Bayes and Lancaster at the Chinese restaurant. Statistical uses of - - PowerPoint PPT Presentation

Bayes and Lancaster at the Chinese restaurant.

Statistical uses of the Fleming-Viot Process. Dario Span`

• University of Warwick

1st Berlin-Padova Young Researchers Workshop 23-25 October, 2014

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Based on joint works with

Bob Griffiths (Oxford) Paul Jenkins (Warwick) Matteo Ruggiero (Torino) and Omiros Papaspiliopoulos (Barcelona)

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Outline

Chinese Restaurant Process and Bayes Computable filters Fleming-Viot Lancaster joins the restaurant

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Dirichlet measures and the Chinese restaurant process

Infinitely many delegates participate to an important probability young researcher workshop. Day 1: dinner at the chinese restaurant. Delegates enter the room one by

• ne, and, if k tables occupied by n1, . . . , nk persons (Pk

i=1 ni = n), then

the (n + 1)-th delegate: joins table with nj people with probability nj/(n + θ) (j = 1, . . . , k); chooses a new table with probability θ/(n + θ); each new table labelled with a color chosen at random from a space E of colors, using a prob. distribution P0.

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Dirichlet measures and the Chinese restaurant process

Infinitely many delegates participate to an important probability young researcher workshop. Day 1: dinner at the chinese restaurant. Delegates enter the room one by

• ne, and, if k tables occupied by n1, . . . , nk persons (Pk

i=1 ni = n), then

the (n + 1)-th delegate: joins table with nj people with probability nj/(n + θ) (j = 1, . . . , k); chooses a new table with probability θ/(n + θ); each new table labelled with a color chosen at random from a space E of colors, using a prob. distribution P0. Let Xn = “color of table occupied by n-th delegate”, n 2 N. Denote X(n) = (X1, . . . , Xn) and en(X(n)) := 1 n

n

X

i=1

δXi, n 2 N.

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Bayes at the Chinese restaurant.

The sequence (X1, X2, . . .) is infinitely exchangeable.

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Bayes at the Chinese restaurant.

The sequence (X1, X2, . . .) is infinitely exchangeable. Prior: e(X(n)) !

n!1 F

a.s. where F ⇠ πθ,P0 Ferguson-Dirichlet : (F(A1), . . . , F(Ad))

πθ,P0

⇠ Dir(θP0(A1), . . . , θP0(Ad)), for every d 2 N and every partition (A1, . . . , Ad) of E.

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Bayes at the Chinese restaurant.

The sequence (X1, X2, . . .) is infinitely exchangeable. Prior: e(X(n)) !

n!1 F

a.s. where F ⇠ πθ,P0 Ferguson-Dirichlet : (F(A1), . . . , F(Ad))

πθ,P0

⇠ Dir(θP0(A1), . . . , θP0(Ad)), for every d 2 N and every partition (A1, . . . , Ad) of E. Likelihood: L

• X(n) | F = µ
• = µ⌦n

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Bayes at the Chinese restaurant.

The sequence (X1, X2, . . .) is infinitely exchangeable. Prior: e(X(n)) !

n!1 F

a.s. where F ⇠ πθ,P0 Ferguson-Dirichlet : (F(A1), . . . , F(Ad))

πθ,P0

⇠ Dir(θP0(A1), . . . , θP0(Ad)), for every d 2 N and every partition (A1, . . . , Ad) of E. Likelihood: L

• X(n) | F = µ
• = µ⌦n

Posterior: L(F | x(n)) ⇠ πθ+n,

n θ+n e(x(n))+ θ θ+n P0.

.

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How crowded is your table?

Under πθ,P0, the pdf of (F(A1), . . . , F(Ad)) is / 2 4

k

Y

j=1

xθP0(Aj)1

j

3 5 I ⇣ (x1, ..., xd) 2 [0, 1]d : |x| = 1 ⌘ If E = {0, 1}, then P0 = p0 2 [0, 1] so πθ,p0 = beta(θp0, θ(1 p0)) If E any polish, then F(A) ⇠ beta(θP0(A), θ(1 P0(A)))

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Diffusion model

The time-evolution of a genetic variant, or allele, is well approximated by a diffusion process on the interval [0, 1].

Time Allele frequency 1 6 / 1

Diffusion model

The time-evolution of a genetic variant, or allele, is well approximated by a diffusion process on the interval [0, 1].

Time Allele frequency 1

Wright-Fisher SDE

dFt = bθ(Ft)dt + p Ft(1 Ft)dWt, F0 = µ, t 0. The infinitesimal drift, bθ(x), encapsulates directional forces such as natural selection, migration, mutation, . . .

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk}

Time Allele frequency 1

How to infer diffusion sample path properties given data?

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Optimal filter.

Assume the diffusion has stationary measure π and transition function Pt(µ, dν). Let fµ(x) the likelihood of data given signal, both at

• stationarity. Two operators.

Update operator (Bayes’ rule): φx(π)(dµ) = fµ(x)π(dµ) Eπ(X) prediction operator (propagator): ψt(π)(dν) = Z

M1

π(dµ)Pt(µ, dν)

Definition

The optimal filter is the solution of the recursion π0 = φxt0(π), πn = φxtn(ψtntn−1(πn)) it is called computable filter if iterating n times update/propagation involves finite sums whose number of terms depends on n.

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk}

Time Allele frequency 1

How to infer diffusion sample path properties given data?

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk}

Time Allele frequency 1

A priori, Ft1 ⇠ π

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk}

Time Allele frequency 1

Update Ft0 | Data at t1

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk}

Time Allele frequency 1

Predict F2 based on posterior update at t1 via Pt2t1(Fx(n1)(t1), ·)

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Filtering with genetic time series data.

We do not observe the path of the frequency diffusion F = (Ft : t 0), but only samples taken at distinct time points t1 < . . . < tk.

Key assumption on likelihood

X1(t), . . . , Xn(t)(t) | Ft

iid

⇠ Ft, t 2 {t1, . . . , tk} Update distribution of Ft2 given Data at t2. Carry on for t3, t4, · · · .

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Tractability of a filter.

dFt = bθ(Ft)dt + p Ft(1 Ft)dWt, F0 = µ, t 0. Ideally we would like to be able to Know the stationary distribution π; Know how to compute posterior P(Ft | Data at t); Know how to compute Pt(µ, dν). Generally all three aspects are intractable. Neutral Fleming-Viot models have them all ! bα,β(x) = 1 2[α(1 x) βx], α, β > 0.

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Tractability of a filter.

dFt = bθ(Ft)dt + p Ft(1 Ft)dWt, F0 = µ, t 0. Ideally we would like to be able to Know the stationary distribution π; Beta(α, β) distribution Know how to compute posterior P(Ft | Data at t);CRP Know how to compute Pt(µ, dν).Lancaster probability Generally all three aspects are intractable. Neutral Fleming-Viot models have them all ! bα,β(x) = 1 2[α(1 x) βx], α, β > 0.

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What are Lancaster probabilities?

Definition

Let (X, Y ) be an exchangeable pair of random variables with (identical) marginal distn. π. The joint distribution of (X, Y ) is a Lancaster probability distribution if, for every n, E [Y n | X = x] = ρnxn + polynomial in x of degree less than n The coefficients {ρn} are termed Canonical Correlation Coefficients. In neutral Fleming-Viot model Ptµn = e 1

2 n(n+θ1)tµn + . . . ,

θ = α + β. Benefit in filtering: Given F0 = µ sample of size n from µ is sufficient to predict sample of size n at time t.

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Genealogy and eigenvalues

The canonical correlation coefficients e 1

2 n(n+θ1)t are the eigenvalues of

the semigroup Pt. A probabilistic interpretation is in terms of the model’s genealogy (dual to the diffusion).

Time Allele frequency 1 16 / 1

Neutral model, a closer look.

Finite population of size N, discrete, non-overlapping generations. at each generation, type of individuals J1, . . . , JN labeled with points in some Polish space E (e.g. E = {0, 1}). At each time k, each individual picks her parent uniformly at random from previous generation k 1. Any individual with probability 1 u inherits her parent’s type. With probability u it mutates to a new type chosen from E according to a probability distribution P0 on E (if E = {0, 1}, then P0{1} 2 [0, 1]). Let F N(k) := 1 N

N

X

i=1

δJi(k), k = 0, 1, . . . .

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Neutral model, a closer look.

Finite population of size N, discrete, non-overlapping generations. at each generation, type of individuals J1, . . . , JN labeled with points in some Polish space E (e.g. E = {0, 1}). At each time k, each individual picks her parent uniformly at random from previous generation k 1. Any individual with probability 1 u inherits her parent’s type. With probability u it mutates to a new type chosen from E according to a probability distribution P0 on E (if E = {0, 1}, then P0{1} 2 [0, 1]). Let F N(k) := 1 N

N

X

i=1

δJi(k), k = 0, 1, . . . . Neutral diffusion obtained as the scaling limit Ft : d = lim

N!1 F N(bNtc),

Nu ! θ

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Neutral model, a closer look.

Finite population of size N, discrete, non-overlapping generations. at each generation, type of individuals J1, . . . , JN labeled with points in some Polish space E (e.g. E = {0, 1}). At each time k, each individual picks her parent uniformly at random from previous generation k 1. Any individual with probability 1 u inherits her parent’s type. With probability u it mutates to a new type chosen from E according to a probability distribution P0 on E (if E = {0, 1}, then P0{1} 2 [0, 1]). Let F N(k) := 1 N

N

X

i=1

δJi(k), k = 0, 1, . . . . Neutral diffusion obtained as the scaling limit Ft : d = lim

N!1 F N(bNtc),

Nu ! θ (if E = {0, 1}, α = θP0{0})

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Lines of descent dual process

Time Allele frequency 1

Pure death process counting “surviving”lineages, starting at {1} Lineage lost by mutation at rate θ/2 or by coalescence at rate n

2

• 18 / 1

Lines of descent dual process

Time Allele frequency 1

Pure death process counting “surviving”lineages, starting at {1} Lineage lost by mutation at rate θ/2 or by coalescence at rate n

2

• Given n surviving lineages at any time, the probability that they will

all survive in the next t generations (time running backwards) is etotal death rate t = e 1

2 n(n+θ1)t 18 / 1

Lines of descent dual process

Time Allele frequency 1

Pure death process counting “surviving”lineages, starting at {1} Lineage lost by mutation at rate θ/2 or by coalescence at rate n

2

• Given n surviving lineages at any time, the probability that they will

all survive in the next t generations (time running backwards) is etotal death rate t = e 1

2 n(n+θ1)t

Given the type-configuration of n surviving lineages at time s and the number of surviving lineages at time s + t, the conditional distribution

• f the survivors’ type configuration does not depend on t !!

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Fleming-Viot diffusion

When E continuous, compact and P0 diffuse,

Properties of Fleming-Viot diffusion.

time-reversible, with Dirichlet stationary measure; Transition function is a Lancaster probability with eigenvalues e 1

2 n(n+θ1)tindependent of dimension of Ft:

Property of dual genealogy

Same as in the two-type. Death process still univariate. Everything else does not depend on t.

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Computable propagation - in one dimension:

Likelihood is a monomial in Ft (binomial).

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Computable propagation - in one dimension:

Likelihood is a monomial in Ft (binomial). For each finite-dimensional projection: Update operator (Bayes’ rule): φx(n)(πα,β)(dµ) = πα+k,β+nk(dµ) = π(dµ) ⇥ Monomial in µ. prediction operator (propagator): ψt(πα+k,β+nk)(dν) = Z 1 πα+k,βn+k(dµ)Pt(µ, dν) = π(dν)Pt φx(n)(π)(dν) π(dν) ! = π(dν) ⇥ n-poly in ν (Lancaster)

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Filtering and duality.

Let g(µ, 1 µ; k, n k) = φn,k(π)(dµ) π(dµ) Then Eµ [g(Ft; k, n k)] = En,k [g(x; Kt, Nt Kt)] .

Time Allele frequency 1 21 / 1

Questions:

Are there other stochastic processes on M(E), reversible, with Dirichlet stationary measure and Lancaster transition functions? If so, filter is computable.

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Questions:

Are there other stochastic processes on M(E), reversible, with Dirichlet stationary measure and Lancaster transition functions? If so, filter is computable. Are all such processes dual to some simpler integer-valued (coalescent-like) process? If so, possible to simulate exactly from transition function. Probabilistic interpretation to eigenvalues.

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Questions:

Are there other stochastic processes on M(E), reversible, with Dirichlet stationary measure and Lancaster transition functions? If so, filter is computable. Are all such processes dual to some simpler integer-valued (coalescent-like) process? If so, possible to simulate exactly from transition function. Probabilistic interpretation to eigenvalues. Applications: filtering for time-series data from infinite-dimensional signal.

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Orthogonal polynomial kernels.

Let F be an a.s. finite measure on E with distribution π. For every d and partition A = (A1, . . . , Ad) of E, let πA be the distribution of (F(A1), . . . , F(Ad)). Assume There exists {QA

κ : κ 2 Zd +}, a system of orthogonal polynomials

complete for L2(πθ,P0,A). The chaotic decomposition of L2(πθ,P0,A) is given by: f (x) =

1

X

n=0

QA

n (f )(x),

QA

n (f )(x) :=

X

|κ|=n

QA

κ (x)hQA κ , f i.

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Orthogonal polynomial kernels.

Let F be an a.s. finite measure on E with distribution π. For every d and partition A = (A1, . . . , Ad) of E, let πA be the distribution of (F(A1), . . . , F(Ad)). Assume There exists {QA

κ : κ 2 Zd +}, a system of orthogonal polynomials

complete for L2(πθ,P0,A). The chaotic decomposition of L2(πθ,P0,A) is given by: f (x) =

1

X

n=0

QA

n (f )(x),

QA

n (f )(x) :=

X

|κ|=n

QA

κ (x)hQA κ , f i.

If fr is a polynomial of degree r, then QA

n (fr) = 0 for n > r.

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Lancaster and Fleming-Viot.

Let Pt be the semigroup of the Fleming-Viot diffusion process. Then f (µ) =

1

X

n=0

Qn(f )(µ), f 2 L2(πθ,P0) implies Ptf (µ) = E [f (Ft) | F0 = µ] =

1

X

n=0

e n(n+θ−1)t

2

Qn(f )(µ),

24 / 1

Lancaster and Fleming-Viot.

Let Pt be the semigroup of the Fleming-Viot diffusion process. Then f (µ) =

1

X

n=0

Qn(f )(µ), f 2 L2(πθ,P0) implies Ptf (µ) = E [f (Ft) | F0 = µ] =

1

X

n=0

e n(n+θ−1)t

2

Qn(f )(µ), In other words: PtQn(f ) = e n(n+θ−1)t

2

Qn(f ), 8f 2 L2(π), n 2 Z+

24 / 1

Lancaster and Fleming-Viot.

Let Pt be the semigroup of the Fleming-Viot diffusion process. Then f (µ) =

1

X

n=0

Qn(f )(µ), f 2 L2(πθ,P0) implies Ptf (µ) = E [f (Ft) | F0 = µ] =

1

X

n=0

e n(n+θ−1)t

2

Qn(f )(µ), In other words: PtQn(f ) = e n(n+θ−1)t

2

Qn(f ), 8f 2 L2(π), n 2 Z+ If fr a polynomial of degree r, then series terminates at r. This makes the propagation step of the filter computable.

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Filtering beyond genetics.

Identify measure-valued Feller processes, reversible, with πθ,P0 stationary measure, whose semigroup satisfies PQn(f ) = ρn(t)Qn(f ), 8f 2 L2(π), n 2 Z+ f 2 L2(πθ,P0) for a sequence of eigenvalues ρn(t).

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Filtering beyond genetics.

Identify measure-valued Feller processes, reversible, with πθ,P0 stationary measure, whose semigroup satisfies PQn(f ) = ρn(t)Qn(f ), 8f 2 L2(π), n 2 Z+ f 2 L2(πθ,P0) for a sequence of eigenvalues ρn(t). Most general form for eigenvalues? (Bochner/Gasper did it for d = 2).

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Two Days at the Chinese Restaurant !

26 / 1

Two Days at the Chinese Restaurant !

Suppose now a dinner at the restaurant is organised for both day 1 and day 2. Each day, same rule for choosing tables. However, on day 1, exactly m individuals keep staying on their seats, up until the start of day 2 dinner.

26 / 1

Two Days at the Chinese Restaurant !

Suppose now a dinner at the restaurant is organised for both day 1 and day 2. Each day, same rule for choosing tables. However, on day 1, exactly m individuals keep staying on their seats, up until the start of day 2 dinner. Let Fi be the distribution of tables of day i, i = 1, 2. To sample from L(F2 | F1 = µ): (i) Sample m labels x1, . . . , xm from m iid(µ) random variables. (ii) Generate F2 from posterior πθ,P0 (dF2 | x1, . . . , xm) .

26 / 1

Chaos at the Workshop.

Let ξm(f )(µ) := E [f (F2) | F1 = µ; m] .

Theorem (Griffiths and S. 13)

For every m, ξm is a Lancaster probability transition kernel ξmQn = cmnQn with cmn = 0 if n > m, so that ξm(f ) =

m

X

n=0

cmnQn(f )

27 / 1

Chaos at the Workshop.

Theorem (cont.)

Moreover: for every n, m, cθ

m,n = E

h Rγ,θγ

n

(Zm) i depends only on θ, where Zm ⇠ beta(γ + m, θ γ) for any 0 < γ < θ, and {Rγ,θγ

n

} are OP wrt beta(γ, θ γ), scaled so that Rγ,θγ

n

(1) = 1.

28 / 1

Multiple Days CRP - constant number of veterans.

If we allow each day and every day m delegates stay until next day, then k-step transition prob.: ξk

mQn = ck m,nQn.

But ck

m,n = E

h Rγ,θγ

n

(ZN(k)) i , where N(k)=number of people who did not leave their seats since day 1.

29 / 1

Multiple Days CRP - constant number of veterans.

If we allow each day and every day m delegates stay until next day, then k-step transition prob.: ξk

mQn = ck m,nQn.

But ck

m,n = E

h Rγ,θγ

n

(ZN(k)) i , where N(k)=number of people who did not leave their seats since day 1. Note (N(k) : k 1) is Markov chain starting at N(1) = m, with negative jumps!

29 / 1

Multiple Days CRP - constant number of veterans.

If we allow each day and every day m delegates stay until next day, then k-step transition prob.: ξk

mQn = ck m,nQn.

But ck

m,n = E

h Rγ,θγ

n

(ZN(k)) i , where N(k)=number of people who did not leave their seats since day 1. Note (N(k) : k 1) is Markov chain starting at N(1) = m, with negative jumps! If we let k = bmtc for t > 0, then as m ! 1, cbmtc

m,n

! e 1

2 n(n+θ1)t

and ξbmtcf !

m!1 Pθ,P0 t

f is the semigroup of a neutral Fleming-Viot process (θ, P0) of Population Genetics.

29 / 1

Multiple Days CRP 2 - iid number of veterans.

Instead of assuming constant m each day, let M(j)= number of people not leaving on j-th night be random, and assume (M(1), M(2), . . .) iid.

30 / 1

Multiple Days CRP 2 - iid number of veterans.

Instead of assuming constant m each day, let M(j)= number of people not leaving on j-th night be random, and assume (M(1), M(2), . . .) iid. Process of surviving veterans (N(k)) still a Markov chain with negative jumps, now with N(1) = M(1).

30 / 1

Multiple Days CRP 2 - iid number of veterans.

Instead of assuming constant m each day, let M(j)= number of people not leaving on j-th night be random, and assume (M(1), M(2), . . .) iid. Process of surviving veterans (N(k)) still a Markov chain with negative jumps, now with N(1) = M(1). The new transition kernel P is so that: PQn = ρnQn where ρn = E h Rγ,θγ

n

(ZM(1)) i .

30 / 1

Multiple Days CRP 2 - iid number of veterans.

Instead of assuming constant m each day, let M(j)= number of people not leaving on j-th night be random, and assume (M(1), M(2), . . .) iid. Process of surviving veterans (N(k)) still a Markov chain with negative jumps, now with N(1) = M(1). The new transition kernel P is so that: PQn = ρnQn where ρn = E h Rγ,θγ

n

(ZM(1)) i . After k days, P(k)Qn = ρk

nQn

30 / 1

Multiple Days CRP 2 - iid number of veterans.

Instead of assuming constant m each day, let M(j)= number of people not leaving on j-th night be random, and assume (M(1), M(2), . . .) iid. Process of surviving veterans (N(k)) still a Markov chain with negative jumps, now with N(1) = M(1). The new transition kernel P is so that: PQn = ρnQn where ρn = E h Rγ,θγ

n

(ZM(1)) i . After k days, P(k)Qn = ρk

nQn

ρk

n = E

h Rγ,θγ

n

(ZNp(k)) i = E h Rγ,θγ

n

(ZN(1)) ik .

30 / 1

Replace k with random K = K(t) ⇠ Po(λt), independent of M. Let e Mt := MK(t).

Theorem

The process (Ft = FK(t) : t 0) is Markov with semigroup PtQn = eψ(n)tQn where ψ(n) = Z 1 (1 Rn(z))λν(dz) Moreover, eψ(n)t = E h Rγ,θγ

n

(Z e

Mt)

i

31 / 1

The most general Markov eigenvalues.

Allowing λ ! 1 and λν ! e ν, we get the most general form of Markov eigenvalues generated by always going to the same Chinese restaurant. Almost L´ evy-Kintchine: ρn(t) = exp ⇢ t ✓ bn(n + θ 1) + Z 1 (1 Rn(z))e ν(dz) ◆ . for some measure e ν.

32 / 1

The most general Markov eigenvalues.

Allowing λ ! 1 and λν ! e ν, we get the most general form of Markov eigenvalues generated by always going to the same Chinese restaurant. Almost L´ evy-Kintchine: ρn(t) = exp ⇢ t ✓ bn(n + θ 1) + Z 1 (1 Rn(z))e ν(dz) ◆ . for some measure e ν. If θ > 2, this is necessarily eigenvalues for the t.f. of a Markov process (Ft) (Bochner/Gasper, 1950-70s), but not clear if there exist a dual (Mt).

32 / 1

The most general Markov eigenvalues.

Allowing λ ! 1 and λν ! e ν, we get the most general form of Markov eigenvalues generated by always going to the same Chinese restaurant. Almost L´ evy-Kintchine: ρn(t) = exp ⇢ t ✓ bn(n + θ 1) + Z 1 (1 Rn(z))e ν(dz) ◆ . for some measure e ν. If θ > 2, this is necessarily eigenvalues for the t.f. of a Markov process (Ft) (Bochner/Gasper, 1950-70s), but not clear if there exist a dual (Mt).

32 / 1

Existence of a one-dimensional dual.

ρn(t) = exp ⇢ t ✓ bn(n + θ 1) + Z 1 (1 Rn(z))e ν(dz) ◆ .

Theorem

Suppose ν has Lebesgue density infinitely differentiable. ρn(t) corresponds to a one-dimensional dual process of surviving CRP veterans if and only if ν(x) = h(x)πγ,θγ(x) for h with all derivatives positive and some γ < θ.

33 / 1

Existence of a one-dimensional dual.

ρn(t) = exp ⇢ t ✓ bn(n + θ 1) + Z 1 (1 Rn(z))e ν(dz) ◆ .

Theorem

Suppose ν has Lebesgue density infinitely differentiable. ρn(t) corresponds to a one-dimensional dual process of surviving CRP veterans if and only if ν(x) = h(x)πγ,θγ(x) for h with all derivatives positive and some γ < θ. Example: h completely monotone makes ρn(t) a canonical correlation sequence of Lancaster eigenvalues, but there is no one-dimensional dual with negative jumps.

33 / 1

Some last observations.

It appears that the FV process is the only Lancaster reversible process with Dirichlet stationary measure and continuous sample paths Existence of one-dimensional dual seems a key factor making it possible to simulate exactly from Pt (Current work with P. Jenkins). If not, the filter is still computable but the t.f. only approximately so. It is not known if general eigenvalues are good independently of any

• dimension. Reasons to doubt.

Not many genetic reversible models. Not many models have explicit stationary distribution. The only other one I know (diploid selection) is under study but eigenfunctions are not polynomials. Possible to use FV as proposal distribution (current work with P. Jenkins).

34 / 1

Some last observations.

It appears that the FV process is the only Lancaster reversible process with Dirichlet stationary measure and continuous sample paths Existence of one-dimensional dual seems a key factor making it possible to simulate exactly from Pt (Current work with P. Jenkins). If not, the filter is still computable but the t.f. only approximately so. It is not known if general eigenvalues are good independently of any

• dimension. Reasons to doubt.

Not many genetic reversible models. Not many models have explicit stationary distribution. The only other one I know (diploid selection) is under study but eigenfunctions are not polynomials. Possible to use FV as proposal distribution (current work with P. Jenkins). Enjoy your (Chinese?) lunch!

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