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Register a new userThe Blackwell relation defines no lattice
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Added by
Johannes Rauh
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Blackwells theorem shows the equivalence of two preorders on the set of information channels. Here, we restate, and slightly generalize, his result in terms of random variables. Furthermore, we prove that the corresponding partial order is not a lattice; that is, least upper bounds and greatest lower bounds do not exist.
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Suppose we have a pair of information channels, $kappa_{1},kappa_{2}$, with a common input. The Blackwell order is a partial order over channels that compares $kappa_{1}$ and $kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, $kappa_{1}$ is said to be Blackwell-inferior to $kappa_{2}$ if and only if $kappa_{1}$ can be constructed by garbling the output of $kappa_{2}$. A related partial order stipulates that $kappa_{2}$ is more capable than $kappa_{1}$ if the mutual information between the input and output is larger for $kappa_{2}$ than for $kappa_{1}$ for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where $kappa_{1}$ is less capable than $kappa_{2}$ but not Blackwell-inferior. We show that this may even happen when $kappa_{1}$ is constructed by coarse-graining the inputs of $kappa_{2}$. Such a coarse-graining is a special kind of pre-garbling of the channel inputs. This example directly establishes that the expected value of the shared utility function for the coarse-grained channel is larger than it is for the non-coarse-grained channel. This contradicts the intuition that coarse-graining can only destroy information and lead to inferior channels. We also discuss our results in the context of information decompositions.
This paper applies machine learning to optimize the transmission policy of cognitive radio inspired non-orthogonal multiple access (CR-NOMA) networks, where time-division multiple access (TDMA) is used to serve multiple primary users and an energy-constrained secondary user is admitted to the primary users time slots via NOMA. During each time slot, the secondary user performs the two tasks: data transmission and energy harvesting based on the signals received from the primary users. The goal of the paper is to maximize the secondary users long-term throughput, by optimizing its transmit power and the time-sharing coefficient for its two tasks. The long-term throughput maximization problem is challenging due to the need for making decisions that yield long-term gains but might result in short-term losses. For example, when in a given time slot, a primary user with large channel gains transmits, intuition suggests that the secondary user should not carry out data transmission due to the strong interference from the primary user but perform energy harvesting only, which results in zero data rate for this time slot but yields potential long-term benefits. In this paper, a deep reinforcement learning (DRL) approach is applied to emulate this intuition, where the deep deterministic policy gradient (DDPG) algorithm is employed together with convex optimization. Our simulation results demonstrate that the proposed DRL assisted NOMA transmission scheme can yield significant performance gains over two benchmark schemes.
This paper presents results pertaining to sequential methods for support recovery of sparse signals in noise. Specifically, we show that any sequential measurement procedure fails provided the average number of measurements per dimension grows slower then log s / D(f0||f1) where s is the level of sparsity, and D(f0||f1) the Kullback-Leibler divergence between the underlying distributions. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n / D(f1||f0), where n is the total dimension of the problem. Lastly, we show that a simple procedure termed sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than (log s + log log n) / D(f0||f1), a mere additive factor more than the lower bound.
Consider a channel ${bf Y}={bf X}+ {bf N}$ where ${bf X}$ is an $n$-dimensional random vector, and ${bf N}$ is a Gaussian vector with a covariance matrix ${bf mathsf{K}}_{bf N}$. The object under consideration in this paper is the conditional mean of ${bf X}$ given ${bf Y}={bf y}$, that is ${bf y} to E[{bf X}|{bf Y}={bf y}]$. Several identities in the literature connect $E[{bf X}|{bf Y}={bf y}]$ to other quantities such as the conditional variance, score functions, and higher-order conditional moments. The objective of this paper is to provide a unifying view of these identities. In the first part of the paper, a general derivative identity for the conditional mean is derived. Specifically, for the Markov chain ${bf U} leftrightarrow {bf X} leftrightarrow {bf Y}$, it is shown that the Jacobian of $E[{bf U}|{bf Y}={bf y}]$ is given by ${bf mathsf{K}}_{{bf N}}^{-1} {bf Cov} ( {bf X}, {bf U} | {bf Y}={bf y})$. In the second part of the paper, via various choices of ${bf U}$, the new identity is used to generalize many of the known identities and derive some new ones. First, a simple proof of the Hatsel and Nolte identity for the conditional variance is shown. Second, a simple proof of the recursive identity due to Jaffer is provided. Third, a new connection between the conditional cumulants and the conditional expectation is shown. In particular, it is shown that the $k$-th derivative of $E[X|Y=y]$ is the $(k+1)$-th conditional cumulant. The third part of the paper considers some applications. In a first application, the power series and the compositional inverse of $E[X|Y=y]$ are derived. In a second application, the distribution of the estimator error $(X-E[X|Y])$ is derived. In a third application, we construct consistent estimators (empirical Bayes estimators) of the conditional cumulants from an i.i.d. sequence $Y_1,...,Y_n$.
In this work a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.
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