20 research outputs found
Context Tree Selection: A Unifying View
The present paper investigates non-asymptotic properties of two popular
procedures of context tree (or Variable Length Markov Chains) estimation:
Rissanen's algorithm Context and the Penalized Maximum Likelihood criterion.
First showing how they are related, we prove finite horizon bounds for the
probability of over- and under-estimation. Concerning overestimation, no
boundedness or loss-of-memory conditions are required: the proof relies on new
deviation inequalities for empirical probabilities of independent interest. The
underestimation properties rely on loss-of-memory and separation conditions of
the process.
These results improve and generalize the bounds obtained previously. Context
tree models have been introduced by Rissanen as a parsimonious generalization
of Markov models. Since then, they have been widely used in applied probability
and statistics
Functional Sequential Treatment Allocation
Consider a setting in which a policy maker assigns subjects to treatments,
observing each outcome before the next subject arrives. Initially, it is
unknown which treatment is best, but the sequential nature of the problem
permits learning about the effectiveness of the treatments. While the
multi-armed-bandit literature has shed much light on the situation when the
policy maker compares the effectiveness of the treatments through their mean,
much less is known about other targets. This is restrictive, because a cautious
decision maker may prefer to target a robust location measure such as a
quantile or a trimmed mean. Furthermore, socio-economic decision making often
requires targeting purpose specific characteristics of the outcome
distribution, such as its inherent degree of inequality, welfare or poverty. In
the present paper we introduce and study sequential learning algorithms when
the distributional characteristic of interest is a general functional of the
outcome distribution. Minimax expected regret optimality results are obtained
within the subclass of explore-then-commit policies, and for the unrestricted
class of all policies
Context tree selection: A unifying view
Context tree models have been introduced by Rissanen in [25] as a parsimonious generalization of Markov models. Since then, they have been widely used in applied probability and statistics. The present paper investigates non-asymptotic properties of two popular procedures of context tree estimation: RissanenĂąs algorithm Context and penalized maximum likelihood. First showing how they are related, we prove finite horizon bounds for the probability of over- and under-estimation. Concerning over-estimation, no boundedness or loss-of-memory conditions are required: the proof relies on new deviation inequalities for empirical probabilities of independent interest. The under-estimation properties rely on classical hypotheses for processes of infinite memory. These results improve on and generalize the bounds obtained in Duarte etĂ al. (2006) [12], Galves etĂ al. (2008) [18], Galves and Leonardi (2008) [17], Leonardi (2010) [22], refining asymptotic results of Böhlmann and Wyner (1999) [4] and CsiszĂĄr and Talata (2006) [9].Algorithm Context Penalized maximum likelihood Model selection Variable length Markov chains Bayesian information criterion Deviation inequalities
Pseudo-regenerative block-bootstrap for hidden Markov chains
International audienc
Bregman superquantiles. Estimation methods and applications
In thiswork,we extend some parameters built on a probability distribution introduced before to the
casewhere the proximity between real numbers is measured by using a Bregman divergence. This leads to the
definition of the Bregman superquantile (thatwe can connect with severalworks in economy, see for example
[18] or [9]). Axioms of a coherent measure of risk discussed previously (see [31] or [3]) are studied in the case
of Bregman superquantile. Furthermore,we deal with asymptotic properties of aMonte Carlo estimator of the
Bregman superquantile. Several numerical tests confirm the theoretical results and an application illustrates
the potential interests of the Bregman superquantile