Duality in a maximum generalized entropy model Shinto Eguchi Osamu - PowerPoint PPT Presentation

MaxEnt, 2014 21-26 September 2014 Chateau Clos Luce, Amboise Duality in a maximum generalized entropy model Shinto Eguchi Osamu Komori Atsumi Ohara

MaxEnt in ecology Presence data obtained by an ecological study Equal mean space Log-entropy (Boltzmann-Gibbs-Shannon) 2

Maximum entropy distribution Problem : find a distribution f to maximize H( f  under the constraints: Solution is Gibbs distribution 3

MAP estimator Loss function Sequential L 1 -regularization (Phillip, Anderson, Schapire 2006) Cf. adaptive Lasso, Zou (2006, JASA) GIS (Geographic Information System) GBIF (Global Biodiversity Inventory Facility) in R 4

Habitat map of right whales 5 http://www.seascapemodeling.org/

6 Area Under ROC Curve

Problems in MaxEnt MaxEnt is useful in presence-only-data in ecological studies MaxEnt jointly suggests the statistical model and estimation, (exponential model and maximum likelihood) Model selection and model validation have simple forms The duality between the model and estimation is not robust (model misspecification, over-fitting to data, cf. Bayes robust) U -entropy leads to extension from MaxEnt to Max U- Ent (The duality is extended in functional degree of freedom) 7

Hill’s diversity numbers (effective number of species) (1973) richness Rare species are highly weighted exp (log-entropy) All species are fairly weighted Simpson index Dominant species are highly weighted 8

Several measures of entropy Simpson (1949) Hill’s diversity (1973) Tsallis entropy (1988) U -entropy (Eguchi, 2006) 9

Generalized entropy Generator Convex conjugate U -entropy Example log entropy Renyi-Tsallis entropy 10

Generalized divergence U -cross entropy U -divergence 11

Max U -Ent Problem : find a distribution f to maximize H U ( f  under the constraints: 12

Minimum divergence geometry Riemannian metric Linear connections 13

Dually flat U -model Riemann metric Linear connections Conjugate convexity: 14

Pythagorean theorem for U -divergence Theorem m-geodesic U -geodesic 15

MaxEnt and MinDiv Max-Entropy model

17 Duality of totally geodesic

18 = Max U -model + Min U -divergence Max U -Ent

Project Hill’s diversity numbers associate with one parameter family of MaxEnt methods, { Max  - Ent :  in R } Can the family improve the classical Maxent? Which  - MaxEnt is the best ? Can we select the best  based on presence data? Test AUC comparison may suggest better Max  - Ent. 19

20 ( Three-toed sloth ) Folivora

Max  -Ent      MaxEnt) 21

Max  -Ent     MaxEnt) 22

A class of Max U -Ent U- MaxEnt U- MaxEnt U -entropy MaxEnt U- MaxEnt U -entropy U- Gibbs distribution Log-entropy Max U -Ent U -entropy U- Gibbs distribution U- estimator Gibbs distribution U -entropy U- Gibbs distribution U- estimator U -divergence KL-divergence U- Gibbs distribution U- estimator U -divergence MLE U -divergence U -divergence Min U- estimator 23

24 Thank you!

A brief tutorial on Maxent 25 (Phillips, 2006) Model validation

( U- estimator , U -model) model U- model exp-model loss function Robust model - log-likelihood MLE Robust estimator U- loss U -estimate 26

Max U -Ent Problem : find a distribution f to maximize H U ( f  under the constraints: 27

Related problems Maxent is equivalent to Poisson point process model ( Renner-Warton, 2013, Loyle et all. 2012, Aarts et al. 2012, Fithian-Hastie, 2012)  - Maxent is equivalent to  - Poisson point process model? 28

 -Poisson distribution 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 5 10 15 20 5 10 15 20 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 5 10 15 20 5 10 15 20 29

Habitat map  = 0  = 0.5 30