Markov Chain Confidence Intervals and Biases

We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

Theoretical Guarantees for Phylogeny Inference from Single-Cell Lineage Tracing

CRISPR-Cas9 lineage tracing technologies have emerged as a powerful tool for investigating develop-ment in single-cell contexts, but exact reconstruction of the underlying clonal relationships in experiment is plagued by data-related complications. These complications are functions of the experimental parameters in these systems, such as the Cas9 cutting rate, the diversity of indel outcomes, and the rate of missing data. In this paper, we develop two theoretically grounded algorithms for reconstruction of the underlying phylogenetic tree, as well as asymptotic bounds for the number of recording sites necessary for exact recapitulation of the ground truth phylogeny at high probability. In doing so, we explore the relationship between the problem difficulty and the experimental parameters, with implications for experimental design. Lastly, we provide simulations validating these bounds and showing the empirical performance of these algorithms. Overall, this work provides a first theoretical analysis of phylogenetic reconstruction in the CRISPR-Cas9 lineage tracing technology.

Tight Polynomial Bounds for Loop Programs in Polynomial Space

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of polynomially-bounded variables, and on a weak programming language for which we have recently shown that tight bounds for polynomially-bounded variables are computable. These bounds are sets of multivariate polynomials. While their computability has been settled, the complexity of this program-analysis problem remained open. In this paper, we show the problem to be PSPACE-complete. The main contribution is a new, space-efficient analysis algorithm. This algorithm is obtained in a few steps. First, we develop an algorithm for univariate bounds, a sub-problem which is already PSPACE-hard. Then, a decision procedure for multivariate bounds is achieved by reducing this problem to the univariate case; this reduction is orthogonal to the solution of the univariate problem and uses observations on the geometry of a set of vectors that represent multivariate bounds. Finally, we transform the univariate-bound algorithm to produce multivariate bounds.

Accuracy Characteristics of the Parametric Burg Method for Spatial Signal Processing in a Nonuniform Array Antenna

Introduction. In the case of a nonuniform (NU) design of the antenna elements (AEs) of the receiving antenna array (AA), the antenna pattern (AP) features sidelobes (SL) with a significantly higher noise level than acceptable values. Under low signal-to-noise ratios (SNR), this noise leads to angular coordinate measuring errors thus worsening the statistical accuracy characteristics (ACs) of the signal. It is of relevance to construct the ACs of angular coordinates when a modified parametric Burg method (BM) is applied to spatial reflected signal processing in a transportable decametre range radar (DRR) with a nonuniform array (NUA) and linear accuracy characteristics. Aim. To analyse the statistical ACs of angular coordinate objects when using a modified BM for spatial reflected signal processing in a DRR with a linear NUA, in which AEs are located with a random step in the range from λ/2 to several λ, where λ is the operating carrier wavelength.Materials and methods. Statistical ACs were constructed by computer modelling in the MatLab software, the reliability of which was confirmed by conventional discrete Fourier transform methods, as well as by comparing the obtained ACs with asymptotic bounds, including Cramer-Rao bounds.Results. The possibility and conditions of using a modified parametric BM for estimating the azimuthal coordinates of reflected radar signals were determined for the case of a nonuniform design of the over-the-horizon DRR receiving AA AEs. Statistical ACs were obtained and compared with the asymptotically optimal ACs of the maximum likelihood estimations corresponding to the uniform AE design.Conclusion. The obtained results confirm the suboptimality of the BM modified for signal processing in the NUA at a random AE spacing step in the range from λ/2 to 2λ, making it applicable for use in transportable DRRs.

Asymptotics of Polynomial Interpolation and the Bernstein Constants

It is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ∞ - 1 , 1 by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when $$\alpha $$ α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in the $$L_{\infty }$$ L ∞ norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

Universality in asymptotic bounds and its saturation in 2D CFT

We study asymptotics of three point coefficients (light-light-heavy) and two point correlators in heavy states in unitary, compact 2D CFTs. We prove an upper and lower bound on such quantities using numerically assisted Tauberian techniques. We obtain an optimal upper bound on the spectrum of operators appearing with fixed spin from the OPE of two identical scalars. While all the CFTs obey this bound, rational CFTs come close to saturating it. This mimics the scenario of bounds on asymptotic density of states and thereby pronounces an universal feature in asymptotics of 2D CFTs. Next, we clarify the role of smearing in interpreting the asymptotic results pertaining to considerations of eigenstate thermalization in 2D CFTs. In the context of light-light-heavy three point coefficients, we find that the order one number in the bound is sensitive to how close the light operators are from the $$ \frac{c}{32} $$ c 32 threshold. In context of two point correlator in heavy state, we find the presence of an enigmatic regime which separates the AdS3 thermal physics and the BTZ black hole physics. Furthermore, we present some new numerical results on the behaviour of spherical conformal block.

On factorizations into coprime parts

Let [Formula: see text] and [Formula: see text] be the number of unordered and ordered factorizations of [Formula: see text] into integers larger than one. Let [Formula: see text] and [Formula: see text] have the additional restriction that the factors are coprime. We establish asymptotic bounds for the sums of [Formula: see text] and [Formula: see text] up to [Formula: see text] for all real [Formula: see text] and the asymptotic bounds for [Formula: see text] and [Formula: see text] for all negative [Formula: see text].