We show that as T→∞, for all t∈[T,2T] outside of a set measure o(T), ∫−logθTlogθT|ζ(1 2+it+ih)|βdh=(logT)fθ(β)+o(1), some explicit exponent fθ(β), where θ>−1 and β>0. This proves an extended version conjecture Fyodorov Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, θ>−1, the moments exhibit phase transition at critical ...

P~ θ : V 7→ [0, 1], where ~ θ is an element of the m-dimensional probability simplex. Hence the probability assigned to a single term vj is defined as: P~ θ (vj) def = θ[j]. Also recall from the previous lecture that the Kullback–Leibler (KL) divergence between two probability distributions P~ θ and P~ θ′ , i.e. the expected log-likelihood ratio with respect to P~ θ, is defined as: D(P~ θ ‖P~ θ...

In [1], it was conjectured that the permanent of a P-lifting θ of a matrix θ of degree M is less than or equal to the M th power of the permanent perm(θ), i.e., perm(θ) 6 perm(θ) and, consequently, that the degree-M Bethe permanent permM,B(θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., permM,B(θ) 6 perm(θ). In this paper, we prove these related conjectures and show ...

In this paper a new class of fuzzy sets, namely r-(τi, τj)-θ-generalized fuzzy closed sets is introduced for smooth bitopological spaces. This class falls strictly in between the class of r-(τj, τi)fuzzy θ-closed sets and the class of r-(τi, τj)generalized fuzzy closed sets. Furthermore, by using the class of r-(τi, τj)-θgeneralized fuzzy closed sets we establish a new fuzzy closure operator wh...

The notions of power of word, periodicity and primitivity are intrinsically connected to the operation of catenation, that dynamically generates word repetitions. When considering generalizations of the power of a word, other operations will be the ones that dynamically generate such pseudo-repetitions. In this paper we define and investigate the operation of θ-catenation that gives rise to the...

Strong convexity can be characterized in a number of ways. The following facts provide some conditions that are equivalent to Definition 1. For the 2-norm, Fact 2 means that the minimum eigenvalue of the Hessian is lower-bounded by κ. Recall that˜µ(θ) and˜µ(θ) are the gradients of˜Φ(θ) and˜Φ(θ), respectively. Since the conjugate function, ˜ Φ * , is assumed to be κ-strongly convex, we have via ...

Many practitioners who use EM and related algorithms complain that they are sometimes slow. When does this happen, and what can be done about it? In this paper, we study the general class of bound optimization algorithms – including EM, Iterative Scaling, Non-negative Matrix Factorization, CCCP – and their relationship to direct optimization algorithms such as gradientbased methods for paramete...

1 Sn lim log P( ∈ A). n→∞ n n Given θ ∈ Rd, define M(θ) = E[exp((θ, X1))] where (·, ·) represents the inner p product of two vectors: (a, b) = i aibi. Define I(x) = supθ∈Rd ((θ, x) − log M(θ)), where again I(x) = ∞ is a possibility. Theorem 1 (Cram ́ Suppose M(θ) < ∞ er’s theorem in multiple dimensions). for all θ ∈ Rd. Then (a) for all closed set F ⊂ Rd , 1 Sn lim sup log P( ∈ F ) ≤ − inf I(x) ...

) | ( j k Y prob pkj = = = θ The jth column of P gives the probability distribution associated with the parameter value j = θ . We consider the problem of estimatingθ , based on an observed sample of Y. The m values of θ are called the hypotheses, and finding the best value of θ is hypothesis testing. A randomized detector of θ is a random variable , with a distribution dependent on the observe...