Cramer Rao-Type Bounds for Sparse Bayesian Learning

In this paper, we derive Hybrid, Bayesian and Marginalized Cram\'{e}r-Rao lower bounds (HCRB, BCRB and MCRB) for the single and multiple measurement vector Sparse Bayesian Learning (SBL) problem of estimating compressible vectors and their prior distribution parameters. We assume the unknown vector to be drawn from a compressible Student-t prior distribution. We derive CRBs that encompass the deterministic or random nature of the unknown parameters of the prior distribution and the regression noise variance. We extend the MCRB to the case where the compressible vector is distributed according to a general compressible prior distribution, of which the generalized Pareto distribution is a special case. We use the derived bounds to uncover the relationship between the compressibility and Mean Square Error (MSE) in the estimates. Further, we illustrate the tightness and utility of the bounds through simulations, by comparing them with the MSE performance of two popular SBL-based estimators. It is found that the MCRB is generally the tightest among the bounds derived and that the MSE performance of the Expectation-Maximization (EM) algorithm coincides with the MCRB for the compressible vector. Through simulations, we demonstrate the dependence of the MSE performance of SBL based estimators on the compressibility of the vector for several values of the number of observations and at different signal powers.Comment: Accepted for publication in the IEEE Transactions on Signal Processing, 11 pages, 10 figure

Secrecy in the 2-User Symmetric Deterministic Interference Channel with Transmitter Cooperation

This work presents novel achievable schemes for the 2-user symmetric linear deterministic interference channel with limited-rate transmitter cooperation and perfect secrecy constraints at the receivers. The proposed achievable scheme consists of a combination of interference cancelation, relaying of the other user's data bits, time sharing, and transmission of random bits, depending on the rate of the cooperative link and the relative strengths of the signal and the interference. The results show, for example, that the proposed scheme achieves the same rate as the capacity without the secrecy constraints, in the initial part of the weak interference regime. Also, sharing random bits through the cooperative link can achieve a higher secrecy rate compared to sharing data bits, in the very high interference regime. The results highlight the importance of limited transmitter cooperation in facilitating secure communications over 2-user interference channels.Comment: 5 pages, submitted to SPAWC 201

On Finding a Subset of Healthy Individuals from a Large Population

In this paper, we derive mutual information based upper and lower bounds on the number of nonadaptive group tests required to identify a given number of "non defective" items from a large population containing a small number of "defective" items. We show that a reduction in the number of tests is achievable compared to the approach of first identifying all the defective items and then picking the required number of non-defective items from the complement set. In the asymptotic regime with the population size $N \rightarrow \infty$, to identify $L$ non-defective items out of a population containing $K$ defective items, when the tests are reliable, our results show that $\frac{C_s K}{1-o(1)} (\Phi(\alpha_0, \beta_0) + o(1))$ measurements are sufficient, where $C_s$ is a constant independent of $N, K$ and $L$, and $\Phi(\alpha_0, \beta_0)$ is a bounded function of $\alpha_0 \triangleq \lim_{N\rightarrow \infty} \frac{L}{N-K}$ and $\beta_0 \triangleq \lim_{N\rightarrow \infty} \frac{K} {N-K}$. Further, in the nonadaptive group testing setup, we obtain rigorous upper and lower bounds on the number of tests under both dilution and additive noise models. Our results are derived using a general sparse signal model, by virtue of which, they are also applicable to other important sparse signal based applications such as compressive sensing.Comment: 32 pages, 2 figures, 3 tables, revised version of a paper submitted to IEEE Trans. Inf. Theor

Solar Jet on 2014 April 16 Modeled by Kelvin--Helmholtz Instability

We study here the arising of Kelvin--Helmholtz Instability (KHI) in one fast jet of 2014 April 16 observed by the Atmospheric Imaging Assembly (AIA) on board Solar Dynamics Observatory (SDO) in different UV and EUV wavelengths. The evolution of jet indicates the blob like structure at its boundary which could be the observational evidence of the KHI. We model the jet as a moving cylindrical magnetic flux tube of radius $a$ embedded in a magnetic field B_i and surrounded by rest magnetized plasma with magnetic field B_e. We explore the propagation of the kink MHD mode along the jet that can become unstable against the KHI if its speed exceeds a critical value. Concerning magnetic fields topology we consider three different configurations, notably of (i) spatially homogeneous magnetic fields (untwisted magnetic flux tube), (ii) internal (label `i') twisted magnetic field and external homogeneous one (label `e') (single-twisted flux tube), and (iii) both internal and external twisted magnetic fields (double-twisted magnetic flux tube). Plasma densities in the two media rho_i and rho_e are assumed to be homogeneous. The density contrast is defined in two ways: first as rho_e/rho_i and second as rho_e/(rho_i + rho_e). Computations show that the KHI can occur at accessible flow velocities in all the cases of untwisted and single-twisted flux tubes. It turns out, however, that in the case of a double-twisted flux tube the KHI can merge at an accessible jet speed only when the density contrast is calculated from the ratio rho_e/(rho_i} + rho_e). Evaluated KHI developing times and kink mode wave phase velocities at wavelength of 4 Mm lie in the ranges of 1--6.2 min and 202--271 km/s, respectively---all being reasonable for the modeled jet.Comment: 35 pages, 11 figure

Hold the Cracks

My medicine has its own special place in our downstairs bathroom. It rests on a little metal shelf by the shower, standing among the bright orange bottles of multivitamins, B12, vitamin C, and calcium chews. My mother is obsessed with natural healing practices – she slathers on bitter goldenseal for infections, feeds us capsules of powdery white willow bark for headaches, and strange clay mixed with water for stomach aches. My little bottle of pink goo looks lost and confused amidst the hand-written labels and bottles of earth-colored liquids. I feel guilty taking it, but almost proud at the same time. It feels so official, taking “real” medicine. It\u27s like the feeling of eating “real” cereal, as opposed to the hot mush my mother always makes when we’re home. It’s like going to tae kwon do class and being a “real” student as opposed to one who learns everything at home. I never felt quite real, quite normal. I knew that I wasn\u27t. As I swallow the thick, candy-flavored substance, I try to block out the voices seeping in from the kitchen. There is nothing more upsetting than those voices – the low, fearful, angry ones that mean they are either displeased with us (my siblings and I), or talking about money

Iatrogenic Specification Error: A Cautionary Tale of Cleaning Data

It is common in empirical research to use what appear to be sensible rules of thumb for cleaning data. Measurement error is often the justification for removing (trimming) or recoding (winsorizing) observations whose values lie outside a specified range. This paper considers identification in a linear model when the dependent variable is mismeasured. The results examine the common practice of trimming and winsorizing to address the identification failure. In contrast to the physical and laboratory sciences, measurement error in social science data is likely to be more complex than simply additive white noise. We consider a general measurement error process which nests many processes including the additive white noise process and a contaminated sampling process. Analytic results are only tractable under strong distributional assumptions, but demonstrate that winsorizing and trimming are only solutions for a particular class of measurement error processes. Indeed, trimming and winsorizing may induce or exacerbate bias. We term this source of bias Iatrogenic' (or econometrician induced) error. The identification results for the general error process highlight other approaches which are more robust to distributional assumptions. Monte Carlo simulations demonstrate the fragility of trimming and winsorizing as solutions to measurement error in the dependent variable.