[3] **viXra:1602.0333 [pdf]**
*replaced on 2016-10-21 05:07:13*

**Authors:** L. Martino, V. Elvira, F. Louzada

**Comments:** 9 Pages.

The Sequential Importance Resampling (SIR) method is the core of the Sequential Monte Carlo (SMC) algorithms (a.k.a., particle filters). In this work, we point out a suitable choice for weighting properly a resampled particle. This observation entails several theoretical and practical consequences, allowing also the design of novel sampling schemes. Specifically, we describe one theoretical result about the sequential estimation of the marginal likelihood. Moreover, we suggest a novel resampling procedure for SMC algorithms called partial resampling, involving only a subset of the current cloud of particles. Clearly, this scheme attenuates the additional variance in the Monte Carlo estimators generated by the use of the resampling.

**Category:** Statistics

[2] **viXra:1602.0112 [pdf]**
*replaced on 2016-09-23 03:15:35*

**Authors:** L. Martino, V. Elvira, F. Louzada

**Comments:** Signal Processing, Volume 131, Pages: 386-401, 2017

The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an approximation $\widehat{ESS}$ of the theoretical ESS definition is widely applied, involving the inverse of the sum of the squares of the normalized importance weights. This formula, $\widehat{ESS}$, has become an essential piece within Sequential Monte Carlo (SMC) methods, to assess the convenience of a resampling step. From another perspective, the expression $\widehat{ESS}$ is related to the Euclidean distance between the probability mass described by the normalized weights and the discrete uniform probability mass function (pmf). In this work, we derive other possible ESS functions based on different discrepancy measures between these two pmfs. Several examples are provided involving, for instance, the geometric mean of the weights, the discrete entropy (including the {\it perplexity} measure, already proposed in literature) and the Gini coefficient among others. We list five theoretical requirements which a generic ESS function should satisfy, allowing us to classify different ESS measures. We also compare the most promising ones by means of numerical simulations.

**Category:** Statistics

[1] **viXra:1602.0053 [pdf]**
*replaced on 2016-02-05 08:42:31*

**Authors:** Jason Lind

**Comments:** 3 Pages. Added preliminary calculations for correcting non-normal distribution

Defines a rated set and uses it to calculated a weight directly from the statistics that enabled broad unified interpretation of data.

**Category:** Statistics