Urn models with multiple drawings Markus Kuba joint work with - - PowerPoint PPT Presentation
Urn models with multiple drawings Markus Kuba joint work with May-Ru Chen; Hosam Mahmoud and Alois Panholzer AofA 2013 Cala Galdana, Menorca, Spain 30.05.2013 Content 1 Urn models - Introduction Urn models with multiple drawings 2 3
Markus Kuba joint work with May-Ru Chen; Hosam Mahmoud and Alois Panholzer AofA 2013 Cala Galdana, Menorca, Spain 30.05.2013
1
Urn models - Introduction
2
Urn models with multiple drawings
3
Analysis using Analytic Combinatorics
AofA 2013, Menorca 2/26
Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: pwhite =
n n+m, pblack = m n+m.
Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a, b, c, d ∈ Z.
2 × 2 ball replacement matrix
M = a b c d
4/26
Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: pwhite =
n n+m, pblack = m n+m.
Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a, b, c, d ∈ Z.
2 × 2 ball replacement matrix
M = a b c d
4/26
Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: pwhite =
n n+m, pblack = m n+m.
Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a, b, c, d ∈ Z.
2 × 2 ball replacement matrix
M = a b c d
4/26
Urn contains n white and m black balls. Every discrete time steps a ball is drawn at random: pwhite =
n n+m, pblack = m n+m.
Color inspection: White - a white and b black balls are added/removed; Black - c white and d black balls are added/removed; a, b, c, d ∈ Z.
2 × 2 ball replacement matrix
M = a b c d
4/26
1
Tenable urns The process of drawing and adding/removing balls can be continued ad infinitum. We start with W0 ∈ N0 white and B0 ∈ N0 black balls: configuration Wn, Bn after n draws?
2
Diminishing urns The process of drawing and adding/removing balls stops after a finite number of steps. We start with n ∈ N0 white and m ∈ N0 black balls, define so-called absorbing states A: what is the probability of reaching a state a ∈ A?
3
Generalizations More than two colors.
AofA 2013, Menorca 5/26
P´
M = 1 0
0 1
(m,n) (m+1,n) (m,n+1)
m m+n n m+n
AofA 2013, Menorca 6/26
P´
M = 1 0
0 1
(m,n) (m+1,n) (m,n+1)
m m+n n m+n
AofA 2013, Menorca 6/26
P´
M = 1 0
0 1
(m,n) (m+1,n) (m,n+1)
m m+n n m+n
AofA 2013, Menorca 6/26
2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.)
FLAJOLET, GABARR ´
O AND PEKARI; BRENNAN AND PRODINGER; STADJE;
DUMAS, FLAJOLET AND PUYHAUBERT; HWANG, K. AND PANHOLZER; FLAJOLET
AND MORCRETTE; MAHMOUD AND MORCRETTE; MORCRETTE; . . .
Probabilistic methods (stochastic processes, martingales)
KINGMAN2; KINGMAN AND VOLKOV; MAHMOUDx; PITTEL; JANSON2; CHAUVIN, POUYANNE ET AL.3; CHEN AND WEI; RENLUND; . . .
Contraction method (for balanced urns)
NEININGER AND KNAPE; CHAUVIN, POUYANNE AND MAILLER; . . .
. . .
An urn model M = a b
c d
Consequently: Tn = Wn + Bn = T0 + n · σ.
AofA 2013, Menorca 7/26
2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.)
FLAJOLET, GABARR ´
O AND PEKARI; BRENNAN AND PRODINGER; STADJE;
DUMAS, FLAJOLET AND PUYHAUBERT; HWANG, K. AND PANHOLZER; FLAJOLET
AND MORCRETTE; MAHMOUD AND MORCRETTE; MORCRETTE; . . .
Probabilistic methods (stochastic processes, martingales)
KINGMAN2; KINGMAN AND VOLKOV; MAHMOUDx; PITTEL; JANSON2; CHAUVIN, POUYANNE ET AL.3; CHEN AND WEI; RENLUND; . . .
Contraction method (for balanced urns)
NEININGER AND KNAPE; CHAUVIN, POUYANNE AND MAILLER; . . .
. . .
An urn model M = a b
c d
Consequently: Tn = Wn + Bn = T0 + n · σ.
AofA 2013, Menorca 7/26
2000-today: several approaches - very interesting developments Analytic Combinatorics (Symbolic methods, generating functions, etc.)
; ; ; ; ; ; ; . . .
Probabilistic methods (stochastic processes, martingales)
; ; ; ; ; ; ; ; . . .
Contraction method (for balanced urns)
; ;. . .
. . .
An urn model M = a b
c d
Consequently: Tn = Wn + Bn = T0 + n · σ.
AofA 2013, Menorca 7/26
Previously: Urn contains w white and b black balls. Draw at random a single ball: pwhite =
w w+b, pblack = b w+b.
p{k times white, (m−k) times black } =
1
(b + w)m m k
with xs = x(x − 1) . . . (x − s + 1). Depending on the drawn multiset
CHEN AND WEI 2005: Generalized P´
MAHMOUD 2008: Tenable balanced linear urns (m = 2). RENLUND 2010: Stochastic approximation for tenable urns.
AofA 2013, Menorca 9/26
c c
Generalization: If we draw {WkSm−k} we add k · c white and (m − k) · c black balls, with c ∈ N. (m + 1) × 2-matrix
M = mc (m − 1)c c . . . . . . c (m − 1)c mc
Urn is balanced Tn = Wn + Bn = nmc + T0.
AofA 2013, Menorca 10/26
c c
Generalization: If we draw {WkSm−k} we add k · c white and (m − k) · c black balls, with c ∈ N. (m + 1) × 2-matrix
M = mc (m − 1)c c . . . . . . c (m − 1)c mc
Urn is balanced Tn = Wn + Bn = nmc + T0.
AofA 2013, Menorca 10/26
c c
Generalization: If we draw {WkSm−k} we add (m − k) · c white and k · c black balls, with c ∈ N. (m + 1) × 2-matrix
M = mc c (m − 1)c . . . . . . (m − 1)c c mc
AofA 2013, Menorca 11/26
c c
Generalization: If we draw {WkSm−k} we add (m − k) · c white and k · c black balls, with c ∈ N. (m + 1) × 2-matrix
M = mc c (m − 1)c . . . . . . (m − 1)c c mc
AofA 2013, Menorca 11/26
Theorem (CHEN AND WEI 2005) For the generalized P´
draws satisfies Wn = Wn Tn
− − → W∞; W∞ is absolutely continuous. Questions (CHEN AND WEI): (1) Is W∞ beta-distributed? (2) Explicit results concerning Wn and W∞;
AofA 2013, Menorca 13/26
Theorem (CHEN AND WEI 2005) For the generalized P´
draws satisfies Wn = Wn Tn
− − → W∞; W∞ is absolutely continuous. Questions (CHEN AND WEI): (1) Is W∞ beta-distributed? (2) Explicit results concerning Wn and W∞;
AofA 2013, Menorca 13/26
Theorem (CHEN AND K.) The expectation and the variance of Wn are given by E(Wn) = W0
T0 (nmc + T0), and V(Wn) = E(W2 n) − E(Wn)2 via
E(W2
n) =
n−1+λ1
n
n−1+λ2
n
mc
n
n−1+ T0−1
mc
n
+ W0c2m T0
n−1
ℓ + T0−m
mc
ℓ + T0−1
mc
ℓ+ T0
mc
ℓ+1
ℓ+ T0−1
mc
ℓ+1
ℓ+1
ℓ+λ2
ℓ+1
with λ1, λ2 given by λ1,2 = −1
2 + mc + T0 ± 1 2
mc
.
AofA 2013, Menorca 14/26
Theorem (CHEN AND K.) For s 1 the moment E(Ws
n) is given by
E(Ws
n) =
n−1
αj,s
0 + n−1
βℓ,s ℓ
j=0 αj,s
with αn,s, βn,s determined by αn,s =
s
cℓ s
ℓ
m
ℓ
ℓ
, βn,s =
s
E(Ws+1−i
n
)
s
s ℓ
ℓ
(−1)j+i−ℓ−1 ℓ
j
ℓ+1−i
m
j
j
Here n
k
n
k
second kind, respectively.
AofA 2013, Menorca 15/26
Wn
(d)
= Wn−1+c·ξn, ξn
(d)
= Hypergeometric(m, Wn−1, Bn−1); in other words: P
Wn−1 k )( Tn−1−Wn−1 m−k
) (Tn−1
m )
.
Hence, E(Ws
n|Fn−1) = Ws n−1+ s
s ℓ
n−1cℓ m
kℓ· Wn−1
k
Tn−1−Wn−1
m−k
m
By Vandermonde’s identity we obtain a recurrence E(Ws
n) = αn−1,s · E(Ws n−1) + βn−1,s,
n, s 1, (1) which leads to the stated result.
AofA 2013, Menorca 16/26
Wn
(d)
= Wn−1+c·ξn, ξn
(d)
= Hypergeometric(m, Wn−1, Bn−1); in other words: P
Wn−1 k )( Tn−1−Wn−1 m−k
) (Tn−1
m )
.
Hence, E(Ws
n|Fn−1) = Ws n−1+ s
s ℓ
n−1cℓ m
kℓ· Wn−1
k
Tn−1−Wn−1
m−k
m
expand kℓ = ℓ
j=1
ℓ
j
By Vandermonde’s identity we obtain a recurrence E(Ws
n) = αn−1,s · E(Ws n−1) + βn−1,s,
n, s 1, (1) which leads to the stated result.
AofA 2013, Menorca 16/26
Wn
(d)
= Wn−1+c·ξn, ξn
(d)
= Hypergeometric(m, Wn−1, Bn−1); in other words: P
Wn−1 k )( Tn−1−Wn−1 m−k
) (Tn−1
m )
.
Hence, E(Ws
n|Fn−1) = Ws n−1+ s
s ℓ
n−1cℓ m
kℓ· Wn−1
k
Tn−1−Wn−1
m−k
m
By Vandermonde’s identity we obtain a recurrence E(Ws
n) = αn−1,s · E(Ws n−1) + βn−1,s,
n, s 1, (1) which leads to the stated result.
AofA 2013, Menorca 16/26
Wn
(d)
= Wn−1+c·ξn, ξn
(d)
= Hypergeometric(m, Wn−1, Bn−1); in other words: P
Wn−1 k )( Tn−1−Wn−1 m−k
) (Tn−1
m )
.
Hence, E(Ws
n|Fn−1) = Ws n−1+ s
s ℓ
n−1cℓ m
kℓ· Wn−1
k
Tn−1−Wn−1
m−k
m
By Vandermonde’s identity we obtain a recurrence E(Ws
n) = αn−1,s · E(Ws n−1) + βn−1,s,
n, s 1, (1) which leads to the stated result.
AofA 2013, Menorca 16/26
Corollary (CHEN AND K.) The limits limn→∞ E(Ws
n/ns) exist and can be recursively
calculated: limn→∞
E(Wn) n
= W0mc
T0
, and
lim
n→∞
E(W2
n)
n2 = Γ( T0
mc)Γ( T0−1 mc )
Γ(λ1)Γ(λ2) ×
0 + W0c2m
T0
∞
ℓ + T0−m
mc
ℓ + T0−1
mc
ℓ+ T0
mc
ℓ+1
ℓ+ T0−1
mc
ℓ+1
ℓ+1
ℓ+λ2
ℓ+1
AofA 2013, Menorca 17/26
(CHEN AND K.): draw m 1 balls with replacement,
p{k times white, (m−k) times black } =
1
(b + w)m m k
Sampling scheme influences the (limiting) distribution. Theorem The expected value coincides: E(Wn) = W0
T0 (nmc + T0); but for the
second moment E(W2
n) =
n−1+µ1
n
n−1+µ2
n
mc
n
2
0 + W0c2m
T0
n−1
ℓ+ T0
mc
ℓ+1
2 ℓ+µ1
ℓ+1
ℓ+µ2
ℓ+1
mc
.
AofA 2013, Menorca 18/26
mc . . . . . . mc We obtain for both sampling schemes a unified result. Theorem (MAHMOUD, PANHOLZER AND K.) The number of white balls Wn after n draws satisfies Wn n
(a.s.)
− − − − → cm
2 ,
n → ∞. Furthermore, Wn − 1
2cmn − 1 2T0
√n
(d)
− − → N
12c2m
with convergence of all moments.
AofA 2013, Menorca 19/26
M = a0 b0 a1 b1
. . . . . .
am−1 bm−1 am bm
.
Draw a multiset {WkSm−k} : we add ak white and bk black balls.
AofA 2013, Menorca 21/26
Case m = 1: draw a single ball 2005 FLAJOLET, GABARR ´
O AND PEKARI: first order PDE for
balanced urns 2006 DUMAS, FLAJOLET AND PUYHAUBERT: differential systems for balanced urns 2010-2011 FLAJOLET AND MORCRETTE: AC for unbalanced urns 2013 MORCRETTE: first order PDE for unbalanced urns! Case m = 2: July 2010 FLAJOLET Second order PDE for Bernoulli-Laplace urn
AofA 2013, Menorca 22/26
Balanced urns: M = a0 b0 a1 b1
. . . . . .
am−1 bm−1 am bm , σ = ai + bi. Approach
differential operators: ∂z: differential operator with respect to z; Θz = z · ∂z. Assume we have w white and b black balls = ⇒ xwyb: ym−k∂m−k
y
xk∂k
x(xwyb) =
xwyb (b + w)m m k
Θk
xΘm−k y
(xwyb) = xwyb (b + w)m m k
AofA 2013, Menorca 23/26
We introduce the differential operators DM =
m
m k
x∂m−k y
,
and DR =
m
m k
xΘm−k y
.
Proposition Starting with W0 white and B0 black balls the generating function
n0 DnxW0yB0 zn (n!)m satisfies
D ∗ H(x, y, z) = 1 zΘm
z ∗ H(x, y, z),
with D = DM (without replacement) or D = DR (with replacement). Further simplifications using Θx + Θy = W0 + B0 + σΘz.
AofA 2013, Menorca 24/26
Similar results for diminishing urn models; some second order PDEs are explicitly solvable (reduction to first order): M =
−1 0 −1
−1 −1 −1
−1 0 1 −2
−1 1 −2
M =
−1 0 −2
1 0
1 0 0 1
explicit results, moments, limit laws, . . . Approach of Morcrette ⇒ higher order PDE for unbalanced urns with multiple drawings. . . Solvable higher order PDEs stemming from urn models
AofA 2013, Menorca 25/26
AofA 2013, Menorca 26/26
AofA 2013, Menorca 26/26