scholarly journals Guessing with a Bit of Help

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 39 ◽  
Author(s):  
Nir Weinberger ◽  
Ofer Shayevitz
Keyword(s):  
Lower Bound ◽  
Data Processing ◽  
Maximum Entropy ◽  
Binary Vector ◽  
Upper Bounds ◽  
Fixed Number ◽  
Binary Symmetric Channel ◽  
Symmetric Channel ◽  
Data Processing Inequality ◽  
Binary Erasure Channel

What is the value of just a few bits to a guesser? We study this problem in a setup where Alice wishes to guess an independent and identically distributed (i.i.d.) random vector and can procure a fixed number of k information bits from Bob, who has observed this vector through a memoryless channel. We are interested in the guessing ratio, which we define as the ratio of Alice’s guessing-moments with and without observing Bob’s bits. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two upper bounds on the guessing ratio by analyzing the performance of the dictator (for general k ≥ 1 ) and majority functions (for k = 1 ). We further provide a lower bound via maximum entropy (for general k ≥ 1 ) and a lower bound based on Fourier-analytic/hypercontractivity arguments (for k = 1 ). We then extend our maximum entropy argument to give a lower bound on the guessing ratio for a general channel with a binary uniform input that is expressed using the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel and conjecture that greedy dictator functions achieve the optimal guessing ratio.

scholarly journals THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS

2009 ◽  
Vol 80 (3) ◽  
pp. 430-453 ◽  
Author(s):  
JOSEF DICK
Keyword(s):  
Lower Bound ◽  
Fractional Order ◽  
Bounded Variation ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Upper Bounds ◽  
Reproducing Kernel Hilbert Spaces ◽  
Smooth Functions ◽  
Kernel Hilbert Spaces

AbstractWe give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and nonperiodic reproducing kernel Hilbert spaces. A lower bound which shows that our results are best possible is also shown.

scholarly journals Full-Duplex Relay with Delayed CSI Elevates the SDoF of the MIMO X Channel

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1484
Author(s):  
Tong Zhang ◽  
Gaojie Chen ◽  
Shuai Wang ◽  
Rui Wang
Keyword(s):  
Lower Bound ◽  
Channel State Information ◽  
Interference Alignment ◽  
Upper Bounds ◽  
Full Duplex ◽  
Multiple Antenna ◽  
Lower And Upper Bounds ◽  
Channel State ◽  
State Information ◽  
Full Duplex Relay

In this article, the sum secure degrees-of-freedom (SDoF) of the multiple-input multiple-output (MIMO) X channel with confidential messages (XCCM) and arbitrary antenna configurations is studied, where there is no channel state information (CSI) at two transmitters and only delayed CSI at a multiple-antenna, full-duplex, and decode-and-forward relay. We aim at establishing the sum-SDoF lower and upper bounds. For the sum-SDoF lower bound, we design three relay-aided transmission schemes, namely, the relay-aided jamming scheme, the relay-aided jamming and one-receiver interference alignment scheme, and the relay-aided jamming and two-receiver interference alignment scheme, each corresponding to one case of antenna configurations. Moreover, the security and decoding of each scheme are analyzed. The sum-SDoF upper bound is proposed by means of the existing SDoF region of two-user MIMO broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). As a result, the sum-SDoF lower and upper bounds are derived, and the sum-SDoF is characterized when the relay has sufficiently large antennas. Furthermore, even assuming no CSI at two transmitters, our results show that a multiple-antenna full-duplex relay with delayed CSI can elevate the sum-SDoF of the MIMO XCCM. This is corroborated by the fact that the derived sum-SDoF lower bound can be greater than the sum-SDoF of the MIMO XCCM with output feedback and delayed CSIT.

scholarly journals Disentangling environmental effects in microbial association networks

Microbiome ◽  
2021 ◽  
Vol 9 (1) ◽  
Author(s):  
Ina Maria Deutschmann ◽  
Gipsi Lima-Mendez ◽  
Anders K. Krabberød ◽  
Jeroen Raes ◽  
Sergio M. Vallina ◽  
...  
Keyword(s):  
Data Processing ◽  
Simulated Data ◽  
Potential Interaction ◽  
Microbial Interactions ◽  
Ecological Interactions ◽  
Sign Pattern ◽  
Interaction Information ◽  
Microbial Association ◽  
Environmental Selection ◽  
Data Processing Inequality

Abstract Background Ecological interactions among microorganisms are fundamental for ecosystem function, yet they are mostly unknown or poorly understood. High-throughput-omics can indicate microbial interactions through associations across time and space, which can be represented as association networks. Associations could result from either ecological interactions between microorganisms, or from environmental selection, where the association is environmentally driven. Therefore, before downstream analysis and interpretation, we need to distinguish the nature of the association, particularly if it is due to environmental selection or not. Results We present EnDED (environmentally driven edge detection), an implementation of four approaches as well as their combination to predict which links between microorganisms in an association network are environmentally driven. The four approaches are sign pattern, overlap, interaction information, and data processing inequality. We tested EnDED on networks from simulated data of 50 microorganisms. The networks contained on average 50 nodes and 1087 edges, of which 60 were true interactions but 1026 false associations (i.e., environmentally driven or due to chance). Applying each method individually, we detected a moderate to high number of environmentally driven edges—87% sign pattern and overlap, 67% interaction information, and 44% data processing inequality. Combining these methods in an intersection approach resulted in retaining more interactions, both true and false (32% of environmentally driven associations). After validation with the simulated datasets, we applied EnDED on a marine microbial network inferred from 10 years of monthly observations of microbial-plankton abundance. The intersection combination predicted that 8.3% of the associations were environmentally driven, while individual methods predicted 24.8% (data processing inequality), 25.7% (interaction information), and up to 84.6% (sign pattern as well as overlap). The fraction of environmentally driven edges among negative microbial associations in the real network increased rapidly with the number of environmental factors. Conclusions To reach accurate hypotheses about ecological interactions, it is important to determine, quantify, and remove environmentally driven associations in marine microbial association networks. For that, EnDED offers up to four individual methods as well as their combination. However, especially for the intersection combination, we suggest using EnDED with other strategies to reduce the number of false associations and consequently the number of potential interaction hypotheses.

scholarly journals Digital Link—A

2021 ◽  
pp. 115-142
Author(s):  
Jean Walrand
Keyword(s):  
Decision Making ◽  
Decision Problem ◽  
False Negative ◽  
Main Tool ◽  
Maximum Likelihood Estimates ◽  
Modulation Scheme ◽  
Bayes Rule ◽  
Binary Symmetric Channel ◽  
Symmetric Channel ◽  
Complex Modulation

AbstractIn a digital link, a transmitter converts bits into signals and a receiver converts the signals it receives into bits. The receiver faces a decision problem that we study in Sect. 7.1. The main tool is Bayes’ Rule. The key notions are maximum a posteriori and maximum likelihood estimates. Transmission systems use codes to reduce the number of bits they need to transmit. Section 7.2 explains the Huffman codes that minimize the expected number of bits needed to transmit symbols; the idea is to use fewer bits for more likely symbols. Section 7.3 explores a commonly used model of a communication channel: the binary symmetric channel. It explains how to calculate the probability of errors. Section 7.4 studies a more complex modulation scheme employed by most smartphones and computers: QAM. Section 7.5 is devoted to a central problem in decision making: how to infer which situation is in force from observations. Does a test reveal the presence of a disease; how to balance the probability of false positive and that of false negative? The main result of that section is the Neyman–Pearson Theorem that the section illustrates with many examples.

scholarly journals Transmission of Hidden Cipher Text over a Binary Symmetric Channel

2012 ◽  
Vol 53 (15) ◽  
pp. 40-43
Author(s):  
Arun Rana ◽  
Nitin Sharma ◽  
Parveen Malik
Keyword(s):  
Binary Symmetric Channel ◽  
Cipher Text ◽  
Symmetric Channel

scholarly journals Inventory Models with Defective Units and Sub-Lot Inspection

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1038
Author(s):  
Han-Wen Tuan ◽  
Gino K. Yang ◽  
Kuo-Chen Hung
Keyword(s):  
Lower Bound ◽  
Optimal Solution ◽  
Upper Bounds ◽  
Interior Solution ◽  
Order Quantity ◽  
Inventory Models ◽  
Defective Items ◽  
Numerical Examples ◽  
Counter Example ◽  
Analytical Foundation

Inventory models must consider the probability of sub-optimal manufacturing and careless shipping to prevent the delivery of defective products to retailers. Retailers seeking to preserve a reputation of quality must also perform inspections of all items prior to sale. Inventory models that include sub-lot sampling inspections provide reasonable conditions by which to establish a lower bound and a pair of upper bounds in terms of order quantity. This should make it possible to determine the conditions of an optimal solution, which includes a unique interior solution to the problem of an order quantity satisfying the first partial derivative. The approach proposed in this paper can be used to solve the boundary. These study findings provide the analytical foundation for an inventory model that accounts for defective items and sub-lot sampling inspections. The numerical examples presented in a previous paper are used to demonstrate the derivation of an optimal solution. A counter-example is constructed to illustrate how existing iterative methods do not necessarily converge to the optimal solution.

scholarly journals A strengthened data processing inequality for the Belavkin–Staszewski relative entropy

2019 ◽  
Vol 32 (02) ◽  
pp. 2050005 ◽  
Author(s):  
Andreas Bluhm ◽  
Ángela Capel
Keyword(s):  
Data Processing ◽  
Relative Entropy ◽  
Quantum Channels ◽  
Equality Case ◽  
Standard Formula ◽  
Conditional Expectations ◽  
Data Processing Inequality ◽  
Equivalent Conditions ◽  
Arbitrary Quantum

In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard [Formula: see text]-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal [Formula: see text]-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.

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