Measurement of aqueous film thickness between charged mercury and mica surfaces : a direct experimental probe of the Poisson-Boltzmann distribution

A stable aqueous electrolyte film is formed between a mercury drop and a flat mica surface due to electrical double-layer repulsion when a negative potential is applied to the mercury. Film thickness has been measured as a function of applied potential while keeping the film pressure constant. By making measurements in this way, it is possible to map the data directly according to the Poisson-Boltzmann equation. An excellent fit to the data is obtained, providing direct evidence for this classical equation and its use as the basis of the Gouy-Chapman model of the diffuse double layer in electrolyte solutions.

A nonparametric Bayesian Poisson Gamma model for count data

We propose a nonparametric Bayesian, linear Poisson gamma model for count data and use it for dictionary learning. A key property of this model is that it captures the parts-based representation similar to nonnegative matrix factorization. We present an auxiliary variable Gibbs sampler, which turns the intractable inference into a tractable one. Combining this inference procedure with the slice sampler of Indian buffet process, we show that our model can learn the number of factors automatically. Using synthetic and real-world datasets, we show that the proposed model outperforms other state-of-the-art nonparametric factor models.

Predicting future purchases with the Poisson log-normal model

The negative binomial distribution (NBD) has been widely used in marketing for modeling purchase frequency counts, particularly in packaged goods contexts. A key managerially relevant use of this model is Conditional Trend Analysis (CTA)—a method of benchmarking future sales utilizing the NBD conditional expectation. CTA allows brand managers to identify whether the sales change in a second period is accounted for by previous non-, light, or heavy buyers of the brand. Although a useful tool, the conditional prediction of the NBD suffers from a bias: it under predicts what the period-one non-buyer class will do in period two and over predicts the sales contribution of existing buyers. In addition, the NBD's assumption of a gamma-distributed mean purchase rate lacks theoretical support—it is not possible to explain why a gamma distribution should hold. This paper therefore proposes an alternative model using a log-normal distribution in place of the gamma distribution, hence creating a Poisson log-normal (PLN) distribution. The PLN distribution has a stronger theoretical grounding than the NBD as it has a natural interpretation relying on the central limit theorem. Empirical analysis of brands in multiple categories shows that the PLN distribution gives better predictions than the NBD.