (and upper) expectation E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) (0,0,1) • More generally, a closed convex set of probability mass functions K ( X ) This is a credal set (Levi, 1980) P ( X ) • Complete ignorance? A vacuous credal set K 0 ( X ) (1,0,0) • Lower (and upper) expectation (0,1,0) E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) (0,0,1) • More generally, a closed convex set of probability mass functions K ( X ) K ( X ) This is a credal set (Levi, 1980) • Complete ignorance? A vacuous credal set K 0 ( X ) (1,0,0) • Lower (and upper) expectation (0,1,0) E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) (0,0,1) • More generally, a closed convex set of probability mass functions K ( X ) This is a credal set (Levi, 1980) • Complete ignorance? A vacuous credal set K 0 ( X ) (1,0,0) K 0 ( X ) • Lower (and upper) expectation (0,1,0) E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
Imprecise probabilities (Walley, 1991) Models of uncertainty about the state of a categorical variable X • A probability mass function P ( X ) (0,0,1) • More generally, a closed convex set of probability mass functions K ( X ) K ( X ) This is a credal set (Levi, 1980) • Complete ignorance? A vacuous credal set K 0 ( X ) (1,0,0) • Lower (and upper) expectation (0,1,0) E K [ f ( X )] = inf P ( X ) ∈ K ( X ) P X P ( x ) f ( X )
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
From Bayesian to credal nets Bayesian nets (Pearl, 1988) Credal nets (Cozman, 2000) • (stochastic) independence by a DAG • strong independence by a DAG • conditional mass functions • conditional credal sets K ( X i | pa( X i )) P ( X i | pa( X i )) • joint probability mass function • joint credal set (strong extension) P ( x 1 , . . . , x n ) = Q n i =1 P ( x i | pa( X i )) K ( X 1 , . . . , X n ) • updating = compute P ( X q | x E ) • updating = compute P ( X q | x E ) NP PP -hard NP-hard (Cooper, 1989) (Campos & Cozman, 2005) • BP efficiently updates polytrees • 2U: fast alg for binary polytrees (Pearl, 1988) (Zaffalon, 1998) • Loopy BP for multi-connected • Loopy 2U for multi-connected binary (Murphy, 1999) (Ide & Cozman, 2002) • . . . ? Updating non-binary credal nets?
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