3 research outputs found
Operational Interpretation of Renyi Information Measures via Composite Hypothesis Testing Against Product and Markov Distributions
We revisit the problem of asymmetric binary hypothesis testing against a
composite alternative hypothesis. We introduce a general framework to treat
such problems when the alternative hypothesis adheres to certain axioms. In
this case we find the threshold rate, the optimal error and strong converse
exponents (at large deviations from the threshold) and the second order
asymptotics (at small deviations from the threshold). We apply our results to
find operational interpretations of various Renyi information measures. In case
the alternative hypothesis is comprised of bipartite product distributions, we
find that the optimal error and strong converse exponents are determined by
variations of Renyi mutual information. In case the alternative hypothesis
consists of tripartite distributions satisfying the Markov property, we find
that the optimal exponents are determined by variations of Renyi conditional
mutual information. In either case the relevant notion of Renyi mutual
information depends on the precise choice of the alternative hypothesis. As
such, our work also strengthens the view that different definitions of Renyi
mutual information, conditional entropy and conditional mutual information are
adequate depending on the context in which the measures are used.Comment: published versio
Universal channel coding for general output alphabet
We propose two types of universal codes that are suited to two asymptotic
regimes when the output alphabet is possibly continuous. The first class has
the property that the error probability decays exponentially fast and we
identify an explicit lower bound on the error exponent. The other class attains
the epsilon-capacity the channel and we also identify the second-order term in
the asymptotic expansion. The proposed encoder is essentially based on the
packing lemma of the method of types. For the decoder, we first derive a
R\'enyi-relative-entropy version of Clarke and Barron's formula the distance
between the true distribution and the Bayesian mixture, which is of independent
interest. The universal decoder is stated in terms of this formula and
quantities used in the information spectrum method. The methods contained
herein allow us to analyze universal codes for channels with continuous and
discrete output alphabets in a unified manner, and to analyze their
performances in terms of the exponential decay of the error probability and the
second-order coding rate.Comment: Several typos are fixe
Equivocations, Exponents and Second-Order Coding Rates under Various R\'enyi Information Measures
We evaluate the asymptotics of equivocations, their exponents as well as
their second-order coding rates under various R\'{e}nyi information measures.
Specifically, we consider the effect of applying a hash function on a source
and we quantify the level of non-uniformity and dependence of the compressed
source from another correlated source when the number of copies of the sources
is large. Unlike previous works that use Shannon information measures to
quantify randomness, information or uniformity, we define our security measures
in terms of a more general class of information measures--the R\'{e}nyi
information measures and their Gallager-type counterparts. A special case of
these R\'{e}nyi information measure is the class of Shannon information
measures. We prove tight asymptotic results for the security measures and their
exponential rates of decay. We also prove bounds on the second-order
asymptotics and show that these bounds match when the magnitudes of the
second-order coding rates are large. We do so by establishing new classes
non-asymptotic bounds on the equivocation and evaluating these bounds using
various probabilistic limit theorems asymptotically.Comment: 47 pages, 9 figures; Presented at the 2015 International Symposium on
Information Theory (Hong Kong); Submitted to the IEEE Transactions on
Information Theory; v3: fixed typos and added some clarifications to the
proof