Variational Dropout and the Local Reparameterization Trick

We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the minibatch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments

Approximation of probability density functions for PDEs with random parameters using truncated series expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored

Meiosis in Phycomyces

A four-factor cross between two strains of Phycomyces involving two auxotrophic, one color, and the mating type marker is described. Samples of 40 germspores from 84 individual fertile germsporangia were characterized. The results show: (i) The germspores of a germsporangium are derived from one meiosis in approximately 78% of the cases. (ii) The four markers are on separate chromosomes. They are nonselective. (iii) Analysis of a large sample of germspores from 106 pooled germsporangia confirms that the four markers are unlinked. (iv) From the ditype/tetratype ratios it is inferred that each marker is located about 15 map units from its centromere

Comparing the performance of geometrically similar airplanes

This note has been prepared for the National Advisory Committee for Aeronautics. It deals with the model rules relating to aeronautical problems, and shows how the characteristics of one airplane can be determined from those of another airplane of different weight or size, and of similar type. If certain rules for the ratios of the dimensions, the weights and the horsepower are followed, a small low-powered airplane can be used for obtaining information as to performance, stability, controllability and maneuverability of a larger prototype, and contrariwise

Low-dose alum application trialled as a management tool for internal nutrient loads in Lake Okaro, New Zealand

Aluminium sulfate (alum) was applied to Lake Okaro, a eutrophic New Zealand lake with recurrent cyanobacterial blooms, to evaluate its suitability for reducing trophic status and bloom frequency. The dose yielded 0.6 g aluminium m–3 in the epilimnion. Before dosing, pH exceeded 8 in epilimnetic waters but was optimal for flocculation (6–8) below 4 m depth. After dosing, there was no significant change in water clarity, hypolimnetic pH decreased to 5.5, and soluble aluminium exceeded recommended guidelines for protection of freshwater organisms. Epilimnetic phosphate concentrations decreased from 40 to 5 mg m–3 and total nitrogen (TN):total phosphorus (TP) mass ratios increased from 7:1 to 37:1. The dominant phytoplankton species changed from Anabaena spp. before dosing, to Ceratium hirudinella , then Staurastrum sp. after dosing. Detection of effectiveness of dosing may have been limited by sampling duration and design, as well as the low alum dose. The decrease in hypolimnetic pH and epilimnetic TP, and increase in Al3+ and chlorophyll a, are attributed to the low alkalinity lake water and coincidence of alum dosing with a cyanobacterial bloom and high pH