We introduce the Skipping Sampler, a novel algorithm to efficiently sample from the restriction of an arbitrary probability density to an arbitrary measurable set. Such conditional densities can arise in the study of risk and reliability and are often of complex nature, for example having multiple isolated modes and non-convex or disconnected support. The sampler can be seen as an instance of the Metropolis-Hastings algorithm with a particular proposal structure, and we establish sufficient conditions under which the Strong Law of Large Numbers and the Central Limit Theorem hold. We give theoretical and numerical evidence of improved performance relative to the Random Walk Metropolis algorithm.
In many stochastic systems, however, the relationship between exogenous inputs and system behaviour is implicit and non-linear. Examples include markets or distributed systems, where complexity arises from the actions of multiple players, or systems involving complex optimisation procedures such as security constrained optimal power flow @cite_21 . In the present work (and unlike @cite_36 ) we therefore do not assume that the set @math is known but, rather, that any sampling procedure may only test pointwise whether @math . As illustrated in Figure , the areas of zero density induced by conditioning on @math can create multiple isolated modes. We therefore now briefly recall the literature on sampling from rare events and from multimodal targets.
@cite_36 @cite_21
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We consider in this chapter a class of two-player nonzero-sum stochastic games with incomplete information, which is inspired by recent applications of game theory in network security. We develop fully distributed reinforcement learning algorithms, which require for each player a minimal amount of information regarding the other player. At each time, each player can be in an active mode or in a sleep mode. If †This material is based upon work supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) under grant number AFOSR MURI FA9550-09-1-0249. D R A F T October 2, 2011, 1:59am D R A F T 2 HYBRID LEARNING IN STOCHASTIC GAMES AND ITS APPLICATION IN NETWORK SECURITY a player is in an active mode, she updates her strategy and estimates of unknown quantities using a specific pure or hybrid learning pattern. The players’ intelligence and rationality are captured by the weighted linear combination of different learning patterns. We use stochastic approximation techniques to show that, under appropriate conditions, the pure or hybrid learning schemes with random updates can be studied using their deterministic ordinary differential equation (ODE) counterparts. Convergence to state-independent equilibria is analyzed for special classes of games, namely, games with two actions, and potential games. Results are applied to network security games between an intruder and an administrator, where the noncooperative behaviors are well characterized by the features of distributed hybrid learning. This paper extends Khatri (1964, 1969) distribution of the largest eigenvalue of central complex Wishart matrices to the noncentral case. It then applies the resulting new statistical results to obtain closed-form expressions for the outage probability of multiple-input-multiple-output (MIMO) systems employing maximal ratio combining (known also as "beamforming" systems) and operating over Rician-fading channels. When applicable these expressions are compared with special cases previously reported in the literature dealing with the performance of (1) MIMO systems over Rayleigh-fading channels and (2) single-input-multiple-output (SIMO) systems over Rician-fading channels. As a double check these analytical results are validated by Monte Carlo simulations and as an illustration of the mathematical formalism some numerical examples for particular cases of interest are plotted and discussed. These results show that, given a fixed number of total antenna elements and under the same scattering condition (1) SIMO systems are equivalent to multiple-input-single-output systems and (2) it is preferable to distribute the number of antenna elements evenly between the transmitter and the receiver for a minimum outage probability performance. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. Many systems in nature can be described using discrete input–output maps. Without knowing details about a map, there may seem to be no a priori reason to expect that a randomly chosen input would be more likely to generate one output over another. Here, by extending fundamental results from algorithmic information theory, we show instead that for many real-world maps, the a priori probability P(x) that randomly sampled inputs generate a particular output x decays exponentially with the approximate Kolmogorov complexity @math K ( x ) of that output. These input–output maps are biased towards simplicity. We derive an upper bound P(x) ≲ @math 2 - a K ( x ) - b , which is tight for most inputs. The constants a and b, as well as many properties of P(x), can be predicted with minimal knowledge of the map. We explore this strong bias towards simple outputs in systems ranging from the folding of RNA secondary structures to systems of coupled ordinary differential equations to a stochastic financial trading model.