Robustness of the nonlinear filter: the correlated case

We consider the question of robustness of the optimal nonlinear filter when the signal process X and the observation noise are possibly correlated. The signal X and observations Y are given by a SDE where the coefficients can depend on the entire past. Using results on pathwise solutions of stochastic differential equations we express X as a functional of two independent Brownian motions under the reference probability measure P0. This allows us to write the filter p as a ratio of two expectations. This is the main step in proving robustness. In this framework we show that when (Xn,Yn) converge to (X,Y) in law, then the corresponding filters also converge in law. Moreover, when the signal and observation processes converge in probability, so do the filters. We also prove that the paths of the filter are continuous in this framework

Robustness of the nonlinear filter

In the nonlinear filtering model with signal and observation noise independent, we show that the filter depends continuously on the law of the signal. We do not assume that the signal process is Markov and prove the result under minimal integrability conditions. The analysis is based on expressing the nonlinear filter as a Wiener functional via the Kallianpur-Striebel Bayes formula

A prospective comparative study of intestinal anastomosis, single layer extramucosal versus double layer

Background: Intestinal anastomosis is an operative procedure that is of central importance in the practice of surgery. Intestinal anastomosis after resection of bowel may be of various types and techniques. This prospective comparative study is performed to evaluate the safety in term of anastomotic leak of single layer interrupted extramucosal technique as compared to conventional double layer technique.Methods: The patients selected for this study are those who were admitted with various clinical conditions requiring resection and anastomosis of small or large bowel presented to P.D.U. Medical College & Hospital, Rajkot between a period of August 2012 to December 2014. A total of 50 patients were included in the study. All the patients above the age of 18 years and less than 60 years, requiring intestinal anastomosis on emergency or electively, were included in the study and those requiring anastomosis to esophageal, gastric and duodenal anastomosis were excluded and randomly allotted single layer and double layer groups and results such as anastomotic leak rate, duration for anastomosis, number of suture material required noted.Results: Mean duration required for single layer anastomosis was 19.6 minutes and for double layer anastomosis was 29.5 minutes and double number of suture material used in double layer anastomosis with equal anastomotic leak rate (6%) in each group.Conclusions: Single layer interrupted extramucosal technique required significantly less duration for anastomosis, is cost effective with no significant difference in anastomotic leak rate and  as safe as conventional double layer technique.

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

On a random directed spanning tree

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square

Weak convergence to a Markov process the martingale approach

In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X n } to X, when X arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processes X n , X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved