256,106 research outputs found
Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces
We prove that intersections and unions of independent random sets in finite
spaces achieve a form of Lipschitz continuity. More precisely, given the
distribution of a random set , the function mapping any random set
distribution to the distribution of its intersection (under independence
assumption) with is Lipschitz continuous with unit Lipschitz constant if
the space of random set distributions is endowed with a metric defined as the
norm distance between inclusion functionals also known as commonalities.
Moreover, the function mapping any random set distribution to the distribution
of its union (under independence assumption) with is Lipschitz continuous
with unit Lipschitz constant if the space of random set distributions is
endowed with a metric defined as the norm distance between hitting
functionals also known as plausibilities.
Using the epistemic random set interpretation of belief functions, we also
discuss the ability of these distances to yield conflict measures. All the
proofs in this paper are derived in the framework of Dempster-Shafer belief
functions. Let alone the discussion on conflict measures, it is straightforward
to transcribe the proofs into the general (non necessarily epistemic) random
set terminology
Density evolution for SUDOKU codes on the erasure channel
Codes based on SUDOKU puzzles are discussed,
and belief propagation decoding introduced for the erasure
channel. Despite the non-linearity of the code constraints, it
is argued that density evolution can be used to analyse code
performance due to the invariance of the code under alphabet
permutation. The belief propagation decoder for erasure channels
operates by exchanging messages containing sets of possible
values. Accordingly, density evolution tracks the probability
mass functions of the set cardinalities. The equations governing
the mapping of those probability mass functions are derived
and calculated for variable and constraint nodes, and decoding
thresholds are computed for long SUDOKU codes with random
interleavers.Funded in part by the European Research Council under ERC grant
agreement 259663 and by the FP7 Network of Excellence NEWCOM# under
grant agreement 318306.This is the accepted manuscript. The final version is available from IEEE at http://dx.doi.org/10.1109/ISTC.2014.6955120
Choice functions as a tool to model uncertainty
Our aim is to develop a tool for modelling different types of assessments about the uncertain value of some random variable. One well-know and widely used way to model uncertainty is using probability mass functions. However, such probability mass functions are not general enough to model, for instance, a total lack of knowledge. A very successful tool for modelling more general types of assessments is coherent sets of desirable gambles. These have many applications in credal networks, predictive inference, conservative reasoning, and so on. However, they are not capable of modelling beliefs corresponding to 'or' statements, for example the belief that a coin has two equal sides of unknown type: either twice heads or twice tails. Such more general assessments can be modelled with coherent choice functions.
The first thing we do is relate coherent choice functions to coherent sets of desirable gambles, which yields an expression for the most conservative coherent choice function compatible with a coherent set of desirable gambles. Next, we study the order-theoretic properties of coherent choice functions. In order for our theory of choice functions to be successful, we need a good conditioning rule. We propose a very intuitive one, and show that it coincides with the usual one for coherent sets of desirable gambles, and therefore also leads to Bayes’s rule. To conclude, we show how to elegantly deal with assessments of indifference
The total belief theorem
In this paper, motivated by the treatment of conditional constraints in the data association problem, we state and prove the generalisation of the law of total probability to belief functions, as finite random sets. Our results apply to the case in which Dempster’s conditioning is employed. We show that the solution to the resulting total belief problem is in general not unique, whereas it is unique when the a-priori belief function is Bayesian. Examples and case studies underpin the theoretical contributions. Finally, our results are compared to previous related work on the generalisation of Jeffrey’s rule by Spies and Smets
On the Privacy of Sublinear-Communication Jaccard Index Estimation via Min-hash Sketching
The min-hash sketch is a well-known technique for low-communication approximation of the Jaccard index between two input sets. Moreover, there is a folklore belief that min-hash sketch based protocols protect the privacy of the inputs. In this paper, we investigate this folklore to quantify the privacy of the min-hash sketch.
We begin our investigation by considering the privacy of min-hash in a centralized setting where the hash functions are chosen by the min-hash functionality and are unknown to the participants. We show that in this case the min-hash output satisfies the standard definition of differential privacy (DP) without any additional noise. This immediately yields a privacy-preserving sublinear-communication semi-honest 2-PC protocol based on FHE where the hash function is evaluated homomorphically.
To improve the efficiency of this protocol, we next consider an implementation in the random oracle model. Here, the protocol participants jointly sample public prefixes for domain separation of the random oracle, and locally evaluate the resulting hash functions on their input sets. Unfortunately, we show that in this public hash function setting, the min-hash output is no longer DP. We therefore consider the notion of distributional differential privacy (DDP) introduced by Bassily et al.~(FOCS 2013). We show that if the honest party\u27s set has sufficiently high min-entropy then the output of the min-hash functionality achieves DDP, again without any added noise. This yields a more efficient semi-honest two-party protocol in the random oracle model, where parties first locally hash their input sets and then perform a 2PC for comparison.
By proving that our protocols satisfy DP and DDP respectively, our results formally confirm and qualify the folklore belief that min-hash based protocols protect the privacy of their inputs
Random sets and exact confidence regions
An important problem in statistics is the construction of confidence regions
for unknown parameters. In most cases, asymptotic distribution theory is used
to construct confidence regions, so any coverage probability claims only hold
approximately, for large samples. This paper describes a new approach, using
random sets, which allows users to construct exact confidence regions without
appeal to asymptotic theory. In particular, if the user-specified random set
satisfies a certain validity property, confidence regions obtained by
thresholding the induced data-dependent plausibility function are shown to have
the desired coverage probability.Comment: 14 pages, 2 figure
Inferential models: A framework for prior-free posterior probabilistic inference
Posterior probabilistic statistical inference without priors is an important
but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of
belief functions, and Bayesian inference with default priors are attempts to
achieve this goal but, to date, none has given a completely satisfactory
picture. This paper presents a new framework for probabilistic inference, based
on inferential models (IMs), which not only provides data-dependent
probabilistic measures of uncertainty about the unknown parameter, but does so
with an automatic long-run frequency calibration property. The key to this new
approach is the identification of an unobservable auxiliary variable associated
with observable data and unknown parameter, and the prediction of this
auxiliary variable with a random set before conditioning on data. Here we
present a three-step IM construction, and prove a frequency-calibration
property of the IM's belief function under mild conditions. A corresponding
optimality theory is developed, which helps to resolve the non-uniqueness
issue. Several examples are presented to illustrate this new approach.Comment: 29 pages with 3 figures. Main text is the same as the published
version. Appendix B is an addition, not in the published version, that
contains some corrections and extensions of two of the main theorem
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