One Observation behind Two-Envelope Puzzles

In two famous and popular puzzles a participant is required to compare two numbers of which she is shown only one. In the first one there are two envelopes with money in them. The sum of money in one of the envelopes is twice as large as the other sum. An envelope is selected at random and handed to you. If the sum in this envelope is x, then the sum in the other one is (1/2)(2x) + (1/2)(0.5x) = 1.25x. Hence, you are better off switching to the other envelope no matter what sum you see, which is paradoxical. In the second puzzle two distinct numbers are written on two slips of paper. One of them is selected at random and you observe it. How can you guess, with probability greater than 1/2 of being correct, whether this number is the larger or the smaller? We show that there is one principle behind the two puzzles: The ranking of n random variables X1, ... , Xn cannot be independent of each of them, unless the ranking is fixed. Thus, unless there is nothing to be learned about the ranking, there must be at least one variable the observation of which conveys information about it.two envelope paradox

Bayesianism without Learning

According to the standard definition, a Bayesian agent is one who forms his posterior belief by conditioning his prior belief on what he has learned, that is, on facts of which he has become certain. Here it is shown that Bayesianism can be described without assuming that the agent acquires any certain information; an agent is Bayesian if his prior, when conditioned on his posterior belief, agrees with the latter. This condition is shown to characterize Bayesian models.Bayesian updating, prior and posterior

Basis Criteria for Generalized Spline Modules via Determinant

Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. The set of generalized splines has a ring and an R-module structure. We study the module structure of generalized splines where the base ring is a greatest common divisor domain. We give basis criteria for generalized splines on cycles, diamond graphs and trees by using determinantal techniques. In the last section of the paper, we define a graded module structure for generalized splines and give some applications of the basis criteria for cycles, diamond graphs and trees.Comment: 20 pages, 10 figure

Nonlinear Channel Estimation for OFDM System by Complex LS-SVM under High Mobility Conditions

A nonlinear channel estimator using complex Least Square Support Vector Machines (LS-SVM) is proposed for pilot-aided OFDM system and applied to Long Term Evolution (LTE) downlink under high mobility conditions. The estimation algorithm makes use of the reference signals to estimate the total frequency response of the highly selective multipath channel in the presence of non-Gaussian impulse noise interfering with pilot signals. Thus, the algorithm maps trained data into a high dimensional feature space and uses the structural risk minimization (SRM) principle to carry out the regression estimation for the frequency response function of the highly selective channel. The simulations show the effectiveness of the proposed method which has good performance and high precision to track the variations of the fading channels compared to the conventional LS method and it is robust at high speed mobility.Comment: 11 page

Recovering Jointly Sparse Signals via Joint Basis Pursuit

This work considers recovery of signals that are sparse over two bases. For instance, a signal might be sparse in both time and frequency, or a matrix can be low rank and sparse simultaneously. To facilitate recovery, we consider minimizing the sum of the $\ell_1$-norms that correspond to each basis, which is a tractable convex approach. We find novel optimality conditions which indicates a gain over traditional approaches where $\ell_1$ minimization is done over only one basis. Next, we analyze these optimality conditions for the particular case of time-frequency bases. Denoting sparsity in the first and second bases by $k_1,k_2$ respectively, we show that, for a general class of signals, using this approach, one requires as small as $O(\max\{k_1,k_2\}\log\log n)$ measurements for successful recovery hence overcoming the classical requirement of $\Theta(\min\{k_1,k_2\}\log(\frac{n}{\min\{k_1,k_2\}}))$ for $\ell_1$ minimization when $k_1\approx k_2$. Extensive simulations show that, our analysis is approximately tight.Comment: 8 pages, 1 figure, submitted to ISIT 201