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 ...

Here we will consider the problem min f ( ) θ∈Θ θ where, for a given value of θ , we are not able to evaluate analytically or numerically, but must obtain (noisy) estimates of f ( f ( ) θ ) θ using simulation. We will assume for now that the set of possible solutions Θ is uncountably infinite. In particular, suppose that Θ is an interval [ θ θ, θ Θ ] of real numbers. One approach to solving the...

I will not describe the underlying assumptions in details. These are the usual sorts of assumptions one makes for parametric models, in order to be able to establish sensible results. See Page 11 of Chapter 3 of Wellner’s notes for a detailed description of the conditions involved. For a multidimensional parametric model {p(x, θ) : θ ∈ Θ ⊂ Rk}, the information matrix I(θ) is given by: I(θ) = Eθ...

Let X(n) be an observation sampled from a distribution Pθ(n) with unknown parameter θ, θ being vector in Banach space E (most often, high-dimensional of dimension d). We study the problem estimation f(θ) for functional f:E↦R some smoothness s>0 based on X(n)∼Pθ(n). Assuming that there exists estimator θˆn=θˆn(X(n)) such n(θˆ n−θ) is sufficiently close to mean zero Gaussian random E, we construc...

Proof. Let M = sup θ∈Θ max{{h(θ), |v(θ)|} and V ε = {θ : d(θ, L) ≤ ε}. Applying Taylor's expansion formula (Folland, 1990), we have v(θ t+1) = v(θ t) + γ n+1 v h (θ t+1) + R t+1 , t ≥ 0, which implies that t i=0 γ i+1 v h (θ i) = v(θ t+1) − v(θ 0) − t i=0 R i+1 ≥ −2M − t i=0 R i+1. Since t i=0 R i+1 converges (owing to Lemma A.2), t i=0 γ i+1 v h (θ i) also converges. Furthermore, v(θ t) = v(θ ...

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...

• Let Ak denote the set of methods that observe a sequence of data pairs Y t = (F (θ, X ), F (τ , X )), 1 ≤ t ≤ k, and return an estimate θ̂(k) ∈ Θ. • Let FG denote the class of functions we want to optimize, where for each (F, P ) ∈ FG the subgradient g(θ;X) satisfies EP [‖g(θ;X)‖2∗] ≤ G. • For each A ∈ Ak and (F, P ) ∈ FG, consider the optimization gap: k(A, F, P,Θ) := f (θ̂(k))− inf θ∈Θ f (θ) ...

and Applied Analysis 3 Lemma 2.3. Besides the properties of the argument functions, the Hamiltonian structure of the problem is essential for us to obtain variational periodic and antiperiodic halfeigenvalues, see Lemma 3.2. After introducing the rotation number function, we can easily obtain the ordering of these variational periodic half-eigenvalues, see 3.38 . For regular self-adjoint linear...

Let the matching polynomial of a graph G be denoted by μ(G, x). A graph G is said to be θ-super positive if μ(G, θ) 6= 0 and μ(G \ v, θ) = 0 for all v ∈ V (G). In particular, G is 0-super positive if and only if G has a perfect matching. While much is known about 0-super positive graphs, almost nothing is known about θsuper positive graphs for θ 6= 0. This motivates us to investigate the struct...

Now, let (u, v) = Rα(θ, φ′). Here, Rα = Ry(α). We omit the z rotation since that does not affect Yl0 which has no azimuthal dependence. The vector corresponding to coordinates (u, v) is then given by   sinu cos v sinu sin v cos u   =   cosα 0 sinα 0 1 0 − sinα 0 cosα     sin θ′ cosφ′ sin θ′ sinφ′ cos θ′   =   cosα sin θ′ cosφ′ + sinα cos θ′ sin θ′ sinφ′ cosα cos θ′ + sinα sin θ′ (...