This document summarizes dependent processes in Bayesian nonparametrics. It motivates the need for dependent random probability measures to accommodate temporal dependence structures beyond the exchangeability assumption. It describes modeling collections of random probability measures indexed by time as either discrete-time or continuous-time processes. The diffusive Dirichlet process is introduced as a dependent Dirichlet process with Dirichlet marginal distributions at each time point and continuous sample paths. Simulation and estimation methods are discussed for this model.

1. Dependent processes
in Bayesian nonparametrics
Matteo Ruggiero
University of Torino and Collegio Carlo Alberto
Moncalieri, Feb 19 2016
0.0 0.2 0.4 0.6 0.8 1.0
time 1
0
0.029
0.059
0.088

2. 1. Motivation and general setting
BNP and discrete random probability measures
p = (p1, p2, . . .) frequencies in
∆∞ = p ∈ [0, 1]∞
:
i
pi = 1
p↓
= (p(1), p(2), . . .) ordered frequencies in
∞ = p ∈ [0, 1]∞
: p1 ≥ p2 ≥ · · · ≥ 0,
i
pi = 1
Assign law to p, which induces a distribution
on ∆∞, ∞
Otherwise assign to the indices unique labels
X1, X2, . . .
iid
∼ P0 continuous on X and define
the discrete measure
∞
i=1
piδXi
which induces a distribution on P(X)
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 3

3. 1. Motivation and general setting
BNP and discrete random probability measures
Approach 1:
model observations Yj directly with
p = (p1, p2, . . .) or P =
∞
i=1
piδXi
where Yj = Xi w.p. pi, and the (Xi, pi) are random
Approach 2:
use mixtures to yield more flexibility and possibly aim at continuous
distributions
f(y) =
X
f(y | x)P(dx) ⇒ f(y) =
∞
i=1
pif(y | Xi)
i.e. Yj ∼ f(y | Xi) w.p. pi and the (Xi, pi) are random
Use either approach as a base for estimation, uncertainty quantification,
forecasting, clustering, . . .
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 4

4. 1. Motivation and general setting
Motivation for dependent processes
Assumptions in classical BNP approach:
observations are excheangeable
observations depend on a fixed environment/state of the world
inference is static (fixed time)/carried out on single environment
Data may not satisfy these assumptions (e.g. prices dynamics)
Need for more general types of dependence
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 5

5. 1. Motivation and general setting
Partial exchangeability
Natural extension is partial exchangeability (de Finetti sense), e.g.




X1,1 X1,2 X1,3 · · ·
X2,1 X2,2 X2,3 · · ·
X3,1 X3,2 X3,3 · · ·
· · · · · · · · · · · ·




row-wise exchangeability (not overall): given i, Xi,j are exchangeable
Accommodates e.g. temporal structures
Collection of random probability measures, indexed by some covariate
Can be extended to an uncountable family
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 6

6. 1. Motivation and general setting
Dependent densities: discrete time
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 7

7. 1. Motivation and general setting
Dependent densities: discrete time
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 8

8. 1. Motivation and general setting
Dependent densities: continuous time
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 9

9. 1. Motivation and general setting
Dependent densities: continuous time
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 10

10. 1. Motivation and general setting
Dependent densities: continuous time
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 11

11. 1. Motivation and general setting
Modelling and inference with time-dependent processes
Temporal dependence structure
Partial exchangeability, for any t we have a distribution (possibly a mixture)
(Possibly multiple) data available at discrete time points
Model collection of random probability measures, forming
a discrete time process, or
a continuous-time process, with continuous paths or jumps
Nonparametric approach to allow for full flexibility
Analyse properties of the resulting model
Devise suitable strategies for
posterior computation
Carry out inference on desired quantities
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 12

12. 1. Motivation and general setting
General setting
X1, X2, . . .
iid
∼ P0 unique labels or locations in X
We are interested in time-dependent random probability measures of type
p(t) = (p1(t), p2(t), . . .) ∈ ∆∞
p↓
(t) = (p(1)(t), p(2)(t), . . .) ∈ ∞
P(t) =
∞
i=1
pi(t)δXi(t) ∈ P(X)
where t ≥ 0 represents time.
Discrete sample paths:
p, p↓
, P are countable collections of distributions, t ∈ N
Continous sample paths:
p, p↓
, P are (random) t-continuous functions from [0, ∞) to ∆∞, ∞ or P(X)
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 13

13. 2. Diffusive Dirichlet mixture models
Dirichlet process
The Dirichlet process [Ferguson 1973] extends the Dirichlet distribution from K
to infinitely many types
Can be defined via stick-breaking [Sethuraman 1994]
Vi
iid
∼ Beta(1, θ), pi = Vi
i−1
k=1
(1 − Vk)
0 1
p1 = V1 1 − V1
V2
p2 (1 − V1)(1 − V2)
V3
...
s.t. pi → 0 as i → ∞ and i≥1 pi = 1
Take Xi
iid
∼ P0 with P0 continuous on X.
Then P = ∞
i=1 piδXi is a Dirichlet process
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 15

14. 2. Diffusive Dirichlet mixture models
Dirichlet process
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
x
p
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 16

15. 2. Diffusive Dirichlet mixture models
Dependent Dirichlet process
Basic idea [MacEachern, 1999]
We aim at defining a process
P(t) =
∞
i=1
pi(t)δXi(t), t ≥ 0,
with Dirichlet process marginals
Handling both (p1(t), p2(t), . . .) and (X1(t), X2(t), . . .) can be non trivial.
Consider instead
P(t) =
∞
i=1
pi(t)δXi , t ≥ 0, Xi
iid
∼ P0
Atoms are fixed, but there are infinitely many of them
In practice, as many as you need
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 17

16. 2. Diffusive Dirichlet mixture models
Diffusive Dirichlet process
Take the Dirichlet stick-breaking weights
pi = Vi
i−1
k=1
(1 − Vk), Vi ∼iid
Beta(1, θ)
Substitute each component Vi ∈ [0, 1] with a diffusion {Vi(t)}t≥0 on [0, 1]
Then take
pi(t) = Vi(t)
i−1
k=1
(1 − Vk(t))
Each component needs to have Beta marginals, Vi(t) ∼ Beta(1, θ)
One-dimensional Wright–Fisher diffusions satisfy this
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 18

17. 2. Diffusive Dirichlet mixture models
Wright–Fisher diffusions
0 2 4 6 8 10
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 19

18. 2. Diffusive Dirichlet mixture models
Wright–Fisher diffusions
% of type 1 individuals (mutation rates: theta_1 = 2 , theta_2 = 8 )
Time (50K steps)
Statespace
0 2 4 6 8 10
0
1
Ergodic frequencies against Stationary Distribution Beta( 2 , 8 )
State space
0.0 0.2 0.4 0.6 0.8 1.0
0123
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 20

19. 2. Diffusive Dirichlet mixture models
Wright–Fisher diffusions
% of type 1 individuals (mutation rates: theta_1 = 8 , theta_2 = 8 )
Time (50K steps)
Statespace
0 2 4 6 8 10
0
1
Ergodic frequencies against Stationary Distribution Beta( 8 , 8 )
State space
0.0 0.2 0.4 0.6 0.8 1.0
0.01.53.0
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 21

20. 2. Diffusive Dirichlet mixture models
Wright–Fisher diffusions
% of type 1 individuals (mutation rates: theta_1 = 0.4 , theta_2 = 0.4 )
Time (50K steps)
Statespace
0 2 4 6 8 10
0
1
Ergodic frequencies against Stationary Distribution Beta( 0.4 , 0.4 )
State space
0.0 0.2 0.4 0.6 0.8 1.0
048
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 22

21. 2. Diffusive Dirichlet mixture models
Diffusive Dirichlet process [Mena and R. 2016]
The resulting object
P(t) =
∞
i=1
Vi(t)
i−1
k=1
(1 − Vk(t))
pi(t)
δXi , Vi(t) ∼ WF(a, b)
has Dirichlet marginals for (a, b) = (1, θ), i.e. P(t) is a DP for all t
has GEM marginals for (a, b) ∈ R2
+
has diffusive behaviour, P(t) is t-continuous in total variation
See also
Gutierrez, Mena and & R. 2016 (version with jumps)
Mena, R. & Walker 2011 (geometric weights, different marginals)
for related models
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 23

22. 2. Diffusive Dirichlet mixture models
Diffusive Dirichlet process
0.0 0.2 0.4 0.6 0.8 1.0
time 1
0
0.029
0.059
0.088
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 24

23. 2. Diffusive Dirichlet mixture models
Estimation
At each time ti we have observations (yi,1, . . . , yi,ni ).
Set up the hierarchical mixture
{Pt, t ≥ 0} ∼ diff-DP or GSB
xti | Pti ∼ Pti
yi,j | ti, xti
iid
∼ f(· | xti )
equivalently yi is drawn from the time-dependent nonparametric mixture model
fti (y) =
X
f(y|x)Pti (dx) =
∞
i=1
pti f(y | xi)
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 25

24. 2. Diffusive Dirichlet mixture models
Simulated data
True model
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 26

25. 2. Diffusive Dirichlet mixture models
Simulated data
Single data points
0 2 4 6 8 10
−202468
True model (heat map), posterior mode (solid), 95% credible intervals for the mean (dashed), 95%
quantiles of posterior density estimate (dotted).
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 27

26. 2. Diffusive Dirichlet mixture models
Simulated data
Multiple data points
0 2 4 6 8 10
−202468
True model (heat map), posterior mode (solid), 95% credible intervals for the mean (dashed), 95%
quantiles of posterior density estimate (dotted).
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 28

27. 2. Diffusive Dirichlet mixture models
Real data: S&P 500 (03/08 - 02/09)
Dependent density estimate
Heat map of estimated density (red), and mean estimate (solid)
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 29

28. 2. Diffusive Dirichlet mixture models
Real data: S&P 500 (03/08 - 02/09)
Dependent density estimate
160 170 180 190 200
80090010001100
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 30

29. 2. Diffusive Dirichlet mixture models
Real data: S&P 500 (03/08 - 02/09)
Dependent density estimate
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 31

30. populations
A different view: modelling evolving populations
A sample path of p↓
(t) = (p(1), . . . , p(7))
Time
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency
Dynamic frenquencies of 7 species
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 33

31. populations
A different view: modelling evolving populations
Distinct values X1, X2, . . . are interpreted as
allelic types in genetics
plant or animal species
unique identifiers of some evolving groups
Large population → species abundances approximate diffusive behaviours
If cannot provide an a priori upper bound, assume infinitely many species
Two different approaches:
constructing stochastic models for pseudo-realistic evolutionary mechanisms
(mutation, selection, recombination, migration, . . . )
studying the association between certain
distributions and connected dynamics
Dynamics in figure are related to
a Dirichlet distribution
Can we extend them? To what extent?
With what interpretation?
Time
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency
Dynamic frenquencies of 7 species
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 34

32. populations
Wright–Fisher signals: Dirichlet-Multinomial model
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 35

33. populations
Poisson-Dirichlet case
No. species Markov chain
(N individuals)
K
Wright-Fisher(N, K, θ)
Fisher (1930), Wright (1931)
Diffusion
(∞ individuals)
d
N → ∞
Wright-Fisher(K, θ)
Sato (1976)
stationary
w.r.t.
Dir θ
K , . . . , θ
K
Random measure
(t fixed)
∞ IMNA(θ)
Ethier and Kurtz (1981)
d K → ∞
PD(θ)
Kingman (1975)
d K → ∞
stationary
w.r.t.
Moran(N, θ)
Watterson (1976)
d
N → ∞
“
d
−→” = convergence in distribution
IMNA = infinitely many neutral alleles
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 36

34. populations
Two-parameter Poisson-Dirichlet case
No. species
∞ PD(θ, α)
Pitman (1995)
Random measure
(t fixed)
Diffusion
(∞ individuals)
IMNA(θ, α)
Petrov (2009)
stationary
w.r.t.
?? Moran(N, θ, α)
R. and Walker (2009)
d
N → ∞
Markov chain
(N individuals)
?? WF(K, θ, α)
Costantini, De Blasi,
Ethier, R., Span`o (2016)
d K → ∞
K ?? WF(N, K, θ, α)
Costantini, De Blasi,
Ethier, R., Span`o (2016)
d
N → ∞
stationary
w.r.t. ??
d K → ∞
Remarks:
IMNA = infinitely many neutral allelesBased on Pitman’s generalized P´olya urn schemeMutation and immigration
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 37

35. Continuous-time Gamma-Poisson model
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CIR path X_t
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Poisson(X_t) likelihood
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 39

36. Continuous-time Gamma-Poisson model
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 40

37. Continuous-time Gamma-Poisson model
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 41

38. The propagation mixture
Prior X ∼ πα := Gamma(α1, α2)
Likelihood Y | X ∼ Poisson(X)
Posterior X | Y1, . . . , Yn ∼ πα,n := Gamma α1 +
n
i=1
yi, α2 + n
Propagation mixture [Papaspiliopoulos & R. 2014]
ψt(πα,n) := πα,n(x)Pt(x, dx )
is given by
ψt(πα,n) =
n
j=0
pt(n, j)Gamma α1 +
n
i=0
yi − j, α2 + n − st
for appropriate time-varying weights pt(n, j)
Can be extended to infinite dimensional models [Papaspiliopoulos, R. & Span`o
2016]
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 42

39. Continuous-time Gamma-Poisson model
0 1 2 3 4 5 6 7
0.1
0.2
0.3
0.4
0.5
t t0
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 43

40. Continuous-time Gamma-Poisson model
0 1 2 3 4 5 6 7
0.1
0.2
0.3
0.4
0.5
t t0
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 44

41. Some references
Costantini, De Blasi, Ethier, R. and Span`o (2016).
Wright–Fisher construction of the two-parameter Poisson–Dirichlet diffusion.
arXiv:1601.06064
Gutierrez, Mena & R. (2016).
A time dependent Bayesian nonparametric model for air quality analysis.
Comput. Statist. Data Anal.
Mena & R. (2016).
Dynamic density estimation with diffusive Dirichlet mixtures. Bernoulli
Mena, R. & Walker (2011).
Geometric stick-breaking processes for continuous-time Bayesian nonparametric modeling.
J. Statist. Plann. Inf.
Papaspiliopoulos & R. (2014).
Optimal filtering and the dual process. Bernoulli
Papaspiliopoulos, R. & Span`o (2014).
Filtering hidden Markov measures. arXiv:1411.4944
R. & Walker (2009).
Countable representation for infinite dimensional diffusions derived from the
two-parameter Poisson–Dirichlet process. Electr. Comm. Probab.
For more info: www.matteoruggiero.it
Matteo Ruggiero (Unito & CCA) Dependent processes in BNP 45