While some of the enclosed already is a well-known derivation, and the
remaining may have been obtained in earlier publications, this note computes
the first two moments of a Student's variate truncated at zero and of an
absolute (or folded) Student's variate.

For many decades, statisticians have made attempts to prepare the Bayesian
omelette without breaking the Bayesian eggs; that is, to obtain probabilistic
likelihood-based inferences without relying on informative prior distributions.
A recent example is Murray Aitkin's recent book, {\em Statistical Inference},
which presents an approach to statistical hypothesis testing based on
comparisons of posterior distributions of likelihoods under competing models.
Aitkin develops and illustrates his method using some simple examples of
inference from iid data and two-way tests of independence.

Approximate Bayesian computation (ABC) have become a essential tool for the
analysis of complex stochastic models. Earlier, Grelaud et al. (2009) advocated
the use of ABC for Bayesian model choice in the specific case of Gibbs random
fields, relying on a inter-model sufficiency property to show that the
approximation was legitimate.

The Adaptive Multiple Importance Sampling (AMIS) algorithm is aimed at an
optimal recycling of past simulations in an iterated importance sampling
scheme. The difference with earlier adaptive importance sampling
implementations like Population Monte Carlo is that the importance weights of
all simulated values, past as well as present, are recomputed at each
iteration, following the technique of the deterministic multiple mixture
estimator of Owen and Zhou (2000).

In this paper, we show how a complete and exact Bayesian analysis of a
parametric mixture model is possible in some cases when components of the
mixture are taken from exponential families and when conjugate priors are used.
This restricted set-up allows us to show the relevance of the Bayesian approach
as well as to exhibit the limitations of a complete analysis, namely that it is
impossible to conduct this analysis when the sample size is too large, when the
data are not from an exponential family, or when priors that are more complex
than conjugate priors are used.

We propose a global noninformative approach for Bayesian variable selection
that builds on Zellner's g-priors and is similar to Liang et al. (2008). Our
proposal does not require any kind of calibration. In the case of a benchmark,
we compare Bayesian and frequentist regularization approaches under a low
informative constraint when the number of variables is almost equal to the
number of observations. The simulated and real dataset experiments we present
here highlight the appeal of Bayesian regularization methods, when compared
with alternatives.

This document is the aggregation of several discussions of Lopes et al.
(2010) we submitted to the proceedings of the Ninth Valencia Meeting, held in
Benidorm, Spain, on June 3-8, 2010, in conjunction with Hedibert Lopes' talk at
this meeting. The main point in those discussions is the potential for
degeneracy in the particle learning methodology, related with the exponential
forgetting of the past simulations. We illustrate the resulting difficulties in
the case of mixtures.

The book A Treatise on Probability was published by John Maynard Keynes in
1921. It contains a critical assessment of the foundations of probability and
of the current statistical methodology. As a modern reader, we review here the
aspects that are most related with statistics, avoiding a neophyte's
perspective on the philosophical issues. In particular, the book is quite
critical of the Bayesian approach and we examine the arguments provided by
Keynes, as well as the alternative he proposes.

In this chapter, we will first present the most standard computational
challenges met in Bayesian Statistics, focussing primarily on mixture
estimation and on model choice issues, and then relate these problems with
computational solutions. Of course, this chapter is only a terse introduction
to the problems and solutions related to Bayesian computations. For more
complete references, see Robert and Casella (2004, 2009), or Marin and Robert
(2007), among others.

The book The Search for Certainty published in 2009 by Krzysztof Burdzy
examines the "philosophical duopoly" at the foundation of statistics and find
it missing. We point out in this review the weakness of the scholarly arguments
presented in the book, we question the relevance of introducing a new set of
probability axioms, and we conclude on the lack of impact of this book on
statistical foundations and on Bayesian statistics in particular.

This is the solution manual to the odd-numbered exercises in our book
"Introducing Monte Carlo Methods with R", published by Springer Verlag on
December 10, 2009, and made freely available to everyone.

Every reversible Markov chain defines an operator whose spectrum encodes the
convergence properties of the chain. When the state space is finite, the
spectrum is just the set of eigenvalues of the corresponding Markov transition
matrix. However, when the state space is infinite, the spectrum may be
uncountable, and is nearly always impossible to calculate. In most applications
of the data augmentation (DA) algorithm, the state space of the DA Markov chain
is infinite.

Casella and Robert (1996) presented a general Rao--Blackwellisation principle
for accept-reject and Metropolis-Hastings schemes that leads to significant
decreases in the variance of the resulting estimators, but at a high cost in
computing and storage. Adopting a completely different perspective, we
introduce instead a universal scheme that guarantees variance reductions in all
Metropolis-Hastings based estimators while keeping the computing cost under
control.

This paper surveys some well-established approaches on the approximation of
Bayes factors used in Bayesian model choice, mostly as covered in Chen et al.
(2000). Our focus here is on methods that are based on importance sampling
strategies rather than variable dimension techniques like reversible jump MCMC,
including: crude Monte Carlo, maximum likelihood based importance sampling,
bridge and harmonic mean sampling, as well as Chib's method based on the
exploitation of a functional equality.

We argue here about the relevance and the ultimate unity of the Bayesian
approach in a non-conflicting and non-antagonistic manner. Our main theme is
that Bayesian data analysis is an effective tool for handling complex models,
as proven by the increasing proportion of Bayesian studies in the applied
sciences. We disregard in this essay the philosophical debates on the deeper
meaning of probability and on the random nature of parameters as things of the
past that do a disservice to the approach and are incomprehensible to most
bystanders.

We are grateful to all discussants (Bernardo, Gelman, Kass, Lindley, Senn,
and Zellner) of our re-visitation for their strong support in our enterprise
and for their overall agreement with our perspective. Further discussions with
them and other leading statisticians showed that the legacy of Theory of
Probability is alive and lasting.