Submitted to the Annals of Statistics
arXiv: arXiv:1710.01552

BAYESIAN INFERENCE FOR STATIONARY DATA ON
FINITE STATE SPACES

arXiv:1710.01552v3 [] 23 Oct 2017

By F. Moritz von Rohrscheidt∗ ,
Ruprecht-Karls-Universität Heidelberg
In this work the issue of Bayesian inference for stationary data is
addressed. Therefor, a parametrization of a statistically appropriate
subspace of the shift-ergodic probability measures on a Cartesian
product of some finite state space is given using an inverse limit
construction. Moreover, an explicit model for the prior is given by
taking into account an additional step in the usual stepwise sampling
scheme of data. An update to the posterior is defined by exploiting
this augmented sample scheme. Thereby, its model-step is updated
using a measurement of the empirical distances between the classes
of considered models.

1. Introduction. During the 20th century and beyond, the theory of
stochastic processes has created a huge field of models which study and unify
a wide range of real world problems. This field is also the natural habitat
for the statistician who infers the data-generating mechanism by means of
observed data. Thereby, keeping track of large sample accuracy of inference
procedures, inference for processes evolving in discrete time - in contrast
to high-frequency approaches to data evolving in continuous time - ought
to require stationarity as a minimal assumption on the model. Otherwise,
if the data-generating mechanism changes during time, the statistician either has only access to non-extendable data or must incorporate this change
as prior knowledge which is often not available. However, unless additional
assumptions on the structure of the data is made, the space of probability
measures that could have generated stationary data is overwhelmingly large.
Most prominent assumptions on the structure of the data which reduce the
space of considered probability measures were studied in depth and constitute of symmetry constraints like exchangeability, partial exchangeability
and others; c.f. Kallenberg (2006). As the perhaps greatest advocate of the
concept of subjectivistic probability theory of the 20th century, Bruno de
Finetti has intensively emphasized the idea of exchangeability in his book
de Finetti (1974) as well as in a published conservation with Hugo Hamaker,
∗

Supported by the DFG (German research foundation); grant 1953
MSC 2010 subject classifications: Primary 60M05, 60M09; secondary 28D99
Keywords and phrases: Bayesian statistics, stationary data, parametrization of shiftergodic measures

1
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F.M. VON ROHRSCHEIDT

see de Finetti (1979). The recurring argument is that the data should have
attached some judgment which possesses an actual meaning. Roughly speaking, the judgment of exchangeability means that the order of collected data
is not necessary information and hence does not affect statistical inference.
For the case of binary data de Finetti (1937) proved the celebrated representation theorem which states that any sequence of random variables is
exchangeable if and only if it is representable as the mixture of sequences
of independent and identically distributed random variable. Thereby the
mixture is understood in form of an integral w.r.t. a measure that is interpreted to represent ones uncertainty in the true data-generating measure.
This subjectivistic approach to probability paves the way for an epistemic
kind of inference that enables one to probabilistically predict future outcomes based on past observations. This is mainly due to the fact that exchangeable data, unlike independent data, are in general not uncorrelated,
c.f. Kingman (1978).
Subsequently de Finetti’s theorem was generalized in many respects. Hewitt and Savage (1955) generalized the underlying state space of the data
to Polish spaces. Aldous (1981) introduced the concept of exchangeability to arrays of random numbers in order to prove a conjecture of Dawid
(1978) about the representability of spherically invariant random matrices.
Another important generalization was given with respect to sequences of
random variables whose joint law is only invariant with respect to a suitable subfamily of the symmetric group. This subfamily can be thought of
appropriate block-transformations of the coordinates which do not affect
the number of transitions the process makes between the states. De Finetti
already hinted to the fact that those sequences are representable as mixtures of Markov chains and Diaconis and Freedman (1980) gave the rigorous
proof. This is based on the idea of the exchangeability of the successor blocks
of one recurrent state, showing that recurrence is a necessary condition for
such mixtures. However, recurrence is certainly implied by some overruling
statistical judgment, namely stationarity. Hence, it not surprising that mixtures of Markov chains can be studied from a statistical viewpoint which
was done in Freedman (1962). Freedman showed even more; any stationary
probability measure on a Cartesian product of some countable state space
which is summarized by a statistic possessing S-structure is representable as
an integral mixture of shift-ergodic probabilities being also summarized by
this statistic. Thereby, S-structure is a certain equivalence-invariance under stacking the data. Since transition counts have S-structure and since
Markov measures are shown to equal the shift-ergodic probabilities being
summarized by transition counts, Freedman (1962) obtained the de Finetti
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BAYESIAN INFERENCE FOR STATIONARY DATA

3

theorem for Markov chains. Related ideas dealing with statistical mixtures
in an extended way can be found in Lauritzen (1988), see also Schervish
(1995) for a well accessible source.
More recently, the Italian school took up the issue of mixing up probability measures in further depth. Fortini et al. (2002) confirm a conjecture of
de Finetti which says that a recurrent probability on a Cartesian product
of some countable state space is a mixture of Markov measures if and only
if the associated matrix of successor-states is exchangeable within rows. In
this work they also considered the more challenging case of an uncountable
state space by using suitable weak approximations of the appearing kernels.
Epifani, Fortini and Ladelli (2002) extends related ideas to semi-Markov processes. The works Fortini, Ladelli and Regazzini (2000); Fortini and Petrone
(2014) characterize mixtures of probabilities using predictive distributions.
Moreover, in Fortini and Petrone (2014) it is proven that the push-forward
of a prior under an appropriate parametrization of the Markov measures
samples the rows of the associated stochastic matrix independently if and
only if transition counts are predictively sufficient. In Fortini and Petrone
(2012a,b) the authors shed light on the construction of priors obtained as
weak limits of predictive schemes similar to the one by Blackwell and MacQueen (1973) yielding the Dirichlet process prior. These constructions yield
priors concentrated on the space of Markov measures using reinforced urn
processes. In Diaconis and Rolles (2006) the construction of conjugate priors
for reversible Markov measures is given and Bacallado et al. (2011) extend
these ideas to higher-order Markov measures and provide procedures for
testing the order. Besides of that, Lijoi, Prünster and Walker (2007) give a
prove of a Doob-type consistency result for stationary models which, as one
would expect, rely on martingale convergence arguments. It is interesting
to see that theory can provide such a deep result which, at the same time,
lacks applications in form of highly general models. This lack can best be
explained by the overwhelming richness of the space of shift-ergodic measures that serves as the parameter space. Roughly, this richness relies on the
fact that a shift-ergodic probability measures in general does not allow for a
parametrization in a sequence of stochastic kernels which is ”Noetherian“of
any degree, in the sense that it eventually remains constant. Berti et al.
(2000); Berti and Rigo (2003) underline this arguing that the appropriate
mixing measure (occasionally called the Maitra measure in honor of Ashok
Maitra, see Maitra (1977)) can be obtained as the law of the weak limit of
a suitably chosen sequence of empirical measures increasing in the degree of
dependence. This can also be reflected by considering invariance of probability measures with respect to transformations consisting of interchanging
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F.M. VON ROHRSCHEIDT

blocks (c.f. (Diaconis and Freedman, 1980, Proposition 27)) whose initial and
terminal strings of states, which have to coincide within the blocks to be admissible, increase continuously in their lengths. From a rather philosophical
point of view Bayesian inference for stationary data is quite challenging because one has to express one’s uncertainty in the structure of the data itself
and has to learn for it from one single series of experiments.
Besides calling back the issue to mind, the aim of the present work is
twofold. On the one hand it gives a parametrization of of a statistically
suitable subspace of the shift-ergodic measures on the Cartesian product
of some finite state space. On the other it provides an explicit model for
a prior exhausting the arising parameter space as well as for its update to
the posterior. For the former, above-mentioned idea of Berti et al. (2000)
motivates the construction of the parameter space in terms of an appropriate
limit. The model itself is then constructed by extending the usual step-wise
sampling scheme, i.e. first sample the data-generating mechanism and then
sample the data from it, by adding an additional step incorporating the
degree of dependence among the data. The considered model can be argued
to be non-informative in a specific way.
The paper is organized as follows. In section 2 the necessary preliminaries
and notations are given. Section 3 deals with the parametrization of the
shift-ergodic measures. Section 4 provides the explicit Bayesian model for
stationary data. Section 5 concludes with an outlook to and a discussion of
further issues.
2. Preliminaries. Let S be a finite space with cardinality #S = s. As
usual, equip the Cartesian product S N = ∏n∈N S with the product σ-field S
generated by the semi-algebra consisting of all cylinder sets of the form
[x1 , x2 , . . . , xk ] = {x ∈ S N ∶ πj (x) = xj ; j = 1, . . . , k; k ∈ N} ,
where πj denotes the projection onto the j-th coordinate. Denote by P(T )
the space of all probability measures on a Polish space (T, T ). It is well
known, see e.g. Kechris (1995), that P(T ) itself becomes a Polish space using
the appropriate Borel σ-field BP induced by weak convergence of measures.
Equivalently, BP is the smallest σ-field which makes the mappings µT ∶
P ↦ P (T ) measurable for all T ∈ T , c.f. Ghosh and Ramamoorthi (2003).
Let θ ∶ P(S N ) → P(S N ) be the shift operator on the space of probability
measures defined through
θ

P ([x1 , . . . , xk ]) z→ P ( ⋃ [j, x1 , x2 , . . . , xk ])
j∈S

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BAYESIAN INFERENCE FOR STATIONARY DATA

5

for all cylinder sets, where θ̃ denotes the shift on S N . Furthermore, call
a probability P ∈ P(S N ) stationary if θP = P holds. By Kolmogoroff’s
extension theorem, that is equal to

θP ([x1 , . . . , xk ]) = P ( ⋃ [x1 , x2 , . . . , xk , j]),
j∈S

for all cylinder sets, which amounts to an expression of stationarity in
terms of a symmetry constraint on P . Moreover, call a set I ∈ S shiftinvariant if it holds θI △ I = ∅, where △ denotes the symmetric difference.
It is easy to see that the family of invariant sets forms a σ-field I, known as
the shift-invariant σ-field. A stationary probability P is called shift-ergodic
if it holds P (I)P (I c ) = 0 for all shift-invariant sets I. Denote by P e (S N )
the space of all ergodic probability measures. By the ergodic decomposition
theorem, c.f. Varadarajan (1963); Maitra (1977), any stationary probability
is uniquely representable as an integral mixture of ergodic probabilities, i.e.
for any set A ∈ S it holds
(EDC)

P (A) = ∫

P e (S N )

Q(A)Π(dQ).

Thereby, the distribution Π ∈ P (P e (S N )), interpreted to express one’s uncertainty, is uniquely determined and serves as the prior distribution in
Bayesian statistics.
Let (Ω, A, P) some probability space and (X ∶ Ω → S N ) = (Xn ∶ Ω → S)n∈N
a discrete-time stochastic process with state space S. The law of a random
object Y with respect to P shall be written as L[Y ]. By virtue of Berti
et al. (2000), Π can be thought of as the weak limit of empirical measures
increasing by their degree of dependence N ∈ N0 which are given as
(E)

)
e(N
n (⋅) =

1 n−N
∑ δ(Xj ,...,Xj+N ) (⋅),
n − N k=1

n > N.

(⌊n/2⌋)

Precisely, they show that the sequence (en
× α)n∈N0 ∈ P N (S N ) consisting
of arbitrarily augmented empirical measures (meaning that α ∈ P(S N ) makes
(⌊n/2⌋)
en
a proper probability measure on the sequence state space) converges
weakly to the mixing measure Π as n tends to infinity for any reasonable
choice of α and for L[X]-almost all data. So, asymptotically speaking, the
extension α does not play any role in this construction of Π. Notice that the
choice for N as the integer part of n/2 is quite arbitrary such that above
(N )
sequence can be replaced by any other of the form (en n × α)n∈N0 such that
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F.M. VON ROHRSCHEIDT
n→∞

Nn ∈ N and n/Nn Ð→ ∞. This weak limit construction reflects the fact
that the shift-ergodic probability measures are summarized by higher order
transition counts are higher order Markov chains, which can easily be obtained as a slight generalization of (Freedman, 1962, Theorem 2). Additional
prior knowledge on the data in form of some additional symmetry judgment
lets Π concentrate on a proper subspace of P e (S N ). For instance, if the
data is judged to be exchangeable, the support of Π becomes the space of
i.i.d.-measures. From the viewpoint of functional analysis, Π is given as the
representing measure of the Choquet simplex P(S N ), see e.g. Phelps (2001)
for details.
By a stochastic kernel from one measurable space (S, S) to another (T, T )
it is meant a measurable mapping κ ∶ S → P(T ).
3. Parametrization of shift-ergodic measures. The present section
is devoted to a decent parametrization of a statistically reasonable subset of the the shift-ergodic measures. The need for such a parametrization
arises from the ergodic decomposition (EDC) which pinpoints the space
of shift-ergodic measures as the appropriate parameter space for the issue
of Bayesian statistics for stationary data. Under additional judgments of
the data the parameter space reduces considerably, c.f. Diaconis and Freedman (1980); Diaconis and Ylvisaker (1985); Diaconis and Freedman (1987).
However, besides taking values in a finite state space, such an additional
constraint on the data is not assumed here.
Roughly, the idea of the mentioned parametrization can be described as
follows. The Ionescu-Tulcea theorem (see e.g. Klenke (2013)) is exploited in
order to parametrize a probability measure on a Cartesian product of the
state space in terms of a sequence of kernels. In the case of a Markov measure,
this sequence of kernels is basically constant and the determining kernel of
degree one can be thought of as a stochastic matrix. In full analogy, in the
general case, one can think of the higher-dimensional kernels as multi-linear
mappings which are introduced in this section as so called stochastic tensors.
By flattening these tensors according to some reasonable convention, see e.g.
Landsberg (2012), one can also think of these tensors as a stochastic matrix
determining a Markov chain evolving in a higher-dimensional space. As in
the case of a usual Markov chain it is then shown that stationarity of the
parametrized probability measure together with a statistically reasonable
constraint on the parametrizing tensors implies the existence of a family of
non-linear mappings such that some stochastic tensor is always given as the
image of the one of next higher dimension under the associated mapping. The
parameter space is then defined as the inverse limit alongside the sequence

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BAYESIAN INFERENCE FOR STATIONARY DATA

7

of kernels and the obtained family of mappings. Notice that inverse limit
extensions have been used successfully in stochastics and statistics since
Kolmogoroff’s extension theorem. They always give rise to the existence of
some abstract limiting object which, loosely speaking, can be seen to form
a kind of closure extending finite procedures. More recent work on statistics
using the idea of inverse limits can be found in Orbanz (2010, 2011). In
the first paper, the author generalizes the idea of a natural conjugate prior
for exponential families to the nonparametric case by pinning conjugate
priors for all finite dimensional marginals and subsequently extending it by
an inverse limit argument. The second paper deals with the construction
of random probability measures similar to but in greater generality as the
construction of the Dirichlet process.
Let S = {1, . . . , s} be the state space of the data and let be given a sequence
of kernels
(κN ∶ S N × Pot(S) → [0, 1])N ∈N

0

increasing by the degree of dependence N . Thereby the convention is used
that κ0 is an initial probability measure on the state space. By the IonescuTulcea theorem there is one and only one probability measure Q ∈ P(S N )
on the Cartesian product of the state space such that Q = ⊗m∈N0 κm . In
accordance to the Markov case the stochastic kernels are written as κN =
N
(prs11 ⋯r
⋯sN )ri ,sj ,∈S;i,j=1,...,N , where the contra-variant index (r1 , . . . , rN ) denotes
the position right before and the co-variant index (s1 , . . . , sN ) the position
right after a jump of an N -dimensional process evolving in discrete time.
Thus, in order to model a process of data possessing dependence of length
N , it is assumed to hold
N
prs11 ⋯r
⋯sN

⎧
⎪
if (r1 , . . . , rN −1 , rN ) ≠ (r2 , . . . , rN , sN )
⎪= 0,
.
⎨
⎪
⎪
⎩∈ [0, 1], otherwise
2N

Letting for N ≥ 1 denote KN ⊂ [0, 1]s the space of stochastic tensors
of degree N , equip KN with the topology of coordinate-wise convergence
TN in order to obtain a topological space (KN , TN ) and a measurable space
(KN , BN ), respectively, where BN is the TN -associated Borel σ-field. The
subsequent result establishes the meaning of stationarity in terms of the
parametrizing sequence of kernels.
Theorem 3.1.

Let (κN ∶ S N × Pot(S) → (0, 1))N ∈N be a sequence of pos0

itive stochastic tensors and Q = ⊗N ∈N0 κN ∈ P(S N ). Then, stationarity of Q
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F.M. VON ROHRSCHEIDT

implies the unique existence of a family of continuous mappings
(φN ∶ KN → KN −1 )N ∈N ,
such that φN (κN ) = κN −1 . Moreover, if Q is stationary it is also shiftergodic.
Proof. Let N be a fixed but arbitrarily chosen positive integer, r ∶= sN
and κN a positive stochastic tensor encoding dependence of length N . Applying a suitable flattening ˜⋅, κN can be thought of as a stochastic (r×r)-matrix
κ̃N governing a Markov chain on the N -th higher shift space S [N ] , c.f. Lind
and Marcus (1995). Now, positivity of κN clearly implies positive recurrence
and aperiodicity of κ̃N . From the theory of Markov chains, c.f. Freedman
(1983), it is well known that in this case there is a unique stochastic row
vector v of length r such that vκ̃N = v. Thus, in essence, there is a mapping
γN

κN z→ v,
which is continuous by virtue of Schweitzer (1968). Furthermore, define the
continuous mapping ζN by

(Stat)

⎛
⎞
ζN
vt
⎟
v = (vt )t∈S r z→ ⎜
⎜ v
⎟
∑
⎝ j∈S (j,π[1∶r−1] (t)) ⎠

π

(t)

=∶ (vπ [1∶r−1]
)
(t)
[2∶r]

,

t∈S r

t∈S r

where π[k∶l] denotes the projection onto the coordinates with indexes k ≤ i ≤
l. Now, let φN ∶= ζN ○ γN , κN −1 ∶= φN (κN ) and extending κN −1 by an appropriate amount of zero-entries. In order to close the proof, let A, B ∈ S be two
measurable sets with Q(A), Q(B) > 0, where Q is as in the claim. By positivity of the parametrizing sequence κ these sets must contain proper subsets
A′ = θ−k1 [c1 ] ∩ ⋯ ∩ θ−kn [cn ] and B ′ = θ−l1 [d1 ] ∩ ⋯ ∩ θ−lm [dm ] with positive integers kn , lm , n, m and states c1 , . . . , cn , d1 , . . . , dm . But then, θ−(lm +q) A′ ∩ B ′
is again of the same form as A′ and B ′ for any q ∈ N0 . Thus, again by
positivity of κ, it follows Q (θ−(lm +q) A′ ∩ B ′ ) > 0 which is well known to be
equivalent to ergodicity of Q.
Remark 1.
(i) The proof of Theorem 3.1 shows that the mapping φN
basically amounts to the mapping which maps a decent stochastic matrix onto its Perron-projection.

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BAYESIAN INFERENCE FOR STATIONARY DATA

9

(ii) The construction in (Stat) uses stationarity in the following way. Letting t = (t1 , . . . , tr ) with r and Q as above one has that
r−1
ptt12 ⋯t
⋯tr =

v(t1 ,...,tr )
∑ v(j,t1 ,...,tr−1

j∈S

⎞
⎛
r−1
⇔ ∑ Q([j, t1 , . . . , tr−1 ]) ptt12 ⋯t
⎠ ⋯tr
⎝j∈S
= Q([t1 , . . . , tr ]),

which holds true since it is well known that stationarity of Q is equivalent to
(*) ∑ Q([j, t1 , . . . , tr−1 ]) = Q([t1 , . . . , tr−1 ]) = ∑ Q([t1 , . . . , tr−1 , j]),
j∈S

j∈S

for all possible cylinder sets, where the second identity holds by virtue
of Kolmogorov’s extension theorem. Notice that (*) states stationarity, above defined as an invariant constraint, in form of a symmetry
condition.
(iii) Theorem 3.1 makes obvious the well known fact that the stationary (ergodic) probability measures on a Cartesian product is (weakly) nowheredense in the space of all (the extreme points of all) probability measures
on that sequence space.
(iv) Continuity of the mappings φN will below be used to induce a measurable structure on the parameter space.
The following example establishes the mappings φ1 , φ2 , φ3 in the case of
binary data.
Example 1. Let S = {0, 1} and some sequence κ = (κN )N ∈N0 of positive
stochastic tensors be given such that κN essentially equals κ3 for all N > 3.
Thus, Q = ⊗N ∈N0 κN is a 3-Markov measure. Then, stationarity of Q implies
the following set of non-linear equations.
(N)

ps1 pss12 pss12 ss23 pss12 ss23 ss34 − ps2 pss23 pss23 ss34 = 0; s1 , s2 , s3 , s4 ∈ {0, 1}.

Solving (N) simultaneously for κ0 , κ1 , κ2 , under the stochasticity constraint

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10

F.M. VON ROHRSCHEIDT

of the tensors, gives the following solutions in terms of κ3 .
00
00
0
0 ⎞
⎛p00 (κ3 ) p01 (κ3 )
01
0
p01
(κ
)
p
(κ
⎜ 0
10 3
11 3 )⎟
⎟
κ2 (κ3 ) = ⎜ 10
⎜p00 (κ3 ) p10
0
0 ⎟
01 (κ3 )
11
⎠
⎝ 0
0
p11
10 (κ3 ) p11 (κ3 )
100

p000
001
000
p100
000 +p001

p000
⎛
p100
+p000
000
001
⎜
⎜
0
⎜
=⎜
010
110
⎜ p110
+p101
010 (p100 −p100 )
⎜ 100
001 )(p010 −p110 )
⎜ 1+(p101
−p
010
010
100
100
⎜
0
⎝

0
001 010
110
p110
101 −p010 (p100 −p100 )
001 )(p010 −p110 )
1+(p101
−p
010
010
100
100

0

0

110 001
101
p101
010 +p100 (p010 −p010 )
001 )(p010 −p110 )
1+(p101
−p
010
010
100
100

010 001
101
p101
011 −p100 (p010 −p010 )
001 )(p010 −p110 )
1+(p101
−p
010
010
100
100

0

0

0

p111
110
011
p111
110 +p111

p011
111
011
p111
110 +p111

where it remains to check that all entries of the stochastic tensor are actually
probabilities. Indeed, that is true according to the following. First note that,
by positivity of the stochastic tensor κ3 , all the denominators of the entries
of κ2 (κ3 ) are strictly positive. Hence, e.g. for p10
00 (κ3 ) it holds
110
101 010
p110
101 010
110
100 + p010 (p100 − p100 )
≤ 0 ⇔ p110
100 + p010 (p100 − p100 ) ≤ 0
101
001
010
1 + (p010 − p010 )(p100 − p110
)
100
101 010
101
⇔ p110
100 p011 + p010 p100 ≤ 0,

which clearly contradicted the positivity of κ3 . The claim for p10
00 (κ3 ) follows
since the rows obviously sum up to 1. The other entries follow the same
reasoning. The solutions for κ1 and κ0 can be obtained as
p0 (κ (κ )) p01 (κ2 (κ3 ))
)
κ1 (κ2 (κ3 )) = ( 01 2 3
p0 (κ2 (κ3 )) p11 (κ2 (κ3 ))
⎛ 00 p00 (κ310)
p (κ )+p (κ )
= ⎜ 01 p311 (κ300) 3
⎝ p11 (κ310)+p01 (κ3 )
10

10

=

11

C1
1
( 1+C
1
1+C2

p00
01 (κ3 )
⎞
p00
(κ
)+p10
3
01
00 (κ3 ) ⎟
p01
(κ
)
3
11
⎠
11
p11
10 (κ3 )+p01 (κ3 )

1
1+C1 )
C2
1+C2

and
κ0 (κ3 ) = (p0 (κ1 (κ2 (κ3 ))), p1 (κ1 (κ2 (κ3 ))))
=(

=

p10 (κ2 (κ3 ))
p01 (κ2 (κ3 ))
)
p10 (κ2 (κ3 )) + p01 (κ2 (κ3 )) p10 (κ2 (κ3 )) + p01 (κ2 (κ3 ))

⎛
⎞
1
1
,
,
1+C1 ⎠
2
⎝ 1 + 1+C
1
+
1+C
1+C
1

2

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⎞
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎠

BAYESIAN INFERENCE FOR STATIONARY DATA

11

where
C1 =

101
010
110
000
p110
p100
100 + p010 (p100 − p100 )
000 + p001
>0
×
001
010
110
p000
1 + (p101
001
010 − p010 ) (p100 − p100 )

C2 =

010
001
101
011
p101
p111
011 − p100 (p010 − p010 )
110 + p111
> 0.
×
001
010
110
p111
1 + (p101
110
010 − p010 ) (p100 − p100 )

Notice that these unique solutions also could have been obtained by applying the machinery mentioned in above proof instead of solving (N) simultaneously.
The example of binary data is not only useful to illustrate how to obtain lower-dimensional kernels from higher-dimensional ones but also for
motivating the subsequent definition of a parameter space of the positive
shift-ergodic measures. The following pictures show the embedding of the
binary shift-ergodic 1-Markov measures in the space of all binary 1-Markov
measures.

Fig 1: Embedding of shift-ergodic Markov measures in all Markov measures;
the binary case.
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F.M. VON ROHRSCHEIDT

Each point of the unit cube parametrizes a 1-Markov measure P ∈ P({0, 1}N )
according to the definition
P ([s1 , . . . , sm ]) ∶= ps1 pss12 ⋯pssm−1
,
m
for all cylinder sets [s1 , . . . , sm ]. Thereby κ0 = (p0 , p1 ) denotes the initial
p0 p0
stochastic tensor and κ2 = ( 10 11 ) the tensor of next higher dimension.
p0 p1
However, only the points on the surface depict the shift-ergodic 1-Markov
φ1

measures reflecting the non-linear mapping K1 z→ K0 .
In addition, the Bernoulli measures, can be embedded into the picture by
taking
p p
p0 p0
( 10 11 ) ∶= ( 0 1 ) .
p0 p1
p0 p1

Fig 2: Embedding of i.i.d.-measures in shift-ergodic Markov measures; the
binary case.
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BAYESIAN INFERENCE FOR STATIONARY DATA

13

In a certain way, this embedding will now be generalized to infinite dimensions. However, this cannot be done in a neat graphical way as above
since in higher dimensions or for state spaces with more than two elements,
respectively, the involved simplices become too large. Thus, in the following
the necessary construction is given in an algebraic manner.
Therefor, let K = ∏N ∈N0 KN the space of sequences of stochastic tensors
and equipped with the product topology and denote by K(+) ⊂ K the dense
subset of sequences of positive stochastic tensors. Now, out of the family
of continuous mappings (φN )N ∈N appearing in Theorem 3.1, build another
family of continuous mappings ψ = (ψM,N ∶ KN (+) → KM (+))M ≤N ;M,N ∈N0
by the construction
ψM,N

⎧
⎪
⎪φM +1 ○ ⋯ ○ φN , if M ≠ N
.
∶= ⎨
⎪
otherwise
⎪
⎩idKM (+) ,

For any sequence a = (an )n∈N denote by π = (πn ∶ a ↦ an )n∈N0 the family
of coordinate projections. The next definition establishes the notion of a
parameter space for the space of positive shift-ergodic measures.
Definition 1. Let K(+) and ψ be as above and define the parameter space P to be the inverse limit alongside the families (KN )N ∈N0 and
(ψM,N )M,N ∈N0 . That is
P = {κ ∈ K(+) ∶ πM (κ) = ψM,N ○ πN (κ); M, N ∈ N0 , M ≤ N } .
Put another way, the parameter space consists of all sequences of positive
kernels increasing by their degree of dependence whose coordinates stick
together through the family of mappings φ obtained in Theorem 3.1 meaning
that the diagram
P
ψM
ψN

KN

ψM,N

KM

commutes, where ψN denotes the restriction of πN to P. Of course, all
ergodic probability measures carrying only finite degree of dependence can
be encountered in P by a similar embedding argument as Figure 2 implies. In
particular these are measures which are obtained by solving a finite system
of non-linear equations similar to (N). However, P parametrizes even more.
For instance all hidden Markov measures which are well known to possess
infinite degree of dependence in general, see e.g. Blackwell (1957); Marcus,
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14

F.M. VON ROHRSCHEIDT

Petersen and Weissman (2011). This, in a certain way, is in correspondence
to an idea presented in Ornstein (1973) which, loosely speaking, describes
the truncation of ergodic processes into multi-step Markov processes in terms
of their distances measured by entropy. However, the present work gives rise
to think of such truncations as suitable projections from a space of “infinitestep Markov processes”.
Since the mappings ψM,N are continuous there is a natural notion of a
topology on P as the coarsest topology which makes all the mappings ψN
continuous, see e.g. Bourbaki (1968, 1969) for further details. Taking B as
the Borel σ-field induced by this topology makes (P, B) a measurable space.
Then, basically any probability measure Π ∈ P (P e (S N )) makes (P, B, Π) a
Bayesian model for stationary data taking values in some finite state space.
Thereby, the positivity of the parametrized subspace of the ergodic measures
amounts to a kind of non-informativity about the possible state-to-state
transitions of the process. Basically any information about the inability of
some jump into a certain state after having visited a finite string of states
would lead the statistical modeling to concentrate on an appropriate subspace of P. However, positivity usually preserves one from underestimating
certain transition probabilities especially when the size of the data is small.
4. Inference. In the present section, an explicit model for the mixing measure Π ∈ P(P) will be obtained. Thereby, the focus shall be on
the possibility of a reasonable update to the posterior Π(⋅∣X[1∶n] ), where
X[1∶n] = (X1 , . . . , Xn ) denotes the observed data. Since the data are assumed to be generally conditionally ergodic, there is uncertainty in any of
the summarizing statistics which might be seen as the transition counts of
any finite degree in analogy to (E). Hence, the update must incorporate the
possibility of learning from the data for the degree of dependence of the true
data-generating probability measure which will throughout be assumed to
be finite. However, it will not be assumed that it is of parametric form in
the sense that it is a.s.-Π bounded from above as it is for instance in Strelioff, Crutchfield and Hübler (2007). Also it is omitted to resort on purely
computational methods as in Bacallado, Diaconis and Holmes (2015) who
infer data with respect to their “closeness to exchangeability”. Therefor, the
authors propose the use of a certain prior being motivated from the Central
Limit Theorem and exploit Markov Chain Monte Carlo methods to sample
from the posterior. This posterior makes then a statement about the order
of dependence of the data in terms of its concentration on the respective
subspace of the considered parameter space. One might keep in mind the
visualization of the embedding of the Bernoulli measures into the space of

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BAYESIAN INFERENCE FOR STATIONARY DATA

15

Markov measures as in above pictures. There the posterior would concentrate around the space diagonal of the parameter cube. However, it is unclear
to the author how these methods could be extended to the unbounded case.
The approach considered here is different to the just mentioned ones since
it relies on an idea which mimics the update mechanism of the celebrated
Dirichlet process. This will enable one to deal with the unbounded case,
i.e. where the dependence of the true probability measure generating the
data can be of any order. The rough idea is to extend the usual Bayesian
sampling scheme by one more stage, namely the drawing of the degree of
dependence N ∈ N0 from a distribution ν ∈ P(N0 ). Then, given N , a positive
stochastic tensor κN is sampled from a specific distribution µ ∈ P(KN ). This
distribution will be taken to reflect another non-informativity, namely that
one cannot learn for some segment of the random tensor by observing a
transition associated to another segment. This is reflected by sampling the
segments of the tensor, which might be thought of as the rows of its flattening
analogously as in Example 1, independently from each other. Notice that the
ability of any cross learning implicitly required additional prior information
on the Bayesian model. Each segment itself is sampled from a Dirichlet
distribution Dir(αt(N ) ) with parameter vector αt(N ) > 0, t(N ) ∈ S N , such
that µ = ⊗t(N ) ∈S N Dir(αt(N ) ). By virtue of Theorem 3.1, it suffices to sample
the stochastic tensor of highest order κN in order to determine a positive
shift-ergodic probability measure Q = ⊗M ∈N0 ψM,N (κN ), where ψM,N (κN )
amounts to the element κ̃M of the embedding of KN into KM such that
φM,N (κ̃M ) = κN whenever M > N . Then, in a last step data is sampled
from Q as defined in Theorem 3.1. The subsequent sampling scheme (Smpl)
summarizes above. Let Π be given according to the decomposition
⎤
⎡
⎤
⎡
⎥
⎢
⎥
⎢
Π = ∫ ⎢ ⊗ Dir(αt(N ) )⎥ ν(dN ) = ∑ ⎢ ⊗ Dir(αt(N ) )⎥ ν(N )
⎥
⎢
⎥
⎢ (N ) N
N ∈N0 ⎣t(N ) ∈S N
⎦
⎦
∈S
N0 ⎣t
such that data is sampled as
(1.) N ∼ ν
(2.) κN ∣ [N, (αt(N ) )t(N ) ∈S N ] ∼
(Smpl)

⊗
t(N ) ∈S N

Dir(αt(N ) )

(3.) X[1∶∞] ∣ [N, (αt(N ) )t(N ) ∈S N , κN ] ∼ ⊗ κN .
N ∈N0

As the way the data are sampled, the update to the posterior is split in
two separate stages, too. Thereby, updating the laws of the stochastic tensors is fairly straight forward. That is basically due to the assumption of
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F.M. VON ROHRSCHEIDT

independence among the segments of the tensors, which corresponds to the
assumption that transition counts of appropriate order are predictively sufficient, c.f. Fortini and Petrone (2014). Thus, given the order of dependence
N , the update of the law of κN is accomplished by separately updating the
Dirichlet distributions associated to the segments of the tensor by the respective transition counts. Since the Dirichlet distribution is a conjugate prior
for the multinomial likelihoods which emerge from these considerations, the
updated law of κN can be expressed as
n−N

κN ∣ [N, (αt(N ) )t(N ) ∈S N , X[1∶n] ] ∼

⊗
t(N ) ∈S N +1

Dir (αt(N ) + ∑ δX[k,k+N ] ({t(N ) })) .
k=1

In contrast, the update of the distribution of the degree of dependence
is not as straight forward as that of the laws of the tensors. That is due to
two main reasons. Firstly, one needs to infer the structure of the data from
the observations. Secondly, the Bayes theorem implies a procedure which
cannot be accomplished by a finite algorithm. For the latter, note that the
Bayes theorem would be obtained as
(B)

P (N ∣X[1∶n] ) =

P (X[1∶n] ∣N ) P(N )
P (X[1∶n] )

=

P (X[1∶n] ∣N ) ν(N )

,

P (X[1∶n] )

where
P(X[1∶n] ∣N ) = ∑ P(X[1∶n] ∣N, κN )P(κN ∣N )
κN

=

⎡
⎤
⎢
⎥
∑ P(X[1∶n] ∣N, κN ) ⎢⎢ ⊗ Dir(αt(N ) )⎥⎥ .
κN ∈KN
⎣t(N ) ∈S N
⎦

Hence, the denominator in (B) is given as
⎡
⎤
⎢
⎥
∑ P (X[1∶n] ∣N, κN ) ⎢⎢ ⊗ Dir (αt(N ) )⎥⎥ ν(N ),
(N
)
N
N ∈N0 κN ∈KN
⎣t ∈S
⎦
∑

requiring the calculation of the likelihood P(X[1∶n] ∣N, κN ) for any N ∈ N0
and κN ∈ KN . From a theoretical point of view this can be done using
the family of mappings ψ. However, from a practical perspective this seems
hopeless.
Hence, another more direct and more pragmatic method is proposed here
with particular focus on posterior consistency. In a specific manner, this will
consist in mimicking the update mechanism of the Dirichlet process. More
precisely this means that the atoms of ν are updated by weights obtained
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BAYESIAN INFERENCE FOR STATIONARY DATA

from the raw data such that the prior guess vanishes by increasing the size of
the data. However, there should be an additional mechanism which decreases
the mass of atoms which correspond to degrees of dependence that are too
large to be inferred from a relatively small size of observations. Moreover,
this mechanism should be able to increase such masses when the size of the
observations increases and the data give evidence.
A second difference to the update mechanism of the Dirichlet process
is that the additional weights can certainly not stem from an assumption
of partially exchangeability of any degree and thus cannot be expressed in
terms of the empirical measure. Instead, these updating weights shall be
obtained as a kind of empirical measurements of the distances between the
tensor spaces in terms of symmetry which are indicated by the observed
data. To make this precise, let for n, N ∈ N0 , N < n and a string s(N ) ∈ S N +1
)
(X[1∶n] ; s(N ) ) =
e(N
n

1 n−N
(s(N ) )
∑ δX
n − N k=1 [k∶k+N ]

be the empirical measure consisting of a properly normalized version of the
order-N transition counts tN = (tN (X[1∶n] ; s(N ) ))s(N ) ∈S N +1 . The following
thought experiment is given in order to clarify the basic idea of what follows.
If e.g. the stochastic process X is known to be sampled from a Markov
measure of order one then all the stochastic tensors of order greater than
one fulfill a certain symmetry condition representing the truncation of the
dependence of the process. That is it holds
2 ⋯tN −1 sN
2 ⋯sN −1 sN
pss12 s⋯s
= ptt12 t⋯t
,
N −1 sN sN +1
N −1 sN sN +1

for all (s1 , . . . , sN +1 ) ∈ S N +1 , (t1 , . . . , tN −1 ) ∈ S N −1 and for all N > 1. When
an outcome of process X[1∶n] is observed this means that one will observe
nearly as many jumps (s1 , . . . , sN ) → (s2 , . . . , sN +1 ) as jumps (t1 , . . . , tN −1 , sN ) →
(t2 , . . . , tN −1 , sN , sN +1 ), at least if n >> N is large enough. The argument is
the same for orders of dependency N > 1. From a statistical viewpoint, this
argumentation is now reversed and exploited in order to find appropriate
update weights for the posterior. Therefor, a suitable estimates for all the
transition probabilities is used. These estimates are given for n > N as
2 ⋯sN −1 sN
(X[1∶n] ) =
p̂ss12 s⋯s
N −1 sN sN +1

=

tN (X[1∶n] ; s1 , . . . , sN +1 )
∑sN ∈S tN (X[1∶n] ; s1 , . . . , sN +1 )
∑n−N
k=1 δX[k∶k+N ] (s1 , . . . , sN +1 )
∑n−N
k=1 δX[k∶k+N −1] (s1 , . . . , sN )

,

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18

F.M. VON ROHRSCHEIDT

which is in full analogy of the order one Markov case for which Diaconis and
Freedman (1980) argue that (

t1 (X[1∶n] ;i,j)

)

∑j∈S t1 (X[1∶n] ;i,j) i,j∈S

converges to a stochastic

(s × s)-matrix for almost all sequences of data X[1∶∞] as n tends to infinity.
See also Fortini et al. (2002) for related results taking into account a decent
extension of the state space. The updating weights will be obtained regarding
(N )
the differences dˆn [i, j, s(N −1) ] which are defined as
n−N

n−N

∣ ∑ δX[k∶k+N −1] (i, s1 , . . . , sN −1 ) × ∑ δX[k∶k+N ] (j, s1 , . . . , sN )
k=1

k=1
n−N

n−N

− ∑ δX[k∶k+N −1] (j, s1 , . . . , sN −1 ) × ∑ δX[k∶k+N ] (i, s1 , . . . , sN ) ∣
k=1

k=1

for i, j ∈ S in order to detect dependencies of order N . Considering again
(N )
above thought experiment dˆn [i, j, s(N −1) ] is expected to decay for all N > 1
(1)
and all strings s(N −1) if n increases, while dˆn [i, j, s] is expected to go to
infinity for all s ∈ S. Those differences will serve as the building blocks of
(N )
the update. However, since, using Landaus notation, it holds dˆn = O(n2 )
for all N ≥ 1 it is necessary to modify these differences in order to achieve
posterior consistency of the update scheme to be presented. Roughly, that is
because without such a modification the update scheme won’t pin point the
tensor space of the highest order as the true one because it treated the lower
ones, which do not possess symmetry neither, similarly in an asymptotic
(N )
(N )
manner. The modification of dˆn will be given as dn defined through
n−N

n−N

)
d(N
[i, j, s(N −1) ] ∶= ∣ ∑ δX[k∶k+N −1] (i, s1 , . . . , sN −1 ) × ∑ δX[k∶k+N ] (j, s1 , . . . , sN )
n
k=1

k=1
n−N

n−N

− ∑ δX[k∶k+N −1] (j, s1 , . . . , sN −1 ) × ∑ δX[k∶k+N ] (i, s1 , . . . , sN ) ∣
k=1

k=1

In order to detect the entire symmetry in the stochastic tensors above
(N )
differences are accumulated in form of Dn defined by
Dn(N ) =

∑

)
d(N
[i, j, s(N −1) ] .
n

i,j∈S
s(N −1) ∈S N

Now, the update scheme is introduced as follows. The prior ν reflecting
β(⋅)
knowledge about the order of dependence is chosen as ν(⋅) = β(N0 ) , where β
is a finite measure supported by the non-negative integers. Then define the
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N

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BAYESIAN INFERENCE FOR STATIONARY DATA

updated version of the prior after having seen data X[1∶n] by
(N )

ν (N ∣X[1∶n] ) ∶=

(U)
(0)

β(N ) + Dn

(k)

,

β(N0 ) + ∑k∈N0 Dn

(N )

where Dn ∶= n and Dn ∶= 0 for N ≥ n. Then we have the following lemma
considering posterior consistency.
β(⋅)

Lemma 4.1. Let ν(⋅) = β(N0 ) be a prior for the order of dependence of the
data X, where β is a finite measure on the non-negative integers. Moreover
let the true order of dependence of X be N0 . Then it holds that
n→∞

ν (N0 ∣X[1∶n] ) Ð→ 1,
n→∞

ν (N ∣X[1∶n] ) Ð→ 1,

∀N ≠ N0 ,

where ν (⋅∣X[1∶n] ) is defined as in (U)
(N )

Proof. Since Dn

(N )

∈ o (n2N0 ), the posterior weight Dn

vanishes in the

(N )
Dn

limit for all N > N0 , while
tends to infinity for all N ≥ N0 . Hence
ν (N ∣X[1∶n] ) decays for N > N0 . Now, considering N < N0 it follows for large
sample sizes
(N )

ν (N ∣X[1∶n] ) ∝

β(N ) + Dn
(1)

(N0 −1)

β(N0 ) + Dn + . . . + Dn
=

1
n2N0
β(N0 )
n2N0

+

(N0 )

+ Dn

(N )

[β(N ) + Dn ]

(1)
Dn
1
2
2(N
−1)
n
0
n

+ ... +

(N −1)
1 Dn 0
n2 n2(N0 −1)

n→∞

+

(N )
Dn 0
n2N0

Ð→ 0.

(N )

since
that

Dn 0
n2N0

attains a non-zero limit. Essentially the same reasoning shows
n→∞

ν (N0 ∣X[1∶n] ) Ð→ 1
such that the lemma is proven.
Lemma 4.1 along with well known results about posterior consistency of
Dirichlet priors do imply the following result.
β(⋅)

Corollary 4.2. Let ν (⋅) = β(N0 ) be a prior for the order of dependence,
where β is a finite measure on the non-negative integers. Moreover, given
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20

F.M. VON ROHRSCHEIDT

the order of dependence, let the rows of the flattened stochastic tensor representing this order be sampled independently according to Dirichlet priors
Dir (αt(N ) ) with. Then, for almost all sequences of data X[1∶∞] , the sequence
of posterior measures defined by
⎡
⎤
⎥
⎢
Π (⋅∣X[1∶n] ) = ∑ ⎢ ⊗ Dir (αt(N ) ∣X[1∶n] )⎥ ν (N ∣X[1∶n] )
⎢
⎥
N ∈N0 ⎣t(N ) ∈S N +1
⎦
centers around the true data generating measure which samples data according to scheme (Smpl).
5. Remarks. The wish of having a better understanding about the
foundations of Bayesian statistics for general stationary data arose from
thinking from a Bayesian perspective about a problem concerning the M /G/∞
queue which came up in Brown (1970). Therein, the author studies an estimation problem in queueing theory from the frequentist viewpoint. By the
statistical setup of this paper, one has only access to general stationary and,
according to the frequentist interpretation of probability, ergodic data. To
approach this issue, Brown pretends as the data was i.i.d. and gives an estimator which resembles the empirical distribution function. However, one
can find some words of criticism in his work which reads as
Although we have obtained an estimator for the c.d.f. G, it is clearly not the
best estimator in any sense because we do not use all the information. The
problem of finding a best estimator (according to any criterion) is still open.

The present work gives an idea of what the suppressed information is as
well as what an improved estimator might incorporate. However, in order to
overcome rather non-telling technicalities, it was necessary to break down
the problem into a discrete-time setting as well as to a finite state space.
However, the questions about generalizing to a more general state spaces
and to a continuous-time domain, respectively, are still open.
It seems possible to obtain a similar construction exploiting theory of
Markov chains on countably infinite state spaces. It is well known that for
irreducible and positive recurrent stochastic matrices there exists a unique
invariant distribution which might serve as the building blocks of a generalization of the theory just presented. However, from a theoretical viewpoint
it seems more difficult to obtain a measurable structure on the generalized inverse limit construction since continuity would have to be achieved
by studying perturbations of linear operators, see e.g. Kato (1966). From a
rather applied viewpoint the question is how the system of non-linear equations (N) can be reasonably approximated in case of countably infinite state
spaces.
The explicit model which was introduced in the present work represents a
specific sampling scheme which bases on several independence assumptions.
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BAYESIAN INFERENCE FOR STATIONARY DATA

21

These assumptions, in a certain sense, reflect the lack of information about
the possibility of cross-learning for the transitions of the observed process.
At this point the model might be further specified if one has gained additional information about this cross-learning in advance. Notice that such
information can be given by certainty of the specific process one does collect
data from, which in turn shrinks the class of models. The M /G/1 queueing model might serve as an example for the possibility of cross-inference
which even takes into account a countably infinite state space. In queueing
theory it is well known that the M /G/1 queue is driven by an embedded
Markov chain governed by a so called stochastic ∆-matrix, c.f. Abolnikov
and Dukhovny (1991). Observing data that stems from an M /G/1 queue,
one can make cross-inference based on the specific shape of these stochastic
matrices. For more details see also von Rohrscheidt and Wichelhaus (2017).
Considering the more general case of an arbitrary infinite stochastic matrix
techniques of Gibson and Seneta (1987a,b) might be of interest in order to
obtain Bayesian models for making inference for Markov chains (possibly of
higher order) on countably infinite state spaces.
The question about generalizations with respect to some uncountable
state space is an even deeper one. Therefor, the work of Fortini et al. (2002)
may give a first idea how complicated things become considering general
Markov kernels and how overwhelmingly large the possible class of Bayesian
models gets. More precisely, again the question of cross-learning for transitions come up here but in a even more involved manner. This is because
one would need to clarify within the modeling how much one can learn for
transitions ”being close to observed ones. One might think about a stone
being thrown onto a surface of some fluid. The question is then how fast do
the waves, representing the ”strength of cross-learning, have to decay? The
fact that the probability of an observation of any of such transitions is zero
doesn’t make it any easier.
Another difficult question is about Bayesian inference for processes evolving in continuous time. The first work on mixtures of additive processes
is Bühlmann (1960) followed by Freedman (1963, 1996) which additionally
study mixtures of continuous-time Markov processes. Further approaches to
symmetries of continuous-time processes can be found in Kallenberg (2006).
Although the results obtained therein might be understood as the probabilistic fundament for Bayesian statistics for continuous-time stochastic processes, there is still a lack of decently working statistical models. One reason
of that fact might be that collecting data from a continuous-time stochastic
process at fixed discrete time points would lead to a rather discrete-time
model in advance. On the other hand, collecting data in a high-frequency
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22

F.M. VON ROHRSCHEIDT

manner, similarly to the consideration of an uncountable state space, might
cause cross-learning effects which would have to be incorporated in the statistical model as prior information.
Acknowledgments: The present work is an excerpt of my doctoral thesis
Bayesian Nonparametric Inference for Queueing Systems which I prepared
at the Ruprecht-Karls-Universität Heidelberg within the DFG-project (German research foundation) grant 1953. I would like to thank my supervisors
Prof. Rainer Dahlhaus and Dr. Cornelia Wichelhaus for their patience in
letting me unfold the problem. I am also grateful to Prof. Jan Johannes who
helped to improve the work in numerous discussions.
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Fritz Moritz von Rohrscheidt
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E-mail: moe0405@gmx.de

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