High-dimensional Ising model and Monte Carlo methods Wojciech - PowerPoint PPT Presentation

Introduction Model selection consistency Experiments References High-dimensional Ising model and Monte Carlo methods Wojciech Rejchel Nicolaus Copernicus University in Toruń Joint work with Błażej Miasojedow Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Markov random field Undirected graph ( V , E ) Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Markov random field Undirected graph ( V , E ) V = { 1 , . . . , d } - set of vertices Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Markov random field Undirected graph ( V , E ) V = { 1 , . . . , d } - set of vertices E ⊂ V × V - set of edges Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Markov random field Undirected graph ( V , E ) V = { 1 , . . . , d } - set of vertices E ⊂ V × V - set of edges Y = ( Y ( 1 ) , . . . , Y ( d )) - random vector Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Markov random field Undirected graph ( V , E ) V = { 1 , . . . , d } - set of vertices E ⊂ V × V - set of edges Y = ( Y ( 1 ) , . . . , Y ( d )) - random vector Y ( s ) is associated with vertex s ∈ V Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Joint distribution of Y is given by �� � 1 p ( y | θ ⋆ ) = θ ⋆ C ( θ ⋆ ) exp rs y ( r ) y ( s ) r < s Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Joint distribution of Y is given by �� � 1 p ( y | θ ⋆ ) = θ ⋆ C ( θ ⋆ ) exp rs y ( r ) y ( s ) r < s d ( d − 1 ) θ ⋆ ∈ R - true parameter 2 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Joint distribution of Y is given by �� � 1 p ( y | θ ⋆ ) = θ ⋆ C ( θ ⋆ ) exp rs y ( r ) y ( s ) r < s d ( d − 1 ) θ ⋆ ∈ R - true parameter 2 Intractable norming constant �� � C ( θ ⋆ ) = � θ ⋆ exp rs y ( r ) y ( s ) r < s y ∈{ 0 , 1 } d Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Joint distribution of Y is given by �� � 1 p ( y | θ ⋆ ) = θ ⋆ C ( θ ⋆ ) exp rs y ( r ) y ( s ) r < s d ( d − 1 ) θ ⋆ ∈ R - true parameter 2 Intractable norming constant �� � C ( θ ⋆ ) = � θ ⋆ exp rs y ( r ) y ( s ) r < s y ∈{ 0 , 1 } d J ( y ) = ( y ( r ) y ( s )) r < s Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model Y ( s ) ∈ {− 1 , 1 } Joint distribution of Y is given by �� � 1 p ( y | θ ⋆ ) = θ ⋆ C ( θ ⋆ ) exp rs y ( r ) y ( s ) r < s d ( d − 1 ) θ ⋆ ∈ R - true parameter 2 Intractable norming constant �� � C ( θ ⋆ ) = � θ ⋆ exp rs y ( r ) y ( s ) r < s y ∈{ 0 , 1 } d J ( y ) = ( y ( r ) y ( s )) r < s 1 p ( y | θ ⋆ ) = � ( θ ⋆ ) ′ J ( y ) � C ( θ ⋆ ) exp Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model θ ⋆ rs = 0 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model θ ⋆ rs = 0 means that Y ( r ) and Y ( s ) are conditionally independent Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model θ ⋆ rs = 0 means that Y ( r ) and Y ( s ) are conditionally independent Finding conditional independence Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model θ ⋆ rs = 0 means that Y ( r ) and Y ( s ) are conditionally independent Finding conditional independence ⇔ recognizing structure of graph Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Ising model θ ⋆ rs = 0 means that Y ( r ) and Y ( s ) are conditionally independent Finding conditional independence ⇔ recognizing structure of graph ⇔ estimation of θ ⋆ Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Likelihood estimation Y 1 , . . . , Y n - independent random vectors from p ( ·| θ ⋆ ) Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Likelihood estimation Y 1 , . . . , Y n - independent random vectors from p ( ·| θ ⋆ ) Negative log-likelihood n ℓ n ( θ ) = − 1 θ ′ J ( Y i ) + log C ( θ ) � n i = 1 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Likelihood estimation Y 1 , . . . , Y n - independent random vectors from p ( ·| θ ⋆ ) Negative log-likelihood n ℓ n ( θ ) = − 1 θ ′ J ( Y i ) + log C ( θ ) � n i = 1 Pseudolikelihood approximation Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Likelihood estimation Y 1 , . . . , Y n - independent random vectors from p ( ·| θ ⋆ ) Negative log-likelihood n ℓ n ( θ ) = − 1 θ ′ J ( Y i ) + log C ( θ ) � n i = 1 Pseudolikelihood approximation Monte Carlo (MC) approximation Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Pseudolikelihood approximation d � p ( y | θ ) = p ( y ( s ) | y ( s − 1 ) , . . . , y ( 1 ) , θ ) s = 1 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Pseudolikelihood approximation d � p ( y | θ ) = p ( y ( s ) | y ( s − 1 ) , . . . , y ( 1 ) , θ ) s = 1 d � ≈ p ( y ( s ) | y ( − s ) , θ ) s = 1 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References Pseudolikelihood approximation d � p ( y | θ ) = p ( y ( s ) | y ( s − 1 ) , . . . , y ( 1 ) , θ ) s = 1 d � ≈ p ( y ( s ) | y ( − s ) , θ ) s = 1 y ( − s ) = ( y ( 1 ) , . . . , y ( s − 1 ) , y ( s + 1 ) , . . . , y ( d )) Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References MC approximation h ( y ) - importance sampling distribution Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References MC approximation h ( y ) - importance sampling distribution Norming constant � θ ′ J ( y ) � � C ( θ ) = exp y ∈{ 0 , 1 } d Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References MC approximation h ( y ) - importance sampling distribution Norming constant exp [ θ ′ J ( y )] � = � θ ′ J ( y ) � � C ( θ ) = exp h ( y ) h ( y ) y ∈{ 0 , 1 } d y ∈{ 0 , 1 } d Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References MC approximation h ( y ) - importance sampling distribution Norming constant exp [ θ ′ J ( y )] � = � θ ′ J ( y ) � � C ( θ ) = exp h ( y ) h ( y ) y ∈{ 0 , 1 } d y ∈{ 0 , 1 } d exp [ θ ′ J ( Y )] = E Y ∼ h h ( Y ) Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods

Introduction Model selection consistency Experiments References MC approximation h ( y ) - importance sampling distribution Norming constant exp [ θ ′ J ( y )] � = � θ ′ J ( y ) � � C ( θ ) = exp h ( y ) h ( y ) y ∈{ 0 , 1 } d y ∈{ 0 , 1 } d exp [ θ ′ J ( Y )] = E Y ∼ h h ( Y ) Norming constant approximation � � θ ′ J ( Y k ) m exp 1 � h ( Y k ) m k = 1 Wojciech Rejchel High-dimensional Ising model and Monte Carlo methods