Bayesian inference with Stan: A tutorial on adding custom distributions.

When evaluating cognitive models based on fits to observed data (or, really, any model that has free parameters), parameter estimation is critically important. Traditional techniques like hill climbing by minimizing or maximizing a fit statistic often result in point estimates. Bayesian approaches instead estimate parameters as posterior probability distributions, and thus naturally account for the uncertainty associated with parameter estimation; Bayesian approaches also offer powerful and principled methods for model comparison. Although software applications such as WinBUGS (Lunn, Thomas, Best, & Spiegelhalter, Statistics and Computing, 10, 325-337, 2000) and JAGS (Plummer, 2003) provide "turnkey"-style packages for Bayesian inference, they can be inefficient when dealing with models whose parameters are correlated, which is often the case for cognitive models, and they can impose significant technical barriers to adding custom distributions, which is often necessary when implementing cognitive models within a Bayesian framework. A recently developed software package called Stan (Stan Development Team, 2015) can solve both problems, as well as provide a turnkey solution to Bayesian inference. We present a tutorial on how to use Stan and how to add custom distributions to it, with an example using the linear ballistic accumulator model (Brown & Heathcote, Cognitive Psychology, 57, 153-178. doi: 10.1016/j.cogpsych.2007.12.002 , 2008).

Thermodynamic Integration and Steppingstone Sampling Methods for Estimating Bayes Factors: A Tutorial.

One of the more principled methods of performing model selection is via Bayes factors. However, calculating Bayes factors requires marginal likelihoods, which are integrals over the entire parameter space, making estimation of Bayes factors for models with more than a few parameters a significant computational challenge. Here, we provide a tutorial review of two Monte Carlo techniques rarely used in psychology that efficiently compute marginal likelihoods: thermodynamic integration (Friel & Pettitt, 2008; Lartillot & Philippe, 2006) and steppingstone sampling (Xie, Lewis, Fan, Kuo, & Chen, 2011). The methods are general and can be easily implemented in existing MCMC code; we provide both the details for implementation and associated R code for the interested reader. While Bayesian toolkits implementing standard statistical analyses (e.g., JASP Team, 2017; Morey & Rouder, 2015) often compute Bayes factors for the researcher, those using Bayesian approaches to evaluate cognitive models are usually left to compute Bayes factors for themselves. Here, we provide examples of the methods by computing marginal likelihoods for a moderately complex model of choice response time, the Linear Ballistic Accumulator model (Brown & Heathcote, 2008), and compare them to findings of Evans and Brown (2017), who used a brute force technique. We then present a derivation of TI and SS within a hierarchical framework, provide results of a model recovery case study using hierarchical models, and show an application to empirical data. A companion R package is available at the Open Science Framework: https://osf.io/jpnb4.

Residues SFQ (173-175) in the large extracellular loop of CD9 are required for gamete fusion.

Gamete fusion is the fundamental first step initiating development of a new organism. Female mice with a gene knockout for the tetraspanin CD9 (CD9 KO mice) produce mature eggs that cannot fuse with sperm. However, nothing is known about how egg surface CD9 functions in the membrane fusion process. We found that constructs including CD9's large extracellular loop significantly inhibited gamete fusion when incubated with eggs but not when incubated with sperm, suggesting that CD9 acts by interaction with other proteins in the egg membrane. We also found that injecting developing CD9 KO oocytes with CD9 mRNA restored fusion competence to the resulting CD9 KO eggs. Injecting mRNA for either mouse CD9 or human CD9, whose large extracellular loops differ in 18 residues, rescued fusion ability of the injected CD9 KO eggs. However, when the injected mouse CD9 mRNA contained a point mutation (F174 to A) the gamete fusion level was reduced fourfold, and a change of three residues (173-175, SFQ to AAA) abolished CD9's activity in gamete fusion. These results suggest that SFQ in the CD9 large extracellular loop may be an active site which associates with and regulates the egg fusion machinery.