Robert

Bayes 250th versus Bayes 2.5.0
Christian P. Robert
Universit´ Paris-Dauphine, University of Warwick, & CREST, Paris
e

written for EMS 2013, Budapest

Outline

Bayes, Thomas (1702–1761)
Jeﬀreys, Harold (1891–1989)
Lindley, Dennis (1923– )
Besag, Julian (1945–2010)
de Finetti, Bruno (1906–1985)

Bayes, Price and Laplace

Bayes’ 1763 paper
Bayes’ example
Laplace’s 1774 derivation

a first Bayes 250
Took place in Edinburgh, Sept. 5–7, 2011:
Sparse Nonparametric Bayesian Learning from
Big Data David Dunson, Duke University
Classification Models and Predictions for Ordered
Data Chris Holmes, Oxford University
Bayesian Variable Selection in Markov Mixture
Models Luigi Spezia, Biomathematics
& Statistics Scotland, Aberdeen
Bayesian inference for partially observed Markov
processes, with application to systems biology
Darren Wilkinson, University of Newcastle
Coherent Inference on Distributed Bayesian
Expert Systems Jim Smith, University of Warwick

Bayesian Priors in the Brain Peggy Series,
University of Edinburgh
Approximate Bayesian Computation for model
selection Christian Robert, Universit´
e
Paris-Dauphine
ABC-EP: Expectation Propagation for
Likelihood-free Bayesian Computation Nicholas
Chopin, CREST–ENSAE
Bayes at Edinburgh University - a talk and tour
Dr Andrew Fraser, Honorary Fellow, University of
Edinburgh

Probabilistic Programming John Winn, Microsoft
Research

Intractable likelihoods and exact approximate
MCMC algorithms Christophe Andrieu,
University of Bristol

How To Gamble If You Must (courtesy of the
Reverend Bayes) David Spiegelhalter, University
of Cambridge

Bayesian computational methods for intractable
continuous-time non-Gaussian time series Simon
Godsill, University of Cambridge

Inference and computing with decomposable
graphs Peter Green, University of Bristol

Efficient MCMC for Continuous Time Discrete
State Systems Yee Whye Teh, Gatsby
Computational Neuroscience Unit, University
College London

Nonparametric Bayesian Models for Sparse
Matrices and Covariances Zoubin Gharamani,
University of Cambridge
Latent Force Models Neil Lawrence, University of
Sheffield
Does Bayes Theorem Work? Michael Goldstein,
Durham University

Adaptive Control and Bayesian Inference Carl
Rasmussen, University of Cambridge
Bernstein - von Mises theorem for irregular
statistical models Natalia Bochkina, University of
Edinburgh

Why Bayes 250?

Publication on Dec. 23, 1763 of
“An Essay towards solving a
Problem in the Doctrine of
Chances” by the late
Rev. Mr. Bayes, communicated
by Mr. Price in the Philosophical
Transactions of the Royal Society
of London.
c 250th anniversary of the Essay

Breaking news!!!
An accepted paper by Stephen Stigler in Statistical Science
uncovers the true title of the Essay:
A Method of
Calculating the Exact
Probability of All
Conclusions founded on
Induction
Intended as a reply to
Hume’s (1748) evaluation
of the probability of
miracles

Breaking news!!!
may have been written as early as 1749: “we may hope to
determine the Propositions, and, by degrees, the whole Nature
of unknown Causes, by a sufficient Observation of their
effects” (D. Hartley)
in 1767, Richard Price used
Bayes’ theorem as a tool to
attack Hume’s argument,
refering to the above title
Bayes’ offprints available at
Yale’s Beinecke Library (but
missing the title page) and
at the Library Company of
Philadelphia (Franklin’s
library)
[Stigler, 2013]

Bayes Theorem

Bayes theorem = Inversion of causes and eﬀects
Continuous version for random
variables X and Y
fX |Y (x|y ) =

fY |X (y |x) × fX (x)
fY (y )

Who was Thomas Bayes?
Reverend Thomas Bayes (ca. 1702–1761), educated in London
then at the University of Edinburgh (1719-1721), presbyterian
minister in Tunbridge Wells (Kent) from 1731, son of Joshua
Bayes, nonconformist minister.
“Election to the Royal Society based on
a tract of 1736 where he defended the
views and philosophy of Newton.
A notebook of his includes a method of
ﬁnding the time and place of
conjunction of two planets, notes on
weights and measures, a method of
diﬀerentiation, and logarithms.”
[Wikipedia]

Bayes’ 1763 paper:

Billiard ball W rolled on a line of length one, with a uniform
probability of stopping anywhere: W stops at p.
Second ball O then rolled n times under the same assumptions. X
denotes the number of times the ball O stopped on the left of W .



Bayes’ question:
Given X , what inference can we make on p?



Bayes’ wording:
“Given the number of times in which an unknown event has
happened and failed; Required the chance that the probability of
its happening in a single trial lies somewhere between any two
degrees of probability that can be named.”



Modern translation:
Derive the posterior distribution of p given X , when
p ∼ U([0, 1]) and X |p ∼ B(n, p)

Resolution

Since
P(X = x|p) =

n x
p (1 − p)n−x ,
x
b

P(a < p < b and X = x) =
a

and

1

P(X = x) =
0

n x
p (1 − p)n−x dp
x

n x
p (1 − p)n−x dp,
x

Resolution (2)

then
P(a < p < b|X = x) =
=

b
a
1
0
b
a

n x
n−x dp
x p (1 − p)
n x
n−x dp
x p (1 − p)
p x (1 − p)n−x dp

B(x + 1, n − x + 1)

,

i.e.
p|x ∼ Be(x + 1, n − x + 1)
[Beta distribution]

Resolution (2)
then
P(a < p < b|X = x) =
=

b
a
1
0
b
a

n x
n−x dp
x p (1 − p)
n x
n−x dp
x p (1 − p)
p x (1 − p)n−x dp

B(x + 1, n − x + 1)

,

i.e.
p|x ∼ Be(x + 1, n − x + 1)
In Bayes’ words:
“The same things supposed, I guess that the probability
of the event M lies somewhere between 0 and the ratio of
Ab to AB, my chance to be in the right is the ratio of
Abm to AiB.”

Laplace’s version

Pierre Simon (de) Laplace (1749–1827):
“Je me propose de d´terminer la probabilit´
e
e
des causes par les ´vńements mati`re neuve `
e e
e
a
bien des ´gards et qui m´rite d’autant plus
e
e
d’ˆtre cultivé que c’est principalement sous ce
e
e
point de vue que la science des hasards peut
ˆtre utile ` la vie civile.”
e
a
[M´moire sur la probabilit´ des causes par les ´vńemens, 1774]
e
e
e e

Laplace’s version

“Si un ´v`nement peut ˆtre produit par un
e e
e
nombre n de causes diﬀ`rentes, les probabilit´s
e
e
de l’existence de ces causes prises de
l’´v`nement, sont entre elles comme les
e e
probabilit´s de l’´v`nement prises de ces
e
e e
causes, et la probabilit´ de l’existence de
e
chacune d’elles, est ´gale ` la probabilit´ de
e
a
e
l’´v`nement prise de cette cause, divise´ par la
e e
e
somme de toutes les probabilit´s de
e
l’´v`nement prises de chacune de ces causes.”
e e
e
e
e e

Laplace’s version

In modern terms: Under a uniform prior,
P(Ai |E )
P(E |Ai )
=
P(Aj |E )
P(E |Aj )
and
f (x|y ) =

f (y |x)
f (y |x) dy

e
e
e e

Laplace’s version

Later Laplace acknowledges Bayes by
“Bayes a cherch´ directement la probabilit´
e
e
que les possibilit´s indiqués par des
e
e
exp´riences d´j` faites sont comprises dans les
e
ea
limites donnés et il y est parvenu d’une
e
mani`re fine et tr`s ingńieuse”
e
e
e
[Essai philosophique sur les probabilit´s, 1810]
e

Another Bayes 250

Meeting that took place at the Royal Statistical Society, June
19-20, 2013, on the current state of Bayesian statistics
G. Roberts (University of Warwick) “Bayes for
differential equation models”
N. Best (Imperial College London) “Bayesian
space-time models for environmental
epidemiology”
D. Prangle (Lancaster University) “Approximate
Bayesian Computation”
P. Dawid (University of Cambridge), “Putting
Bayes to the Test”
M. Jordan (UC Berkeley) “Feature Allocations,
Probability Functions, and Paintboxes”
I. Murray (University of Edinburgh) “Flexible
models for density estimation”
M. Goldstein (Durham University) “Geometric
Bayes”
C. Andrieu (University of Bristol) “Inference with
noisy likelihoods”

A. Golightly (Newcastle University), “Auxiliary
particle MCMC schemes for partially observed
diffusion processes”
S. Richardson (MRC Biostatistics Unit)
“Biostatistics and Bayes”
C. Yau (Imperial College London)
“Understanding cancer through Bayesian
approaches”
S. Walker (University of Kent) “The Misspecified
Bayesian”
S. Wilson (Trinity College Dublin), “Linnaeus,
Bayes and the number of species problem”
B. Calderhead (UCL) “Probabilistic Integration
for Differential Equation Models”
P. Green (University of Bristol and UT Sydney)
“Bayesian graphical model determination”

The search for certain π

Keynes’ treatise
Jeﬀreys’ prior distributions
Jeﬀreys’ Bayes factor
expected posterior priors

Keynes’ dead end

In John Maynard Keynes’s A Treatise on Probability (1921):

“I do not believe that there is
any direct and simple method by
which we can make the transition
from an observed numerical
frequency to a numerical measure
of probability.”
[Robert, 2011, ISR]

Keynes’ dead end

In John Maynard Keynes’s A Treatise on Probability (1921):
“Bayes’ enunciation is strictly
correct and its method of arriving
at it shows its true logical
connection with more
fundamental principles, whereas
Laplace’s enunciation gives it the
appearance of a new principle
specially introduced for the
solution of causal problems.”
[Robert, 2011, ISR]

Who was Harold Jeﬀreys?

Harold Jeﬀreys (1891–1989)
mathematician, statistician,
geophysicist, and astronomer.
Knighted in 1953 and Gold
Medal of the Royal Astronomical
Society in 1937. Funder of
modern British geophysics. Many
of his contributions are
summarised in his book The
Earth.

[Wikipedia]

Theory of Probability

The first modern and comprehensive treatise on (objective)
Bayesian statistics
Theory of Probability (1939)
begins with probability, refining
the treatment in Scientific
Inference (1937), and proceeds to
cover a range of applications
comparable to that in Fisher’s
book.
[Robert, Chopin & Rousseau, 2009, Stat. Science]

Jeﬀreys’ justiﬁcations

All probability statements are conditional
Actualisation of the information on θ by extracting the
information on θ contained in the observation x
The principle of inverse probability does correspond
to ordinary processes of learning (I, §1.5)
Allows incorporation of imperfect information in the decision
process
A probability number can be regarded as a
generalization of the assertion sign (I, §1.51).

Posterior distribution
Operates conditional upon the observations
Incorporates the requirement of the Likelihood Principle
...the whole of the information contained in the
observations that is relevant to the posterior
probabilities of diﬀerent hypotheses is summed up in
the values that they give the likelihood (II, §2.0).
Avoids averaging over the unobserved values of x
Coherent updating of the information available on θ,
independent of the order in which i.i.d. observations are
collected
...can be used as the prior probability in taking
account of a further set of data, and the theory can
therefore always take account of new information (I,
§1.5).
Provides a complete inferential scope

Subjective priors

Subjective nature of priors
Critics (...) usually say that the prior probability is
‘subjective’ (...) or refer to the vagueness of previous
knowledge as an indication that the prior probability
cannot be assessed (VIII, §8.0).
Long walk (from Laplace’s principle of insuﬃcient reason) to a
reference prior:
A prior probability used to express ignorance is merely the
formal statement of ignorance (VIII, §8.1).

The fundamental prior

...if we took the prior probability density for the
parameters to be proportional to ||gik ||1/2 [= |I (θ)|1/2 ], it
could stated for any law that is differentiable with respect
to all parameters that the total probability in any region
of the αi would be equal to the total probability in the
corresponding region of the αi ; in other words, it satisfies
the rule that equivalent propositions have the same
probability (III, §3.10)
Note: Jeffreys never mentions Fisher information in connection
with (gik )

The fundamental prior

In modern terms:
if I (θ) is the Fisher information matrix associated with the
likelihood (θ|x),
∂ ∂
I (θ) = Eθ
∂θT ∂θ
the reference prior distribution is
π∗ (θ) ∝ |I (θ)|1/2

Note: Jeﬀreys never mentions Fisher information in connection
with (gik )

Objective prior distributions

reference priors (Bayarri, Bernardo, Berger, ...)
not supposed to represent complete ignorance (Kass
& Wasserman, 1996)
The prior probabilities needed to express ignorance
of the value of a quantity to be estimated, where
there is nothing to call special attention to a
particular value are given by an invariance theory
(Jeffreys, VIII, §8.6).
often endowed with or seeking frequency-based properties
Jeffreys also proposed another Jeffreys prior dedicated to
testing (Bayarri & Garcia-Donato, 2007)

Jeffreys’ Bayes factor
Definition (Bayes factor, Jeffreys, V, §5.01)
For testing hypothesis H0 : θ ∈ Θ0 vs. Ha : θ ∈ Θ0

B01 =

π(Θ0 |x)
π(Θc |x)
0

π(Θ0 )
=
π(Θc )
0

f (x|θ)π0 (θ)dθ
Θ0
Θc
0

f (x|θ)π1 (θ)dθ

Equivalent to Bayes rule: acceptance if
B01 > {(1 − π(Θ0 ))/a1 }/{π(Θ0 )/a0 }
What if... π0 is improper?!
[DeGroot, 1973; Berger, 1985; Marin & Robert, 2007]

Expected posterior priors (example)
Starting from reference priors πN and πN , substitute by prior
0
1
distributions π0 and π1 that solve the system of integral equations
π0 (θ0 ) =

X

πN (θ0 | x)m1 (x)dx
0

and
π1 (θ1 ) =

X

πN (θ1 | x)m0 (x)dx,
1

where x is an imaginary minimal training sample and m0 , m1 are
the marginals associated with π0 and π1 respectively
m0 (x) = f0 (x|θ0 )π0 (dθ0 )

m1 (x) = f1 (x|θ1 )π1 (dθ1 )
[Perez & Berger, 2000]

Existence/Unicity
Recurrence condition
When both the observations and the parameters in both models
are continuous, if the Markov chain with transition
Q θ0 | θ0 =

g θ0 , θ0 , θ1 , x, x

dxdx dθ1

where
g θ0 , θ0 , θ1 , x, x

= πN θ0 | x f1 (x | θ1 ) πN θ1 | x
0
1

f0 x | θ0 ,

is recurrent, then there exists a solution to the integral equations,
unique up to a multiplicative constant.
[Cano, Salmer´n, & Robert, 2008, 2013]
o

Bayesian testing of hypotheses
Lindley’s paradox
dual versions of the paradox
“Who should be afraid of the
Lindley–Jeﬀreys paradox?”
Bayesian resolutions

Who is Dennis Lindley?

British statistician, decision theorist and
leading advocate of Bayesian statistics.
Held positions at Cambridge,
Aberystwyth, and UCL, retiring at the
early age of 54 to become an itinerant
scholar. Wrote four books and
numerous papers on Bayesian statistics.
c “Coherence is everything”

Lindley’s paradox

In a normal mean testing problem,
¯
xn ∼ N(θ, σ2 /n) ,

H0 : θ = θ 0 ,

under Jeffreys prior, θ ∼ N(θ0 , σ2 ), the Bayes factor
2
B01 (tn ) = (1 + n)1/2 exp −ntn /2[1 + n] ,

where tn =

√
n|¯n − θ0 |/σ, satisfies
x
n−→∞

B01 (tn ) −→ ∞
[assuming a fixed tn ]

Lindley’s paradox

Often dubbed Jeﬀreys–Lindley paradox...
In terms of
t=
K∼

√
n − 1¯/s ,
x
πν
2

1+

ν=n−1
t2
ν

−1/2ν+1/2

.

(...) The variation of K with t is much more important
than the variation with ν (Jeﬀreys, V, §5.2).

Two versions of the paradox
“the weight of Lindley’s paradoxical result (...) burdens
proponents of the Bayesian practice”.
[Lad, 2003]
oﬃcial version, opposing frequentist and Bayesian assessments
[Lindley, 1957]
intra-Bayesian version, blaming vague and improper priors for
the Bayes factor misbehaviour:
if π1 (·|σ) depends on a scale parameter σ, it is often the case
that
σ−→∞
B01 (x) −→ +∞
for a given x, meaning H0 is always accepted
[Robert, 1992, 2013]

Evacuation of the first version

Two paradigms [(b) versus (f)]
one (b) operates on the parameter space Θ, while the other
(f) is produced from the sample space
one (f) relies solely on the point-null hypothesis H0 and the
corresponding sampling distribution, while the other
(b) opposes H0 to a (predictive) marginal version of H1
one (f) could reject “a hypothesis that may be true (...)
because it has not predicted observable results that have not
occurred” (Jeffreys, VII, §7.2) while the other (b) conditions
upon the observed value xobs
one (f) resorts to an arbitrary fixed bound α on the p-value,
while the other (b) refers to the boundary probability of 1
2

More arguments on the ﬁrst version

observing a constant tn as n increases is of limited interest:
under H0 tn has limiting N(0, 1) distribution, while, under H1
tn a.s. converges to ∞
behaviour that remains entirely compatible with the
consistency of the Bayes factor, which a.s. converges either to
0 or ∞, depending on which hypothesis is true.
Consequent literature (e.g., Berger & Sellke,1987) has since then
shown how divergent those two approaches could be (to the point
of being asymptotically incompatible).
[Robert, 2013]

Nothing’s wrong with the second version
n, prior’s scale factor: prior variance n times larger than the
observation variance and when n goes to ∞, Bayes factor
goes to ∞ no matter what the observation is
n becomes what Lindley (1957) calls “a measure of lack of
conviction about the null hypothesis”
when prior diﬀuseness under H1 increases, only relevant
information becomes that θ could be equal to θ0 , and this
overwhelms any evidence to the contrary contained in the data
mass of the prior distribution in the vicinity of any ﬁxed
neighbourhood of the null hypothesis vanishes to zero under
H1
[Robert, 2013]
c deep coherence in the outcome: being indecisive about the
alternative hypothesis means we should not chose it

“Who should be afraid of the Lindley–Jeﬀreys paradox?”

Recent publication by A. Spanos with above title:
the paradox demonstrates against
Bayesian and likelihood resolutions of the
problem for failing to account for the
large sample size.
the failure of all three main paradigms
leads Spanos to advocate Mayo’s and
Spanos’“postdata severity evaluation”
[Spanos, 2013]

“Who should be afraid of the Lindley–Jeﬀreys paradox?”
Recent publication by A. Spanos with above title:
“the postdata severity evaluation
(...) addresses the key problem with
Fisherian p-values in the sense that
the severity evaluation provides the
“magnitude” of the warranted
discrepancy from the null by taking
into account the generic capacity of
the test (that includes n) in question
as it relates to the observed
data”(p.88)
[Spanos, 2013]

On some resolutions of the second version
use of pseudo-Bayes factors, fractional Bayes factors, &tc,
which lacks proper Bayesian justiﬁcation
[Berger & Pericchi, 2001]
use of identical improper priors on nuisance parameters, a
notion already entertained by Jeﬀreys
[Berger et al., 1998; Marin & Robert, 2013]
use of the posterior predictive distribution, which uses the
data twice (see also Aitkin’s (2010) integrated likelihood)
[Gelman, Rousseau & Robert, 2013]
use of score functions extending the log score function
log B12 (x) = log m1 (x) − log m2 (x) = S0 (x, m1 ) − S0 (x, m2 ) ,
that are independent of the normalising constant
[Dawid et al., 2013]

Bayesian computing (R)evolution

Besag’s early contributions
MCMC revolution and beyond

computational jam

In the 1970’s and early 1980’s, theoretical foundations of Bayesian
statistics were sound, but methodology was lagging for lack of
computing tools.
restriction to conjugate priors
limited complexity of models
small sample sizes
The ﬁeld was desperately in need of a new computing paradigm!
[Robert & Casella, 2012]

MCMC as in Markov Chain Monte Carlo

Notion that i.i.d. simulation is deﬁnitely not necessary, all that
matters is the ergodic theorem
Realization that Markov chains could be used in a wide variety of
situations only came to mainstream statisticians with Gelfand and
Smith (1990) despite earlier publications in the statistical literature
like Hastings (1970) and growing awareness in spatial statistics
(Besag, 1986)
Reasons:
lack of computing machinery
lack of background on Markov chains
lack of trust in the practicality of the method

Who was Julian Besag?

British statistician known chieﬂy for his
work in spatial statistics (including its
applications to epidemiology, image
analysis and agricultural science), and
Bayesian inference (including Markov
chain Monte Carlo algorithms).
Lecturer in Liverpool and Durham, then
professor in Durham and Seattle.
[Wikipedia]

pre-Gibbs/pre-Hastings era

Early 1970’s, Hammersley, Clifford, and Besag were working on the
specification of joint distributions from conditional distributions
and on necessary and sufficient conditions for the conditional
distributions to be compatible with a joint distribution.

[Hammersley and Clifford, 1971]

pre-Gibbs/pre-Hastings era

Early 1970’s, Hammersley, Clifford, and Besag were working on the
specification of joint distributions from conditional distributions
and on necessary and sufficient conditions for the conditional
distributions to be compatible with a joint distribution.
“What is the most general form of the conditional
probability functions that define a coherent joint
function? And what will the joint look like?”
[Besag, 1972]

Hammersley-Cliﬀord[-Besag] theorem

Theorem (Hammersley-Cliﬀord)
Joint distribution of vector associated with a dependence graph
must be represented as product of functions over the cliques of the
graphs, i.e., of functions depending only on the components
indexed by the labels in the clique.
[Cressie, 1993; Lauritzen, 1996]


A probability distribution P with positive and continuous density f
satisﬁes the pairwise Markov property with respect to an
undirected graph G if and only if it factorizes according to G, i.e.,
(F ) ≡ (G )



Under the positivity condition, the joint distribution g satisﬁes
g j (y j |y 1 , . . . , y j−1 , y j+1 , . . . , y p )

p

g (y1 , . . . , yp ) ∝
j=1

for every permutation

g j (y j |y 1 , . . . , y j−1 , y j+1 , . . . , y p )

on {1, 2, . . . , p} and every y ∈ Y.

To Gibbs or not to Gibbs?

Julian Besag should certainly be credited to a large extent of the
(re?-)discovery of the Gibbs sampler.

To Gibbs or not to Gibbs?

Julian Besag should certainly be credited to a large extent of the
(re?-)discovery of the Gibbs sampler.
“The simulation procedure is to consider the sites
cyclically and, at each stage, to amend or leave unaltered
the particular site value in question, according to a
probability distribution whose elements depend upon the
current value at neighboring sites (...) However, the
technique is unlikely to be particularly helpful in many
other than binary situations and the Markov chain itself
has no practical interpretation.”
[Besag, 1974]

Clicking in

After Peskun (1973), MCMC mostly dormant in mainstream
statistical world for about 10 years, then several papers/books
highlighted its usefulness in speciﬁc settings:
Geman and Geman (1984)
Besag (1986)
Strauss (1986)
Ripley (Stochastic Simulation, 1987)
Tanner and Wong (1987)
Younes (1988)

Enters the Gibbs sampler

Geman and Geman (1984), building on
Metropolis et al. (1953), Hastings (1970), and
Peskun (1973), constructed a Gibbs sampler
for optimisation in a discrete image processing
problem with a Gibbs random ﬁeld without
completion.
Back to Metropolis et al., 1953: the Gibbs
sampler is already in use therein and ergodicity
is proven on the collection of global maxima

Besag (1986) integrates GS for SA...

“...easy to construct the transition matrix Q, of a
discrete time Markov chain, with state space Ω and limit
distribution (4). Simulated annealing proceeds by
running an associated time inhomogeneous Markov chain
with transition matrices QT , where T is progressively
decreased according to a prescribed “schedule” to a value
close to zero.”
[Besag, 1986]

...and links with Metropolis-Hastings...

“There are various related methods of constructing a
manageable QT (Hastings, 1970). Geman and Geman
(1984) adopt the simplest, which they term the ”Gibbs
sampler” (...) time reversibility, a common ingredient in
this type of problem (see, for example, Besag, 1977a), is
present at individual stages but not over complete cycles,
though Peter Green has pointed out that it returns if QT
is taken over a pair of cycles, the second of which visits
pixels in reverse order”
[Besag, 1986]

The candidate’s formula
Representation of the marginal likelihood as
m(x) =

π(θ)f (x|θ)
π(θ|x)

or of the marginal predictive as
pn (y |y ) = f (y |θ)πn (θ|y ) πn+1 (θ|y , y )
[Besag, 1989]

Why candidate?
“Equation (2) appeared without explanation in a Durham
University undergraduate ﬁnal examination script of
1984. Regrettably, the student’s name is no longer
known to me.”

Implications

Newton and Raftery (1994) used this representation to derive
the [infamous] harmonic mean approximation to the marginal
likelihood
Gelfand and Dey (1994)
Geyer and Thompson (1995)
Chib (1995)
Marin and Robert (2010) and Robert and Wraith (2009)
[Chen, Shao & Ibrahim, 2000]

Implications

Newton and Raftery (1994)
Gelfand and Dey (1994) also relied on this formula for the
same purpose in a more general perspective
Chib (1995)

Implications

Geyer and Thompson (1995) derived MLEs by a Monte Carlo
approximation to the normalising constant
Chib (1995)

Implications

Chib (1995) uses this representation to build a MCMC
approximation to the marginal likelihood

Implications

Chib (1995)
corrected Newton and Raftery (1994) by restricting the
importance function to an HPD region

Removing the jam

In early 1990s, researchers found that Gibbs and then Metropolis Hastings algorithms would crack almost any problem!
Flood of papers followed applying MCMC:
linear mixed models (Gelfand & al., 1990; Zeger & Karim, 1991;
Wang & al., 1993, 1994)
generalized linear mixed models (Albert & Chib, 1993)
mixture models (Tanner & Wong, 1987; Diebolt & Robert., 1990,
1994; Escobar & West, 1993)
changepoint analysis (Carlin & al., 1992)
point processes (Grenander & Møller, 1994)
&tc

Removing the jam
In early 1990s, researchers found that Gibbs and then Metropolis Hastings algorithms would crack almost any problem!
Flood of papers followed applying MCMC:
genomics (Stephens & Smith, 1993; Lawrence & al., 1993;
Churchill, 1995; Geyer & Thompson, 1995; Stephens & Donnelly,
2000)
ecology (George & Robert, 1992)
variable selection in regression (George & mcCulloch, 1993; Green,
1995; Chen & al., 2000)
spatial statistics (Raftery & Banﬁeld, 1991; Besag & Green, 1993))
longitudinal studies (Lange & al., 1992)
&tc

MCMC and beyond

reversible jump MCMC which impacted considerably Bayesian
model choice (Green, 1995)
adaptive MCMC algorithms (Haario & al., 1999; Roberts
& Rosenthal, 2009)
exact approximations to targets (Tanner & Wong, 1987;
Beaumont, 2003; Andrieu & Roberts, 2009)
particle ﬁlters with application to sequential statistics,
state-space models, signal processing, &tc. (Gordon & al.,
1993; Doucet & al., 2001; del Moral & al., 2006)

MCMC and beyond beyond

comp’al stats catching up with comp’al physics: free energy
sampling (e.g., Wang-Landau), Hamilton Monte Carlo
(Girolami & Calderhead, 2011)
sequential Monte Carlo (SMC) for non-sequential problems
(Chopin, 2002; Neal, 2001; Del Moral et al 2006)
retrospective sampling
intractability: EP – GIMH – PMCMC – SMC2 – INLA
QMC[MC] (Owen, 2011)

Particles

Iterating/sequential importance sampling is about as old as Monte
Carlo methods themselves!
[Hammersley and Morton,1954; Rosenbluth and Rosenbluth, 1955]

Found in the molecular simulation literature of the 50’s with
self-avoiding random walks and signal processing
[Marshall, 1965; Handschin and Mayne, 1969]
Use of the term “particle” dates back to Kitagawa (1996), and Carpenter
et al. (1997) coined the term “particle ﬁlter”.

pMC & pMCMC

Recycling of past simulations legitimate to build better
importance sampling functions as in population Monte Carlo
[Iba, 2000; Capp´ et al, 2004; Del Moral et al., 2007]
e

synthesis by Andrieu, Doucet, and Hollenstein (2010) using
particles to build an evolving MCMC kernel pθ (y1:T ) in state
^
space models p(x1:T )p(y1:T |x1:T )
importance sampling on discretely observed diﬀusions
[Beskos et al., 2006; Fearnhead et al., 2008, 2010]

towards ever more complexity

de Finetti’s exchangeability theorem
Bayesian nonparametrics
Bayesian analysis in a Big Data era

Who was Bruno de Finetti?
“Italian probabilist, statistician and
actuary, noted for the “operational
subjective” conception of probability.
The classic exposition of his distinctive
theory is the 1937 “La pr´vision: ses
e
lois logiques, ses sources subjectives,”
which discussed probability founded on
the coherence of betting odds and the
consequences of exchangeability.”
[Wikipedia]
Chair in Financial Mathematics at Trieste University (1939) and
Roma (1954) then in Calculus of Probabilities (1961). Most
famous sentence:
“Probability does not exist”

Exchangeability
Notion of exchangeable sequences:
A random sequence (x1 , . . . , xn , . . .) is exchangeable if for
any n the distribution of (x1 , . . . , xn ) is equal to the
distribution of any permutation of the sequence
(xσ1 , . . . , xσn )
de Finetti’s theorem (1937):
An exchangeable distribution is a mixture of iid
distributions
n

f (xi |G )dπ(G )

p(x1 , . . . , xn ) =
i=1

where G can be inﬁnite-dimensional
Extension to Markov chains (Freedman, 1962; Diaconis
& Freedman, 1980)

Bayesian nonparametrics

Based on de Finetti’s representation,
use of priors on functional spaces (densities, regression, trees,
partitions, clustering, &tc)
production of Bayes estimates in those spaces
convergence mileage may vary
available eﬃcient (MCMC) algorithms to conduct
non-parametric inference
[van der Vaart, 1998; Hjort et al., 2010; M¨ller & Rodriguez, 2013]
u

Dirichlet processes

One of the earliest examples of priors on distributions
[Ferguson, 1973]

stick-breaking construction of D(α0 , G0 )
generate βk ∼ B(1, α0 )
deﬁne π1 = β1 and πk =

k−1
j=1 (1

− βj )βk

generate θk ∼ G0
derive G =

k

πk δθk ∼ D(α0 , G0 )
[Sethuraman, 1994]

Chinese restaurant process
If we assume
G ∼ D(α0 , G0 )
θi ∼ G

then the marginal distribution of (θ1 , . . .) is a Chinese restaurant
process (P´lya urn model), which is exchangeable. In particular,
o
i−1

θi |θ1:i−1 ∼ α0 G0 +

δθj
j=1

Posterior distribution built by MCMC
[Escobar and West, 1992]

Many alternatives

truncated Dirichlet processes
Pitman Yor processes
completely random measures
normalized random measures with independent increments
(NRMI)
[M¨ller and Mitra, 2013]
u

Theoretical advances

posterior consistency: Seminal work of Schwarz (1965) in iid
case and extension of Barron et al. (1999) for general
consistency
consistency rates: Ghosal & van der Vaart (2000) Ghosal et
al. (2008) with minimax (adaptive ) Bayesian nonparametric
estimators for nonparametric process mixtures (Gaussian,
Beta) (Rousseau, 2008; Kruijer, Rousseau & van der Vaart,
2010; Shen, Tokdar & Ghosal, 2013; Scricciolo, 2013)
Bernstein-von Mises theorems: (Castillo, 2011; Rivoirard
& Rousseau, 2012; Kleijn & Bickel, 2013; Castillo
& Rousseau, 2013)
recent extensions to semiparametric models

Consistency and posterior concentration rates

Posterior
dπ(θ|X n ) =

fθ (X n )dπ(θ)
m(X n )

fθ (X n )dπ(θ)

m(X n ) =
Θ

and posterior concentration: Under Pθ0
Pπ [d(θ, θ0 )

|X n ] = 1+op (1),

Pπ [d(θ, θ0 )

n |X

n

] = 1+op (1)

Given n : consistency
where d(θ, θ ) is a loss function. e.g. Hellinger, L1 , L2 , L∞

Consistency and posterior concentration rates

Posterior
dπ(θ|X n ) =

fθ (X n )dπ(θ)
m(X n )

fθ (X n )dπ(θ)

m(X n ) =
Θ

and posterior concentration: Under Pθ0
Pπ [d(θ, θ0 )

|X n ] = 1+op (1),

Pπ [d(θ, θ0 )

n |X

n

] = 1+op (1)

Setting n ↓ 0: consistency rates
where d(θ, θ ) is a loss function. e.g. Hellinger, L1 , L2 , L∞

Bernstein–von Mises theorems

Parameter of interest
ψ = ψ(θ) ∈ Rd ,
(with dim(θ) = +∞)
BVM:
√
^
π[ n(ψ − ψ)
and

d < +∞,

z|X n ] = Φ(z/

θ∼π

V0 ) + op (1),

√
^
n(ψ − ψ(θ0 )) ≈ N(0, V0 )

Pθ0

under Pθ0

[Doob, 1949; Le Cam, 1986; van der Vaart, 1998]

New challenges

Novel statisticial issues that forces a diﬀerent Bayesian answer:
very large datasets
complex or unknown dependence structures with maybe p
multiple and involved random eﬀects
missing data structures containing most of the information
sequential structures involving most of the above

n

New paradigm?

“Surprisingly, the conﬁdent prediction of the previous
generation that Bayesian methods would ultimately supplant
frequentist methods has given way to a realization that Markov
chain Monte Carlo (MCMC) may be too slow to handle
modern data sets. Size matters because large data sets stress
computer storage and processing power to the breaking point.
The most successful compromises between Bayesian and
frequentist methods now rely on penalization and
optimization.”

[Lange at al., ISR, 2013]

New paradigm?

Observe (Xi , Ri , Yi Ri ) where
Xi ∼ U(0, 1)d , Ri |Xi ∼ B(π(Xi ))

and Yi |Xi ∼ B(θ(Xi ))

(π(·) is known and θ(·) is unknwon)
Then any estimator of E[Y ] that does not depend on π is
inconsistent.
c There is no genuine Bayesian answer producing a consistent
estimator (without throwing away part of the data)
[Robins & Wasserman, 2000, 2013]

New paradigm?

sad reality constraint that
size does matter
focus on much smaller
dimensions and on sparse
summaries
many (fast if non-Bayesian)
ways of producing those
summaries
Bayesian inference can kick
in almost automatically at
this stage

Approximate Bayesian computation (ABC)
Case of a well-deﬁned statistical model where the likelihood
function
(θ|y) = f (y1 , . . . , yn |θ)
is out of reach!
Empirical approximations to the original
Bayesian inference problem
Degrading the data precision down
to a tolerance ε
Replacing the likelihood with a
non-parametric approximation
Summarising/replacing the data
with insuﬃcient statistics

ABC methodology
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:

Foundation
For an observation y ∼ f (y|θ), under the prior π(θ), if one keeps
jointly simulating
θ ∼ π(θ) , z ∼ f (z|θ ) ,
until the auxiliary variable z is equal to the observed value, z = y,
then the selected
θ ∼ π(θ|y)
[Rubin, 1984; Diggle & Gratton, 1984; Griﬃth et al., 1997]

ABC algorithm

In most implementations, degree of approximation:
Algorithm 1 Likelihood-free rejection sampler
for i = 1 to N do
repeat
generate θ from the prior distribution π(·)
generate z from the likelihood f (·|θ )
until ρ{η(z), η(y)}
set θi = θ
end for
where η(y) deﬁnes a (not necessarily suﬃcient) statistic

Comments

role of distance paramount
(because = 0)
scaling of components of η(y) also
capital
matters little if “small enough”
representative of “curse of
dimensionality”
small is beautiful!, i.e. data as a
whole may be weakly informative
for ABC
non-parametric method at core

ABC simulation advances

Simulating from the prior is often poor in eﬃciency
Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y ...
[Marjoram et al, 2003; Beaumont et al., 2009, Del Moral et al., 2012]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002; Blum & Fran¸ois, 2010; Biau et al., 2013]
c
.....or even by including

in the inferential framework [ABCµ ]
[Ratmann et al., 2009]

ABC as an inference machine

Starting point is summary statistic η(y), either chosen for
computational realism or imposed by external constraints
ABC can produce a distribution on the parameter of interest
conditional on this summary statistic η(y)
inference based on ABC may be consistent or not, so it needs
to be validated on its own
the choice of the tolerance level is dictated by both
computational and convergence constraints

How Bayesian aBc is..?

At best, ABC approximates π(θ|η(y)):
approximation error unknown (w/o massive simulation)
pragmatic or empirical Bayes (there is no other solution!)
many calibration issues (tolerance, distance, statistics)
the NP side should be incorporated into the whole Bayesian
picture
the approximation error should also be part of the Bayesian
inference

Noisy ABC
ABC approximation error (under non-zero tolerance ) replaced
with exact simulation from a controlled approximation to the
target, convolution of true posterior with kernel function
π (θ, z|y) =

π(θ)f (z|θ)K (y − z)
,
π(θ)f (z|θ)K (y − z)dzdθ

with K kernel parameterised by bandwidth .
[Wilkinson, 2013]

Theorem
˜
The ABC algorithm based on a randomised observation y = y + ξ,
ξ ∼ K , and an acceptance probability of
K (y − z)/M
gives draws from the posterior distribution π(θ|y).

Which summary?

Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics [except when done by the
experimenters in the field]

Which summary?

Loss of statistical information balanced against gain in data
roughening
Approximation error and information loss remain unknown
Choice of statistics induces choice of distance function
towards standardisation
borrowing tools from data analysis (LDA) machine learning
[Estoup et al., ME, 2012]

Which summary?

may be imposed for external/practical reasons
may gather several non-B point estimates
we can learn about eﬃcient combination
distance can be provided by estimation techniques

Which summary for model choice?
‘This is also why focus on model discrimination typically
(...) proceeds by (...) accepting that the Bayes Factor
that one obtains is only derived from the summary
statistics and may in no way correspond to that of the
full model.’
[Scott Sisson, Jan. 31, 2011, xianblog]

Depending on the choice of η(·), the Bayes factor based on this
insuﬃcient statistic,
η
B12 (y) =

π1 (θ1 )f1η (η(y)|θ1 ) dθ1
,
π2 (θ2 )f2η (η(y)|θ2 ) dθ2

is either consistent or inconsistent
[Robert et al., PNAS, 2012]

Which summary for model choice?
Depending on the choice of η(·), the Bayes factor based on this
insuﬃcient statistic,
η
B12 (y)

=

π1 (θ1 )f1η (η(y)|θ1 ) dθ1
,
π2 (θ2 )f2η (η(y)|θ2 ) dθ2

is either consistent or inconsistent
[Robert et al., PNAS, 2012]
n=100

0.7

1.0

n=100

q

0.6

q

0.8

q
q

0.5

q
q
q

q
q
q
q

0.4

q
q
q

q

0.1

0.2

q

q

0.2

0.3

0.4

0.6

q

q
q
q

q
q

0.0

q
q
q

Gauss

Laplace

0.0

q

Gauss

Laplace

Selecting proper summaries
Consistency only depends on the range of
µi (θ) = Ei [η(y)]
under both models against the asymptotic mean µ0 of η(y)

Theorem
If Pn belongs to one of the two models and if µ0 cannot be
attained by the other one :
0 = min (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2)
< max (inf{|µ0 − µi (θi )|; θi ∈ Θi }, i = 1, 2) ,
η
then the Bayes factor B12 is consistent

[Marin et al., 2012]

Selecting proper summaries
Consistency only depends on the range of
µi (θ) = Ei [η(y)]
0.7
0.6

0.7

0.4

0.4
q

0.5

0.6

q

0.5

0.7
0.4

0.5

0.6

under both models against the asymptotic mean µ0 of η(y)

q
q

M2

M1

M2

1.0

M1
1.0

M2

0.3

0.3

0.3

q

M1

0.8

0.8

0.8

q
q

q

q

q
q

q

0.6

0.6

0.6

q

q
q

q

q

q
q

q
q
q

0.2

q
q
q
q

0.2

0.2

q
q
q

q

0.4

0.4

0.4

q
q

M1

M2

M2

1.0

M1

0.8

0.8

0.8

1.0

M2

0.0

0.0

0.0

q

M1

q
q
q
q

0.4

q
q

q
q
q

q

q
q

0.2

q

0.2

0.2

q
q

q
q
q

0.4

0.4

0.6

0.6

q
q

0.6

q
q

M1

M2

0.0

q
q

0.0

0.0

q

M1

M2

M1

M2

[Marin et al., 2012]

on some Bayesian open problems
In 2011, Michael Jordan, then ISBA President, conducted a
mini-survey on Bayesian open problems:
Nonparametrics and semiparametrics: assessing and validating
priors on inﬁnite dimension spaces with an inﬁnite number of
nuisance parameters
Priors: elicitation mecchanisms and strategies to get the prior
from the likelihood or even from the posterior distribution
Bayesian/frequentist relationships: how far should one reach
for frequentist validation?
Computation and statistics: computational abilities should be
part of the modelling, with some expressing doubts about
INLA and ABC
Model selection and hypothesis testing: still unsettled
opposition between model checking, model averaging and
model selection
[Jordan, ISBA Bulletin, March 2011]

yet another Bayes 250

Meeting that will take place in Duke University, December 17:
Stephen Fienberg, Carnegie-Mellon
University
Michael Jordan, University of
California, Berkeley
Christopher Sims, Princeton University
Adrian Smith, University of London
Stephen Stigler, University of Chicago
Sharon Bertsch McGrayne, author of
“the theory that would not die”

Robert

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