Combinatorial Markov chains on linear extensions

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in terms of discrete time Markov chain

Local conservation scores without a priori assumptions on neutral substitution rates

<p>Abstract</p> <p>Background</p> <p>Comparative genomics aims to detect signals of evolutionary conservation as an indicator of functional constraint. Surprisingly, results of the ENCODE project revealed that about half of the experimentally verified functional elements found in non-coding DNA were classified as unconstrained by computational predictions. Following this observation, it has been hypothesized that this may be partly explained by biased estimates on neutral evolutionary rates used by existing sequence conservation metrics. All methods we are aware of rely on a comparison with the neutral rate and conservation is estimated by measuring the deviation of a particular genomic region from this rate. Consequently, it is a reasonable assumption that inaccurate neutral rate estimates may lead to biased conservation and constraint estimates.</p> <p>Results</p> <p>We propose a conservation signal that is produced by local Maximum Likelihood estimation of evolutionary parameters using an optimized sliding window and present a Kullback-Leibler projection that allows multiple different estimated parameters to be transformed into a conservation measure. This conservation measure does not rely on assumptions about neutral evolutionary substitution rates and little a priori assumptions on the properties of the conserved regions are imposed. We show the accuracy of our approach (KuLCons) on synthetic data and compare it to the scores generated by state-of-the-art methods (phastCons, GERP, SCONE) in an ENCODE region. We find that KuLCons is most often in agreement with the conservation/constraint signatures detected by GERP and SCONE while qualitatively very different patterns from phastCons are observed. Opposed to standard methods KuLCons can be extended to more complex evolutionary models, e.g. taking insertion and deletion events into account and corresponding results show that scores obtained under this model can diverge significantly from scores using the simpler model.</p> <p>Conclusion</p> <p>Our results suggest that discriminating among the different degrees of conservation is possible without making assumptions about neutral rates. We find, however, that it cannot be expected to discover considerably different constraint regions than GERP and SCONE. Consequently, we conclude that the reported discrepancies between experimentally verified functional and computationally identified constraint elements are likely not to be explained by biased neutral rate estimates.</p

Dimensioning a multiple hashing scheme

The number of items of data which are irretrievable without additional effort after hashing can be greatly reduced if several hash tables are used simultaneously. Here we show that, in a multiple hashing scheme, this number has a distribution very close to Poisson. Thus choosing the number and sizes of the tables to minimize the expected number of irretrievable items is the right way to dimension a scheme

Betting Strategies in Horse Races

In this paper we consider the strategies a gambler may employ in situations such as horse races. We assume that the gambler knows which horses have odds which are favourable to him, that he wants to bet in such a way as to have a given positive expectation of win on a race, and that he wishes to minimise the probability of loss of his finite capital. We show that the best strategy is to bet on all the horses whose odds are favourable so as to minimise the probability of loss on a race. We further show that in order to achieve the last objective it is advisable to have a bet on a horse with fair odds, and at times on a horse with unfavourable odds, in addition to a bet on a horse with favourable odds