Variational Inference of Sparse Network from Count Data Julien - PowerPoint PPT Presentation

Variational Inference of Sparse Network from Count Data Julien Chiquet, Mahendra Mariadasou, St´ ephane Robin AgroParisTech/INRA (French National Institute for Agricultural Research) ICML, Long Beach Convention Center, June the 11th 2019 R/C++ PLNmodels package, development version on github install.packages("PLNmodels") https://jchiquet.github.io/PLNmodels/ 1

Sparse multivariate Poisson Lognormal model A sparse latent multivariate Gaussian model Z i iid ∼ N p ( 0 p , Ω − 1 ) , Ω sparse , � Ω � 1 , offdiagonal < c Y i | Z i ∼ P (exp { O i + X ⊤ i B + Z i } ) ( i , j ) / ∈ E ⇔ Z i ⊥ ⊥ Z j | Z \{ i , j } ⇔ Ω ij = 0 . Interpretation ◮ Dependency structure (network) encoded in the latent space ( Ω ) ◮ Additional effects due to covariates X are fixed ◮ Conditional Poisson distribution = noise model 2

Sparse Variational Inference Variational approximation Take q i ≡ N ( m i , diag ( s i )) to approximate of p ( Z i | Y i ) model parameters θ = ( B , Ω ) 1 ) ⊤ . . . ( s 2 variational parameters ψ = ( M , S ) where M = [ m ⊤ 1 . . . m ⊤ n ] ⊤ , S = [( s 2 n ) ⊤ ] ⊤ Sparse lower bound of the likelihood J ( θ , ψ ) − λ � Ω � 1 , off = E q [log p θ ( Y , Z )] + H [ q ψ ( Z )] − λ � Ω � 1 , off . Alternate optimization – objective is biconcave in ( B , M , S ) and Ω 1. ( ˆ B , ˆ M , ˆ S ) : gradient ascent 2. ˆ Ω : graphical-Lasso problem Selection of λ – StARS ( Stability Approach to Regularization Selection ) 3

Illustration: first round of French Presidential 2017 source: https://data.gouv.fr ◮ data: votes cast for each of the 11 candidates in the more than 63,000 polling stations ◮ offset: log-registered population of voter (account for different station sizes) ◮ covariate: ” department”(administrative division, a proxy for geography) � find competing candidates, who appeal to different voters, and compatible candidates Inferred network of partial correlation Latent Positions (PCA) (blue: negative, red: positive) ● DUPONT−AIGNAN CHEMINADE ● FILLON ASSELINEAU ● HAMON ARTHAUD ● LASSALLE ● POUTOU ● LE PEN ● MÉLENCHON ● MACRON 4