Necessary and Sufficient Conditions for Sparsity Pattern Recovery
... Fano’s inequality in [23]. More recent publications with necessary conditions include [24]–[27]. As described in Section III, our new necessary condition is stronger than the previous results in certain important regimes. In contrast to removing all computational strictures, it is also interesting t ...
... Fano’s inequality in [23]. More recent publications with necessary conditions include [24]–[27]. As described in Section III, our new necessary condition is stronger than the previous results in certain important regimes. In contrast to removing all computational strictures, it is also interesting t ...
Composition of Zero-Knowledge Proofs with Efficient Provers
... it was realized that composability is a subtle issue. In particular, this motivated a strengthening of the GMR definition, known as auxiliary-input zero knowledge [21, 19, 9], which was shown to be closed under sequential composition [19]. The need for this stronger definition was subsequently justi ...
... it was realized that composability is a subtle issue. In particular, this motivated a strengthening of the GMR definition, known as auxiliary-input zero knowledge [21, 19, 9], which was shown to be closed under sequential composition [19]. The need for this stronger definition was subsequently justi ...
MARKOV CHAINS: BASIC THEORY 1.1. Definition and First
... of the transition probability matrix on the simplex is a contraction? First, it tells us that if we start the Markov chain in two different initial distributions, then the distributions after one step are closer than they were to start. Consequently, by induction, after n steps they are even closer: ...
... of the transition probability matrix on the simplex is a contraction? First, it tells us that if we start the Markov chain in two different initial distributions, then the distributions after one step are closer than they were to start. Consequently, by induction, after n steps they are even closer: ...
1. Markov chains
... We will let P n (i, j) denote the (i, j) element in the matrix P n . ⊲ Exercise [1.7] gives some basic practice with the definitions. So, in principle, we can find the answer to any question about the probabilistic behavior of a Markov chain by doing matrix algebra, finding powers of matrices, etc. ...
... We will let P n (i, j) denote the (i, j) element in the matrix P n . ⊲ Exercise [1.7] gives some basic practice with the definitions. So, in principle, we can find the answer to any question about the probabilistic behavior of a Markov chain by doing matrix algebra, finding powers of matrices, etc. ...
full version
... all-powerful cheating verifier learns nothing from its interaction with the prover; we denote the class of languages admitting statistical zero-knowledge proofs by SZK. It has recently been established [35] that all languages in SZK have constant-round statistical zero-knowledge proof systems (with ...
... all-powerful cheating verifier learns nothing from its interaction with the prover; we denote the class of languages admitting statistical zero-knowledge proofs by SZK. It has recently been established [35] that all languages in SZK have constant-round statistical zero-knowledge proof systems (with ...
Markov Chains and Queues in Discrete Time
... 2), there are differences in the rate of visits to a recurrent state. In order to describe these, define Ni (n) as the number of visits to state i until time n. Further define for a recurrent state i ∈ E the mean time mi := E(τi |X0 = i) until the first visit to i (after time zero) under the conditi ...
... 2), there are differences in the rate of visits to a recurrent state. In order to describe these, define Ni (n) as the number of visits to state i until time n. Further define for a recurrent state i ∈ E the mean time mi := E(τi |X0 = i) until the first visit to i (after time zero) under the conditi ...
On direct and indirect probabilistic reasoning in legal proof1
... Models of rational legal proof are usually of three kinds: statistical, story-based, and argument-based.2 All three approaches acknowledge that evidence cannot provide watertight support for a factual claim but always leaves room for doubt and uncertainty. Statistical approaches (cf. e.g. Schum 1994 ...
... Models of rational legal proof are usually of three kinds: statistical, story-based, and argument-based.2 All three approaches acknowledge that evidence cannot provide watertight support for a factual claim but always leaves room for doubt and uncertainty. Statistical approaches (cf. e.g. Schum 1994 ...
Lesson 6 Chapter 5: Convolutions and The Central Limit Theorem
... concrete. The height (in inches) of a randomly selected segment is uniformly distributed in (35.5, 36.5). A roadway can be laid across the two towers provided the heights of the two towers are within 4 inches. Find the probability that the roadway can be laid. Solution: The heights of the two towers ...
... concrete. The height (in inches) of a randomly selected segment is uniformly distributed in (35.5, 36.5). A roadway can be laid across the two towers provided the heights of the two towers are within 4 inches. Find the probability that the roadway can be laid. Solution: The heights of the two towers ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.