Inference in dynamical systems and the geometry of learning group - - PowerPoint PPT Presentation

Inference in dynamical systems and the geometry of learning group actions

Geometry and Topology of Data Sayan Mukherjee

Departments of Statistical Science, Mathematics. Computer Science, Biostatistics & Bioinformatics Duke University https://sayanmuk.github.io/ Joint work with: Dynamical systems —

• K. McGoff (UNC Ch) | A. Nobel (UNC CH)

Group actions —

• T. Gao (U Chicago) | J. Brodzki (U Southampton)

Geometry of learning group actions

Modeling variation in shapes

• S. J. Gould

From distances to trees

50 molars from 5 primate genera

5 primate genera

Squirrel Monkey Howler Monkey Spider monkey Black handed spider monkey Titi monkey

Parallel transport in a shape space

Evolution as broccoli

Synchronization

Given a sequence of objects {o1, ..., on} and a group G learn a collection of group elements ρij that transform object oi to oj. If it is possible to find a sequence of group elements that allow for an accurate transformation between objects then this set can be synchronized.

Geometry and cohomology of synchronization

I Geometry: Holonomy and fiber bundle structures in

synchronization problems;

I Cohomology: Twisted de Rham theory and twisted Laplacian

for synchronization;

I Learning group actions algorithm (SynCut); I Classify primate molars to uncover dietary habits.

Problem setup

Γ = (V , E, w): vertex set V , edge set E, and weights wij.

Problem setup

Γ = (V , E, w): vertex set V , edge set E, and weights wij. G is a topological group acting on a normed vector space F.

Problem setup

Γ = (V , E, w): vertex set V , edge set E, and weights wij. G is a topological group acting on a normed vector space F. edge potential – ρ : E ! G with ρij = ρ−1

ji .

vertex potential – f : V ! F

Problem setup

Γ = (V , E, w): vertex set V , edge set E, and weights wij. G is a topological group acting on a normed vector space F. edge potential – ρ : E ! G with ρij = ρ−1

ji .

vertex potential – f : V ! F Goal: given ρ find fi = ρijfj 8i ⇠ j,

Problem setup

Γ = (V , E, w): vertex set V , edge set E, and weights wij. G is a topological group acting on a normed vector space F. edge potential – ρ : E ! G with ρij = ρ−1

ji .

vertex potential – f : V ! F Goal: given ρ find fi = ρijfj 8i ⇠ j,

• r

η = min

f :V !G kf k6=0

1 2 X

i,j∈V

wij kfi ρijfjk2

F

P

i∈V di kfik2 F

, di = X

j∼i

wij.

Vertex and edge potentials

Let Γ = (V , E) be a graph with vertex set V and edge set E. Let G be a group.

Vertex and edge potentials

Let Γ = (V , E) be a graph with vertex set V and edge set E. Let G be a group.

I A vertex potential is a map

f : V → G.

I An edge potential is a function

ρ : E → G where ρji = ρ−1

ij

∀i ∼ j

ρ(i, j)

f(i) f(j)

Vertex and edge potentials

G-valued 0- and 1-cochains vertex potentials: C 0 (Γ; G) := {f : V → G} edge potentials: C 1 (Γ; G) := n ρ : E → G | ρij = ρ−1

ji , ∀i ∼ j

• .

Vertex and edge potentials

G-valued 0- and 1-cochains vertex potentials: C 0 (Γ; G) := {f : V → G} edge potentials: C 1 (Γ; G) := n ρ : E → G | ρij = ρ−1

ji , ∀i ∼ j

• .

Typically F = G but one can decouple.

Fibre bundle framework

The Geometry of Synchronization Problems

I Data:

I graph Γ = (V , E) I linear algebraic group G, equipped with a norm k·k I edge potential ρ : E ! G satisfying ρij = ρ−1

ji , 8 (i, j) 2 E

The Geometry of Synchronization Problems

I Data:

I graph Γ = (V , E) I linear algebraic group G, equipped with a norm k·k I edge potential ρ : E ! G satisfying ρij = ρ−1

ji , 8 (i, j) 2 E

I Observation:

I Let U = {Ui | 1  i  |V |} be an open cover of Γ (viewed as a

1-dimensional simplicial complex), where Ui is the (open) star neighborhood of vertex i.

I The ρ defines a flat principal G-bundle over Γ (denoted as Bρ).

Fibre bundles

Fibre Bundle E = (E, M, F, π)

I M: base manifold I F: fibre manifold I E: total manifold I π : E → M: smooth surjective map (bundle projection) I local triviality: for “small” open set U ⊂ M, π−1 (U) is

diffeomorphic to U × F

Fibre bundles

Fibre Bundle E = (E, M, F, π)

I M: base manifold I F: fibre manifold I E: total manifold I π : E → M: smooth surjective map (bundle projection) I local triviality: for “small” open set U ⊂ M, π−1 (U) is

diffeomorphic to U × F Commutative diagram π−1(Ui) Ui × F Ui π φi Proj1

Fibre bundles and consistency conditions

Theorem (Steenrod 1951, §2). If topological group G acts on Y and {Ui}, {ρij} is a system

• f coordinate transformations in

the space X such that ρii = e 2 G for all Ui ρij = ρ−1

ji

if Ui \ Uj 6= ; ρijρjk = ρik if Ui \ Uj \ Uk 6= ; then there exists a fibre bundle B with base space X, fibre Y , group G, and coordinate transforms {ρij}.

Fibre bundles and consistency conditions

Theorem (Steenrod 1951, §2). If topological group G acts on Y and {Ui}, {ρij} is a system

• f coordinate transformations in

the space X satisfying the cycle-consistency conditions then there exists a fibre bundle B with base space X, fibre Y , group G, and coordinate transforms {ρij}.

Fibre bundles and consistency conditions

Theorem (Steenrod 1951, §2). If topological group G acts on Y and {Ui}, {ρij} is a system

• f coordinate transformations in

the space X satisfying the cycle-consistency conditions then there exists a fibre bundle B with base space X, fibre Y , group G, and coordinate transforms {ρij}. In other words we have constructed a bundle with fibre G over each vertex of the graph. The graph can be regarded as the nerve

• f the cover in the above theorem.

Cycle-consistent edge potentials

Geometric observations

I Denote

C 0 (Γ; G) := {f : V ! G} C 1 (Γ; G) := n ρ : E ! G | ρij = ρ−1

ji , 8i ⇠ j

Geometric observations

I Denote

C 0 (Γ; G) := {f : V ! G} C 1 (Γ; G) := n ρ : E ! G | ρij = ρ−1

ji , 8i ⇠ j

• I Consider the right action of C 0 (Γ; G) on C 1 (Γ; G):

C 1 (Γ; G) ⇥ C 0 (Γ; G) ! C 1 (Γ; G) (ρ, f ) 7 ! τρf defined as (τρf )ij := f −1

i

ρijfj, 8i ⇠ j.

Geometric observations

I Denote

C 0 (Γ; G) := {f : V ! G} C 1 (Γ; G) := n ρ : E ! G | ρij = ρ−1

ji , 8i ⇠ j

• I Consider the right action of C 0 (Γ; G) on C 1 (Γ; G):

C 1 (Γ; G) ⇥ C 0 (Γ; G) ! C 1 (Γ; G) (ρ, f ) 7 ! τρf defined as (τρf )ij := f −1

i

ρijfj, 8i ⇠ j.

I ρ synchronizable , τρ f synchronizable for all f 2 C 0 (Γ; G),

i.e. synchronizability is defined at the level of equivalence classes C 1 (Γ; G) /C 0 (Γ; G)

Holonomy and synchronization

Paths

A path γ connecting the vertices v and w is a sequence of edges γ = (e1, e2, . . . , en), γ1 = (en, en1, . . . , e1).

Paths

A path γ connecting the vertices v and w is a sequence of edges γ = (e1, e2, . . . , en), γ1 = (en, en1, . . . , e1). Maps from edges to group elements holρ (γ) = ρi0,1ρi1,i2 · · · ρiN−2,iN−1ρiN−1,iN 2 G. holρ

• γ1

= holρ (γ)1 , holρ

• γ γ0

= holρ (γ) holρ

• γ0

.

Holonomy and synchronization

Corollary (Gao-Brodzki-M, 2016)

For a connected graph Γ and topological group G, an edge potential ρ ∈ C 1 (Γ; G) is synchronizable if and only if Holρ (Γ) is trivial.

Holonomy and synchronization

Corollary (Gao-Brodzki-M, 2016)

For a connected graph Γ and topological group G, an edge potential ρ ∈ C 1 (Γ; G) is synchronizable if and only if Holρ (Γ) is trivial.

Theorem (Gao-Brodzki-M, 2016)

There exists a one-to-one correspondence between sets C 1 (Γ; G) /C 0 (Γ; G) ∼ = Hom (π1 (Γ) , G) /G where G acts on Hom (π1 (Γ) , G) by conjugations. Also, Hom (π1 (Γ) , G) /G is in one-to-one correspondence with equivalence classes of flat principal G-bundles Bρ defined by ρ ∈ C 1 (Γ; G).

Obstruction to holonomy

Quantification of obstructions

Let F be a vector space such that G ⇢ GL(F), and f 2 C 0 (Γ; F).

Quantification of obstructions

Let F be a vector space such that G ⇢ GL(F), and f 2 C 0 (Γ; F). The frustration of a function f η (f ) = 1 2 X

i,j∈V

kfi ρijfjk2 X

i∈V

kfik2 , can be written as a Raleigh quotient η (f ) = hf , L1f i hf , f i where L1 is the Graph Connection Laplacian (GCL) (L1f )i = 1 deg (i) X

j∼i

(fi ρijfj)

Discrete Hodge theory

Γ = (V , E): Ω0 (Γ) := {f : V → K} , Ω1 (Γ) := {ω : E → K | ωij = −ωji ∀i ∼ j} ,

Discrete Hodge theory

Γ = (V , E): Ω0 (Γ) := {f : V → K} , Ω1 (Γ) := {ω : E → K | ωij = −ωji ∀i ∼ j} , Define cochain complex 0 − − → ← − − Ω0 (Γ)

d

− − → ← − −

δ

Ω1 (Γ) − − → ← − − 0, where (df )ij = fi − fj, ∀f ∈ Ω0 (Γ) , (δω)i = 1 deg (i) X

j∼i

ωij, ∀ω ∈ Ω1 (Γ) .

Discrete Hodge theory

Define cochain complex 0 − − → ← − − Ω0 (Γ)

d

− − → ← − −

δ

Ω1 (Γ) − − → ← − − 0, where (df )ij = fi − fj, ∀f ∈ Ω0 (Γ) , (δω)i = 1 deg (i) X

j∼i

ωij, ∀ω ∈ Ω1 (Γ) . Then (L0f )i := (δdf )i = 1 deg (i) X

j∼i

(fi − fj) ∀i ∈ V , ∀f ∈ Ω0 (Γ) .

Twisted De Rham-Hodge Theory

(L0f )i = 1 deg (i) X

j∼i

(fi − fj) , ∀f : V → K (L1f )i = 1 deg (i) X

j∼i

(fi − ρijfj) ∀f : V → F

Twisted De Rham-Hodge Theory

(L0f )i = 1 deg (i) X

j∼i

(fi − fj) , ∀f : V → K (L1f )i = 1 deg (i) X

j∼i

(fi − ρijfj) ∀f : V → F Na¨ ıvely: (dρf )ij = fi − ρijfj, ∀f ∈ C 0 (Γ; F) (δρω)i = 1 deg (i) X

j∼i

ωij, ∀ω ∈ C 1 (Γ; F) then L1 = δρdρ. There is a problem

(dρf )ij ⇠ fi ρijfj, 8f 2 C 0 (Γ; F) (δρω)i ⇠ 1 deg (i) X

j∼i

ωij, 8ω 2 C 1 (Γ; F) Issue: dρ does not map into C 1 (Γ; F) (no skew-symmetry). fj ρjifj = ρji (fi ρijfj) 6= (fi ρijfj) .

(dρf )ij ⇠ fi ρijfj, 8f 2 C 0 (Γ; F) (δρω)i ⇠ 1 deg (i) X

j∼i

ωij, 8ω 2 C 1 (Γ; F) Issue: dρ does not map into C 1 (Γ; F) (no skew-symmetry). fj ρjifj = ρji (fi ρijfj) 6= (fi ρijfj) . Fix: Interpret fi ρijfj as the “local expression” of (dρf )ij in a local trivialization over U = {Ui | 1  i  |V |} of the associated F-bundle of Bρ, denoted as Bρ [F], such that the extra ρji factor encodes a bundle transformation from Ui to Uj.

Twisted De Rham-Hodge Theory

I Combinatorial Hodge Theory:

0 − − → ← − − Ω0 (Γ)

d

− − → ← − −

δ

Ω1 (Γ) − − → ← − − 0,

I Twisted Combinatorial Hodge Theory:

0 − − → ← − − C 0 (Γ; F)

dρ

− − → ← − −

δρ

Ω1 (Γ; Bρ [F]) − − → ← − − 0.

Twisted De Rham-Hodge Theory

I Combinatorial Hodge Theory:

0 − − → ← − − Ω0 (Γ)

d

− − → ← − −

δ

Ω1 (Γ) − − → ← − − 0,

I Twisted Combinatorial Hodge Theory:

0 − − → ← − − C 0 (Γ; F)

dρ

− − → ← − −

δρ

Ω1 (Γ; Bρ [F]) − − → ← − − 0.

Theorem (Gao, Brodzki, M (2016)))

Define ∆(0)

ρ

:= δρdρ, ∆(1)

ρ

:= dρδρ then the following Hodge-type decomposition holds: C 0 (Γ; F) = ker ∆(0)

ρ

⊕ im δρ = ker dρ ⊕ im δρ, Ω1 (Γ; Bρ [F]) = im dρ ⊕ ker ∆(1)

ρ

= im dρ ⊕ ker δρ.

Recovering the GCL

An O (d)-valued edge potential ξ 2 C 1 (Γ; O (d)) is synchronizable if and only if there exists g 2 C 0 (Γ; O (d)) such that ξij = gig−1

j

for all i ⇠ j.

Recovering the GCL

An O (d)-valued edge potential ξ 2 C 1 (Γ; O (d)) is synchronizable if and only if there exists g 2 C 0 (Γ; O (d)) such that ξij = gig−1

j

for all i ⇠ j. Define frustration ν (Γ) = 1 2d 1 vol (Γ) inf

g∈C 0(Γ;O(d))

X

i,j∈V

wij

• gig−1

j

ρij

• 2

F

= 1 2d 1 vol (Γ) inf

ξ∈C 1

sync(Γ;O(d))

X

i,j∈V

wij kξij ρijk2

F ,

where we define C 1

sync (Γ; O (d))

:=

• ξ 2 C 1 (Γ; O (d))
• ξ synchronizable

=

• ξ 2 C 1 (Γ; O (d))
• Holξ (Γ) is trivial

.

Find cochains in C 0 Γ; Rd “closest to” a global frame of Bρ ⇥ Rd⇤ η (f ) = D f , ∆(0)

ρ f

E kf k2 = 1 2 X

i,j2V

wij kfi ρijfjk2 X

i2V

di kfik2 = 1 2vol (Γ) [f ]> L1 [f ] , 8f 2 C 0 ⇣ Γ; Sd1⌘ , kf k 6= 0.

Learning group actions

Learning group actions

Problem (Learning group actions)

Given a group G acting on a set X, simultaneously learn actions of G on X and a partition of X into disjoint subsets X1, · · · , XK. Each action is cycle-consistent on each Xi (1 ≤ i ≤ K).

Learning group actions

Learning group actions by synchronization

Problem (Learning group actions by synchronization)

Denote XK for all partitions of Γ into K nonempty connected subgroups (K ≤ n) and ν (Si) = inf

f ∈C 0(Γ;G)

X

j,k∈Si

wjkCostG (fj, ρjkfk) , vol (Si) = X

j∈Si

dj, Solve the optimization problem min

{S1,··· ,SK }∈XK

max

1≤i≤K ν (Si)

min

1≤i≤K vol (Si)

(1) and output an optimal partition {S1, · · · , SK} together with the minimizing f ∈ C 0 (Γ; G).

Algorithm: SynCut

Input: Γ = (V , E, w), ρ 2 C 1 (Γ; G), number of partitions K Output: Partitions {S1, · · · , SK}

• 1. Solve synchronization problem over Γ for ρ, obtain

f 2 C 0 (Γ; G)

Algorithm: SynCut

Input: Γ = (V , E, w), ρ 2 C 1 (Γ; G), number of partitions K Output: Partitions {S1, · · · , SK}

• 1. Solve synchronization problem over Γ for ρ, obtain

f 2 C 0 (Γ; G)

• 2. Compute dij = exp (wij kfi ρijfjk) on all edges (i, j) 2 E

Algorithm: SynCut

Input: Γ = (V , E, w), ρ 2 C 1 (Γ; G), number of partitions K Output: Partitions {S1, · · · , SK}

• 1. Solve synchronization problem over Γ for ρ, obtain

f 2 C 0 (Γ; G)

• 2. Compute dij = exp (wij kfi ρijfjk) on all edges (i, j) 2 E
• 3. Spectral clustering on weighted graph (V , E, d) to get

{S1, · · · , Sk}

Algorithm: SynCut

Input: Γ = (V , E, w), ρ 2 C 1 (Γ; G), number of partitions K Output: Partitions {S1, · · · , SK}

• 1. Solve synchronization problem over Γ for ρ, obtain

f 2 C 0 (Γ; G)

• 2. Compute dij = exp (wij kfi ρijfjk) on all edges (i, j) 2 E
• 3. Spectral clustering on weighted graph (V , E, d) to get

{S1, · · · , Sk}

• 4. Solve synchronization problem within each partition Sj, “glue

up” the local solutions to obtain f∗ 2 C 0 (Γ; G)

Algorithm: SynCut

Input: Γ = (V , E, w), ρ 2 C 1 (Γ; G), number of partitions K Output: Partitions {S1, · · · , SK}

• 1. Solve synchronization problem over Γ for ρ, obtain

f 2 C 0 (Γ; G)

• 2. Compute dij = exp (wij kfi ρijfjk) on all edges (i, j) 2 E
• 3. Spectral clustering on weighted graph (V , E, d) to get

{S1, · · · , Sk}

• 4. Solve synchronization problem within each partition Sj, “glue

up” the local solutions to obtain f∗ 2 C 0 (Γ; G)

• 5. f f∗, repeat from Step 2

Dietary habits of primates

Geometric morphometrics

second mandibular molar of a Philippine flying lemur Philippine flying lemur (Cynocephalus volans)

Geometric morphometrics

• Manually put k landmarks

p1, p2, · · · , pk

• Use spatial coordinates of the

landmarks as features

• Represent a shape in R3×k

second mandibular molar of a Philippine flying lemur

Geometric morphometrics

• Manually put k landmarks

p1, p2, · · · , pk

• Use spatial coordinates of the

landmarks as features pj = (xj, yj, zj) , j = 1, · · · , k

• Represent a shape in R3×k

second mandibular molar of a Philippine flying lemur

Geometric morphometrics

• Manually put k landmarks

p1, p2, · · · , pk

• Use spatial coordinates of the

landmarks as features pj = (xj, yj, zj) , j = 1, · · · , k

• Represent a shape in R3×k

second mandibular molar of a Philippine flying lemur

Shape distances: automated landmarks

dcWn (S1, S2): Conformal Wasserstein Distance (CWD) dcP (S1, S2): Continuous Procrustes Distance (CPD) dcKP (S1, S2): Continuous Kantorovich-Procrustes Distance (CKPD)

Continuous Procrustes distance

Define A(S, S0) as the set of area preserving diffeomorphisms, maps a : S − → S0 such that for any measurable subset Ω ⊂ S Z

Ω

dAS = Z

a(Ω)

dAS0.

Continuous Procrustes distance

Define A(S, S0) as the set of area preserving diffeomorphisms, maps a : S − → S0 such that for any measurable subset Ω ⊂ S Z

Ω

dAS = Z

a(Ω)

dAS0. The continuous procrustes distance is dp(S, S0) = inf

a2A(S,S0)

min

R2rigid motion

Z

S

|R(x) − a(x)|2dAS.

Continuous Procrustes distance

Define A(S, S0) as the set of area preserving diffeomorphisms, maps a : S − → S0 such that for any measurable subset Ω ⊂ S Z

Ω

dAS = Z

a(Ω)

dAS0. The continuous procrustes distance is dp(S, S0) = inf

a2A(S,S0)

min

R2rigid motion

Z

S

|R(x) − a(x)|2dAS. Near optimal a are “almost” conformal, so simplify above to searching near conformal maps.

The actions: G

dcP (Si, Sj) = inf

C∈A(Si,Sj)

inf

R∈E(3)

✓Z

Si

kR (x) C (x) k2 dvolSi (x) ◆ 1

2

dij

• !

fij

Si Sj

50 molars from 5 primate genera

5 primate genera

Squirrel Monkey Howler Monkey Spider monkey Black handed spider monkey Titi monkey

Folivorous, frugivorous, and insectivorous

Open questions

(1) Use Hodge structure to design synchronization algorithms (2) Synchronization beyond the regime of linear algebraic groups (3) Statistical complexity of learning group actions (4) Random walks on fibre bundles (5) Provable algorithms (6) Synchronization on simplicial complexes (7) Bayesian model

Learning dynamical systems

Microbial ecology

n.enterobacteriaceae n.oral n.rikenellacae

Day 02 Day 09 Day 16 Day 23 Day 30 −2 2 −8 −4 −6 −4 −2 2

Balance Value Vessel

1 2 3 4

Posterior 95% credible interval

Justin Silverman and Lawerence David

General framework

We consider mathematical models of the form:

I X is the “phase space”; I Xt is the “true state” of bioreactor (your stomach) at time t; I Y is the “observation space”; I Yt is our observation at time t.

General framework

We consider mathematical models of the form:

I X is the “phase space”; I Xt is the “true state” of bioreactor (your stomach) at time t; I Y is the “observation space”; I Yt is our observation at time t.

We only have access to the observations {Ytk}n

k=0.

General questions

Given access to the observations {Ytk}n

k=0, we might want to

ask

I what is the “true state” of the bioreactor at time t? (filtering)

General questions

Given access to the observations {Ytk}n

k=0, we might want to

ask

I what is the “true state” of the bioreactor at time t? (filtering) I what are we likely to observe at time tn+1? (prediction)

General questions

Given access to the observations {Ytk}n

k=0, we might want to

ask

I what is the “true state” of the bioreactor at time t? (filtering) I what are we likely to observe at time tn+1? (prediction) I what are the rules governing the evolution of the system?

(model selection / parameter estimation) We’ll focus on the last type of question.

Basic assumptions

How are the variables {Xtk}n

k=0 and {Ytk}n k=0 related?

We’ll assume the process (Xt, Yt)t has:

I stationarity: the rules governing both the state space and

• ur observations don’t change over time.

Basic assumptions

How are the variables {Xtk}n

k=0 and {Ytk}n k=0 related?

We’ll assume the process (Xt, Yt)t has:

I stationarity: the rules governing both the state space and

• ur observations don’t change over time.

I Markov property: given the microbial population today,

the microbial population tomorrow is independent of the population yesterday.

Basic assumptions

How are the variables {Xtk}n

k=0 and {Ytk}n k=0 related?

We’ll assume the process (Xt, Yt)t has:

I stationarity: the rules governing both the state space and

• ur observations don’t change over time.

I Markov property: given the microbial population today,

the microbial population tomorrow is independent of the population yesterday.

I conditionally independent observations: given the state

• f the population today, today’s observation is independent
• f any other variables.

Such systems are called “hidden Markov models” (HMMs).

HMMs

Microbial ecology

n.enterobacteriaceae n.oral n.rikenellacae

Day 02 Day 09 Day 16 Day 23 Day 30 −2 2 −8 −4 −6 −4 −2 2

Balance Value Vessel

1 2 3 4

Posterior 95% credible interval

Dynamic linear models

xt+1 = At+1xt yt = Btxt + vt, Here: yt is an observation in Rp; xt is a hidden state in Rq; At is a p × p state transition matrix; Bt is a q × p observation matrix; vt is a zero-mean vector in Rq.

Stochastic versus deterministic systems

Should the process (Xt)t be stochastic or deterministic?

Stochastic versus deterministic systems

Should the process (Xt)t be stochastic or deterministic?

I If the conditional distribution of Xtk+1 given Xtk has positive

variance, then we’ll say the process (Xt)t is stochastic.

Stochastic versus deterministic systems

Should the process (Xt)t be stochastic or deterministic?

I If the conditional distribution of Xtk+1 given Xtk has positive

variance, then we’ll say the process (Xt)t is stochastic.

I Otherwise, we’ll say the process (Xt)t is deterministic.

In ecology both types of systems are commonly used.

Deterministic dynamics

Deterministic dynamics: for each θ, there is a map Tθ : X → X such that Xt+1 = Tθ(Xt).

Deterministic dynamics

Deterministic dynamics: for each θ, there is a map Tθ : X → X such that Xt+1 = Tθ(Xt). The Markov transition kernel is degenerate: Qθ(x, y) = δTθ(x)(y).

Deterministic dynamics

Deterministic dynamics: for each θ, there is a map Tθ : X → X such that Xt+1 = Tθ(Xt). The Markov transition kernel is degenerate: Qθ(x, y) = δTθ(x)(y). Such systems do not satisfy the strong stochastic mixing conditions used in previous work for HMMs.

Setting for deterministic dynamics

Suppose that for each θ in Θ (parameter space), we have (X, X, Tθ, µθ), where

I X is a complete separable metric space with Borel

σ-algebra X

Setting for deterministic dynamics

Suppose that for each θ in Θ (parameter space), we have (X, X, Tθ, µθ), where

I X is a complete separable metric space with Borel

σ-algebra X

I Tθ : X → X is a measurable map,

Setting for deterministic dynamics

Suppose that for each θ in Θ (parameter space), we have (X, X, Tθ, µθ), where

I X is a complete separable metric space with Borel

σ-algebra X

I Tθ : X → X is a measurable map, I µθ is a probability measure on (X, X) is Tθ-invariant if

µθ(T −1

θ

A) = µθ(A), ∀A ∈ X

Setting for deterministic dynamics

Suppose that for each θ in Θ (parameter space), we have (X, X, Tθ, µθ), where

I X is a complete separable metric space with Borel

σ-algebra X

I Tθ : X → X is a measurable map, I µθ is a probability measure on (X, X) is Tθ-invariant if

µθ(T −1

θ

A) = µθ(A), ∀A ∈ X

I the measure preserving system (X, X, Tθ, µθ) is ergodic if

T −1

θ

A = A implies µ(A) = {0, 1}.

Setting for deterministic dynamics

Suppose that for each θ in Θ (parameter space), we have (X, X, Tθ, µθ), where

I X is a complete separable metric space with Borel

σ-algebra X

I Tθ : X → X is a measurable map, I µθ is a probability measure on (X, X) is Tθ-invariant if

µθ(T −1

θ

A) = µθ(A), ∀A ∈ X

I the measure preserving system (X, X, Tθ, µθ) is ergodic if

T −1

θ

A = A implies µ(A) = {0, 1}. Family of systems (X, , X, Tθ, µθ)θ∈Θ ≡ (Tθ, µθ)θ∈Θ.

Mixing

Stochastic mixing: Let (Xt) be a stochastic process. Consider the function α(s) α(s) = sup{|P(A ∩ B) − P(A)P(B)|} such that A ∈ X t

−∞, B ∈ X ∞ t+s the process is strongly mixing if

α(s) → 0 as s → ∞

Mixing

Stochastic mixing: Let (Xt) be a stochastic process. Consider the function α(s) α(s) = sup{|P(A ∩ B) − P(A)P(B)|} such that A ∈ X t

−∞, B ∈ X ∞ t+s the process is strongly mixing if

α(s) → 0 as s → ∞ Dynamical mixing: T is strongly mixing if for all A, B ∈ X lim

n→∞ µ(T −nA ∩ B) = µ(A)µ(B).

An example

I X0 ∼ U[0, 1]; I Xk+1 = θXk(1 − Xk);

An example

I X0 ∼ U[0, 1]; I Xk+1 = θXk(1 − Xk); I Yk ∼ N(Xk, σ2 θ).

Classical Bayesian inference

Likelihood: Lik(data | θ)

Classical Bayesian inference

Likelihood: Lik(data | θ) Prior: π(θ)

Classical Bayesian inference

Likelihood: Lik(data | θ) Prior: π(θ) Marginal likelihood: R

θ Lik(data | θ) × π(θ)dθ = Pr(data)

Classical Bayesian inference

Likelihood: Lik(data | θ) Prior: π(θ) Marginal likelihood: R

θ Lik(data | θ) × π(θ)dθ = Pr(data)

Bayes rule: σn(θ | data) = Lik(data | θ) × π(θ) Pr(data) .

Example

Likelihood: X1, ..., Xn

iid

∼ N(θ, 1)

Example

Likelihood: X1, ..., Xn

iid

∼ N(θ, 1) Prior: θ ∼ N(0, 1)

Example

Likelihood: X1, ..., Xn

iid

∼ N(θ, 1) Prior: θ ∼ N(0, 1) Bayes rule σn(θ | X1, ..., Xn) = (2π)−(n+1)/2e− P

i(xi−θ)2/2 × e−θ2/2

R ∞

−∞(2π)−(n+1)/2e− P

i(xi−θ)2/2 × e−θ2/2dθ

. θ | X1, ..., Xn ∼ N ✓ n n + 1 ¯ X, 1 n + 1 ◆ , ¯ X = 1 n X

i

Xi.

Preliminaries

Observation system (Y, T, ν) with T : Y → Y

Preliminaries

Observation system (Y, T, ν) with T : Y → Y Tracking systems: Compact metrizable space X := X × Θ with map S : X → X. S : Θ × X → X, Sθ : X → X.

Preliminaries

Observation system (Y, T, ν) with T : Y → Y Tracking systems: Compact metrizable space X := X × Θ with map S : X → X. S : Θ × X → X, Sθ : X → X. Loss or regret: c : X × Y → R+. Cost of cn(x, y) := cn(xn−1 , yn−1 ) =

n−1

X

k=0

c(xk, yk), xn−1 = (x, Sx, . . . , Sn−1x) and yn−1 = (y, Ty, . . . , T n−1y).

Gibbs posterior

Given observations (y, Ty, . . . , T n−1y) ∈ Yn and a prior π on X.

Gibbs posterior

Given observations (y, Ty, . . . , T n−1y) ∈ Yn and a prior π on X. Consider the probability measure over Borel sets A ⊂ X σn(A | y) = R

A exp

• −cn(x, y)
• dπ(x)

Zn(y) , A ⊂ Θ × X Zn(y) = Z

X

exp

• −cn(x, y)
• dπ(x).

Gibbs posterior

Given observations (y, Ty, . . . , T n−1y) ∈ Yn and a prior π on X. Consider the probability measure over Borel sets A ⊂ X σn(A | y) = R

A exp

• −cn(x, y)
• dπ(x)

Zn(y) , A ⊂ Θ × X Zn(y) = Z

X

exp

• −cn(x, y)
• dπ(x).

Two questions (1) Is limn→∞ σn(· | y) unique.

Gibbs posterior

Given observations (y, Ty, . . . , T n−1y) ∈ Yn and a prior π on X. Consider the probability measure over Borel sets A ⊂ X σn(A | y) = R

A exp

• −cn(x, y)
• dπ(x)

Zn(y) , A ⊂ Θ × X Zn(y) = Z

X

exp

• −cn(x, y)
• dπ(x).

Two questions (1) Is limn→∞ σn(· | y) unique. (2) Does limn→∞ σn(· | y) concentrate around T.

Gibbs posterior

(1) Decision theoretic perspective of Bayesian inference, coherent inference with respect to a utility.

Gibbs posterior

(1) Decision theoretic perspective of Bayesian inference, coherent inference with respect to a utility. (2) If cn is the negative log likelihood then recover standard posterior.

Gibbs posterior

(1) Decision theoretic perspective of Bayesian inference, coherent inference with respect to a utility. (2) If cn is the negative log likelihood then recover standard posterior. (3) Robust to misspecification, robust statistics.

Gibbs posterior

(1) Decision theoretic perspective of Bayesian inference, coherent inference with respect to a utility. (2) If cn is the negative log likelihood then recover standard posterior. (3) Robust to misspecification, robust statistics. (4) Calibration/violation of likelihood principle σn(A | y) =

R

A exp

• −ψcn(x,y)
• dπ(x)

Zn(y)

.

Gibbs measures

Given X, the map S, a potential function h, and a measure µ0 σn(x; µ0, h) = exp Pm

k=0 h(Skx)

• R

X exp

Pm

k=0 h(Skx)

• dµ0

. The Gibbs measure is the limit point of the sequence σn(x; µ0, h) and the Gibbs measure is denoted as µ0(h).

Gibbs measures

Given X, the map S, a potential function h, and a measure µ0 σn(x; µ0, h) = exp Pm

k=0 h(Skx)

• R

X exp

Pm

k=0 h(Skx)

• dµ0

. The Gibbs measure is the limit point of the sequence σn(x; µ0, h) and the Gibbs measure is denoted as µ0(h). Recall σn(x | y) = exp

• − Pm

k=0 c(Skx, T ky)

• R

X exp

• − Pm

k=1 c(Skx, T ky)

• dπ(x).

Joinings and couplings

Definition (Joining)

Let (X, A, µ, T) and (Y, B, ν, S) be two dynamical systems. A joining of T and S is a probability measure λ on X × Y, with marginals µ and ν respectively, and invariant to the product map T × S.

Joinings and couplings

Definition (Joining)

Let (X, A, µ, T) and (Y, B, ν, S) be two dynamical systems. A joining of T and S is a probability measure λ on X × Y, with marginals µ and ν respectively, and invariant to the product map T × S.

Definition (Coupling)

A coupling of two random variable X and X 0 taking values in (E, E) is any pair of random variables (Y, Y 0) taking values in (E × E, E × E) whose marginals have the same distribution as X and X 0, X D = Y and X 0 D = Y 0.

Joinings

J (µ, ν) is the set of all joinings of (X, S, µ) and (Y, T, ν). Define J (S : ν) = S

µ J (µ, ν), where the union is over all

S-invariant Borel probability measures µ, µ ∈ M(X, S).

Variational formulation of Zn(y) – average cost

Recall ν is the measure for T and λ ∈ J (S : ν)

Variational formulation of Zn(y) – average cost

Recall ν is the measure for T and λ ∈ J (S : ν) Define λy ∈ M(X) (λ “projected" onto dνy) λ = Z

Y

λy ⊗ δy dν(y).

Variational formulation of Zn(y) – average cost

Recall ν is the measure for T and λ ∈ J (S : ν) Define λy ∈ M(X) (λ “projected" onto dνy) λ = Z

Y

λy ⊗ δy dν(y). Limiting average cost lim

n→∞

1 n Z

X

cn(x, y) dλy(x) = Z c dλ.

Variational formulation of Zn(y) – entropy term

Given two Borel probability measures π and µ on X and a finite measurable partition ξ of X. Denote µ ξ π as π(C) = 0 ) µ(C) = 0 for C 2 ξ.

Variational formulation of Zn(y) – entropy term

Given two Borel probability measures π and µ on X and a finite measurable partition ξ of X. Denote µ ξ π as π(C) = 0 ) µ(C) = 0 for C 2 ξ. Define L(σ k π, ξ) = ⇢ P

C∈ξ σ(C) log π(C),

if σ ξ π 1,

• therwise,

with 0 · log 0 = 0.

Variational formulation of Zn(y) – entropy term

Given two Borel probability measures π and µ on X and a finite measurable partition ξ of X. Denote µ ξ π as π(C) = 0 ) µ(C) = 0 for C 2 ξ. Define L(σ k π, ξ) = ⇢ P

C∈ξ σ(C) log π(C),

if σ ξ π 1,

• therwise,

with 0 · log 0 = 0. In spirit consider all finite measurable partitions ξ F(σ, π) = sup

ξ

L(σ k π, ξ).

Variational formulation of Zn(y) – weak Gibbs prior

A prior π satisfies the weak Gibbs property if there exists a continuous function φ : X → R and refining sequence of partitions {ξm} such that diam(ξm) → 0 where ξm has zero boundary for all S-invariant measures and for each m, there exists α = α(m) > 0 and K = K(m) such that for all n ∈ N and x ∈ X K −1e−αn ≤ π(ξm

n (x))

eφn(x) ≤ Keαn. and α(m) → 0.

Convergence

Theorem (McGoff-M.-Nobel)

Suppose a weak GIbbs prior, then for ν almost every y, lim

n→∞ −1

n log Zn(y) = inf

λ∈J (S:ν)

⇢Z c dλ + F(λ, π)

• ,

and the infimum in the above expression is attained.

Convergence

Theorem (McGoff-M.-Nobel)

Suppose a weak GIbbs prior, then for ν almost every y, lim

n→∞ −1

n log Zn(y) = inf

λ∈J (S:ν)

⇢Z c dλ + F(λ, π)

• ,

and the infimum in the above expression is attained. Furthermore, 1 n

n−1

X

k=0

ˆ λn

• S−k· | Y n−1
• −

→ Peq(X, S). Peq(X, S) are the set of processes that minimize the variational expression above.

Convergence

Proposition (McGoff-M.-Nobel)

Suppose a weak GIbbs prior and consider the pressure P(µ, ν) = inf

λ∈J (S:ν)

⇢Z c dλ + F(λ, π)

• P(θ : ν)

= inf

µ∈M(Xθ,Sθ) P(µ, ν),

θ∗ = arg min

θ∈Θ P(θ : ν).

Convergence

Proposition (McGoff-M.-Nobel)

Suppose a weak GIbbs prior and consider the pressure P(µ, ν) = inf

λ∈J (S:ν)

⇢Z c dλ + F(λ, π)

• P(θ : ν)

= inf

µ∈M(Xθ,Sθ) P(µ, ν),

θ∗ = arg min

θ∈Θ P(θ : ν).

For all ε > 0 P(d(Sθ∗, T) < ε) → 1a.s as n → ∞.

Toy example: Markov model

I {µθ : ✓ ∈ Θ} is a collection of Gibbs measures on a

common finite state space;

I there exists ✓∗ ∈ Θ such that ˆ

= µθ∗;

I `(✓; yn−1

) = − log µθ(yn−1 ). The standard Variational Principle for Gibbs measures yields that the posterior distribution converges almost surely to ✓∗.

Toy example: Markov model

I {µθ : ✓ ∈ Θ} is a collection of Gibbs measures on a

common finite state space;

I there exists ✓∗ ∈ Θ such that ˆ

= µθ∗;

I `(✓; yn−1

) = − log µθ(yn−1 ). The standard Variational Principle for Gibbs measures yields that the posterior distribution converges almost surely to ✓∗. More generally: convergence analysis for Gibbs posteriors under dependence.

Objective Bayesian inference

For each t ∈ N and x, y ∈ X, let dt(x, y) = max

• d(Skx, Sky) : 0 ≤ k < t

, where d is some metric on X.

Objective Bayesian inference

For each t ∈ N and x, y ∈ X, let dt(x, y) = max

• d(Skx, Sky) : 0 ≤ k < t

, where d is some metric on X. Nsep(✏, t) is packing number and Un,✏ is the packing set. Consider the distribution 1 Nsep(✏, t) X

x∈Un,✏

x.

Objective Bayesian inference

For each t ∈ N and x, y ∈ X, let dt(x, y) = max

• d(Skx, Sky) : 0 ≤ k < t

, where d is some metric on X. Nsep(✏, t) is packing number and Un,✏ is the packing set. Consider the distribution 1 Nsep(✏, t) X

x∈Un,✏

x. The Gibbs posterior distribution is n,✏(· | y) = 1 Zn,✏(y) X

x∈Un,✏

exp

• −cn(x, y)
• x.

Contributions

Reframes posterior consistency as two-stage process: first find the limiting variational problem, and then analyze this problem to address consistency. Provides general framework and suite of tools from the thermodynamic formalism for analyzing asymptotic behavior of Gibbs posteriors.

Questions

Statistics questions.

I What types of observations and models are amenable to

this analysis?

Questions

Statistics questions.

I What types of observations and models are amenable to

this analysis?

I For which combinations of observations and models can

• ne establish posterior consistency?

Questions

Statistics questions.

I What types of observations and models are amenable to

this analysis?

I For which combinations of observations and models can

• ne establish posterior consistency?

Dynamics questions.

I How far can the thermodynamic formalism be pushed?

Questions

Statistics questions.

I What types of observations and models are amenable to

this analysis?

I For which combinations of observations and models can

• ne establish posterior consistency?

Dynamics questions.

I How far can the thermodynamic formalism be pushed? I Under what conditions is there a limiting variational

characterization?

Questions

Statistics questions.

I What types of observations and models are amenable to

this analysis?

I For which combinations of observations and models can

• ne establish posterior consistency?

Dynamics questions.

I How far can the thermodynamic formalism be pushed? I Under what conditions is there a limiting variational

characterization?

I Under what conditions is there a unique equilibrium

joining?

Open problems

(1) Rates of convergence for a family of dynamical systems F.

Open problems

(1) Rates of convergence for a family of dynamical systems F. (2) General conditions for learnability in dynamical systems.

Open problems

(1) Rates of convergence for a family of dynamical systems F. (2) General conditions for learnability in dynamical systems. (3) Extension to continuous time dynamics, differential equations.

Open problems

(1) Rates of convergence for a family of dynamical systems F. (2) General conditions for learnability in dynamical systems. (3) Extension to continuous time dynamics, differential equations. (4) Computational issues.

Open problems

(1) Rates of convergence for a family of dynamical systems F. (2) General conditions for learnability in dynamical systems. (3) Extension to continuous time dynamics, differential equations. (4) Computational issues. (5) Integration of ideas from statistical models of time series and dynamical systems theory.

Acknowledgements

Thanks: Part I: Ingrid Debauchies, Misha Belkin, Lek-Heng Lim, Leo Guibas, Doug Boyer, Shmuel Weinberger Part II: Konstantin Mischaikow, Ramon van Handel, Steve Lalley, Jonathan Mattingly, Karl Petersen, Ioanna Manolopoulou, Jim Berger.

Acknowledgements

Thanks: Part I: Ingrid Debauchies, Misha Belkin, Lek-Heng Lim, Leo Guibas, Doug Boyer, Shmuel Weinberger Part II: Konstantin Mischaikow, Ramon van Handel, Steve Lalley, Jonathan Mattingly, Karl Petersen, Ioanna Manolopoulou, Jim Berger. Funding:

I Center for Systems Biology at Duke I NSF DMS, CCF

, CISE

I AFOSR I DARPA I NIH

New journal

195,

Evolution of cooperation in mammals

Evolution of cooperation in mammals

Study population. Brock’s fi tasks create “competition” at the transcriptional level, which is resolved differently Brock’s field site

Evolution of cooperation in mammals

Brock’s fi tasks create “competition” at the transcriptional level, which is resolved differently Brock’s field site