Sum Product Algorithm

Algorithm sum product updates variable to local

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Unformatted text preview: ates Variable to local function: µx →f (x ) = µh → x h∈n(x )\f Instructor: Arindam Banerjee The Sum-Product Algorithm Sum Product Updates Variable to local function: µx →f (x ) = µh → x h∈n(x )\f Local function to variable: µf →x (x ) = f (x ) ∼x Instructor: Arindam Banerjee µy →f (y ) y ∈n(f )\{x } The Sum-Product Algorithm Example: Step 1 µfA →x1 (x1 ) = fA (x1 ) µfB →x2 (x2 ) = fB (x2 ) µx4 →fD (x4 ) = 1 µx5 →fE (x5 ) = 1 Instructor: Arindam Banerjee The Sum-Product Algorithm Example: Step 2 µx1 →fC (x1 ) = µfA →x1 (x1 ) µx2 →fC (x2 ) = µfB →x2 (x2 ) µfD →x3 (x3 ) = fD (x3 , x4 )µx4 →fD (x4 ) ∼x3 µfE →x3 (x3 ) = fD (x3 , x5 )µx5 →fE (x5 ) ∼x3 Instructor: Arindam Banerjee The Sum-Product Algorithm Example: Step 3 µfC →x3 (x3 ) = fC (x1 , x2 , x3 )µx1 →fC (x1 )µx2 →fC (x2 ) ∼x3 µx3 →fC (x3 ) = µfD →x3 (x3 )µfE →x3 (x3 ) Instructor: Arindam Banerjee The Sum-Product Algorithm Example: Step 4 µfC →x1 (x1 ) = fC (x1 , x2 , x3 )µx2 →fC (x2 )µx3 →fC (x3 ) ∼x1 µfC →x2 (x2 ) = fC (x1 , x2 , x3 )µx1 →fC (x1 )µx3 →fC (x3 ) ∼x2 µx3 →fD (x3 ) = µfC →x3 (x3 )µfE →x3 (x3 ) µx3 →fE (x3 ) = µfC →x3 (x3 )µfD →x3 (x3 ) Instructor: Arindam Banerjee The Sum-Product Algorithm Example: Step 5 µx1 →fA (x1 ) = µfC →x1 (x1 ) µx2 →fB (x2 ) = µfC →x2 (x2 ) µfD →x4 (x4 ) = fD (x3 , x4 )µx3 →fD (x4 ) ∼x4 µfE →x5 (x5 ) = fD (x3 , x5 )µx3 →fE (x5 ) ∼x5 Instructor: Arindam Banerjee The Sum-Product Algorithm Example: Termination Marginal function is the product of all incoming messages g1 (x1 ) = µfA →x1 (x1 )µfC →x1 (x1 ) g2 (x2 ) = µfB →x2 (x2 )µfC →x2 (x2 ) g3 (x3 ) = µfC →x3 (x3 )µfD →x3 (x3 )µfE →x3 (x3 ) g2 (x2 ) = µfD →x4 (x4 ) g5 (x5 ) = µfE →x5 (x5 ) Instructor: Arindam Banerjee The Sum-Product Algorithm Belief Propagation in Bayes Nets Instructor: Arindam Banerjee The Sum-Product Algorithm Belief Propagation in Bayes Nets fA (x1 ) = p (x1 ) fB (x2 ) = p (x2 ) Instructor: Arindam Banerjee fC (x1 , x2 , x3 ) = p (x3 |x1 , x2 ) The Sum-Product Algorithm Belief Propagation in Bayes Nets fA (x1 ) = p (x1 ) fB (x2 ) = p (x2 ) fD (x3 , x4 ) = p (x4 |x3 ) Instructor: Arindam Banerjee fC (x1 , x2 , x3 ) = p (x3 |x1 , x2 ) fE (x3 , x5 ) = p (x5 |x3 ) The Sum-Product Algorithm...
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