Hidden Markov Models of Evidence Accumulation in Speeded Decision Tasks

Speeded decision tasks are usually modeled within the evidence accumulation framework, enabling inferences on latent cognitive parameters, and capturing dependencies between the observed response times and accuracy. An example is the speed-accuracy trade-off, where people sacrifice speed for accuracy (or vice versa). Different views on this phenomenon lead to the idea that participants may not be able to control this trade-off on a continuum, but rather switch between distinct states (Dutilh et al., Cognitive Science 35(2):211–250, 2010). Hidden Markov models are used to account for switching between distinct states. However, combining evidence accumulation models with a hidden Markov structure is a challenging problem, as evidence accumulation models typically come with identification and computational issues that make them challenging on their own. Thus, an integration of hidden Markov models with evidence accumulation models has still remained elusive, even though such models would allow researchers to capture potential dependencies between response times and accuracy within the states, while concomitantly capturing different behavioral modes during cognitive processing. This article presents a model that uses an evidence accumulation model as part of a hidden Markov structure. This model is considered as a proof of principle that evidence accumulation models can be combined with Markov switching models. As such, the article considers a very simple case of a simplified Linear Ballistic Accumulation. An extensive simulation study was conducted to validate the model’s implementation according to principles of robust Bayesian workflow. Example reanalysis of data from Dutilh et al. (Cognitive Science 35(2):211–250, 2010) demonstrates the application of the new model. The article concludes with limitations and future extensions or alternatives to the model and its application.

Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

Hidden Markov Models of Evidence Accumulation in Speeded Decision Tasks

Speeded decision tasks are usually modeled within the evidence accumulation framework, enabling inferences on latent cognitive parameters, and capturing dependencies between the observed response times and accuracy. An example is the speed-accuracy trade-off, where people sacrifice speed for accuracy (or vice versa). Different views on this phenomenon lead to the idea that participants may not be able to control this trade-off on a continuum, but rather switch between distinct states (Dutilh, et al., 2010).Hidden Markov models are used to account for switching between distinct states. However, combining evidence accumulation models with a hidden Markov structure is a challenging problem, as evidence accumulation models typically come with identification and computational issues that make them challenging on their own. Thus, hidden Markov models have not used the evidence accumulation framework, giving up on the inference on the latent cognitive parameters, or capturing potential dependencies between response times and accuracy within the states.This article presents a model that uses an evidence accumulation model as part of a hidden Markov structure. This model is considered as a proof of principle that evidence accumulation models can be combined with Markov switching models. As such, the article considers a very simple case of a simplified Linear Ballistic Accumulation. An extensive simulation study was conducted to validate the model's implementation according to principles of robust Bayesian workflow. Example reanalysis of data from Dutilh, et al. (2010) demonstrates the application of the new model. The article concludes with limitations and future extensions or alternatives to the model and its application.

Data-driven analysis on the simulations of the spread of COVID-19 under different interventions of China

ABSTRACTSince February 2020, COVID-19 has spread rapidly to more than 200 countries in the world. During the pandemic, local governments in China have implemented different interventions to efficiently control the spread of the epidemic. Characterizing transmission of COVID-19 under some typical interventions is essential to help countries develop appropriate interventions. Based on the pre-symptomatic transmission patterns of COVID-19, we established a novel compartmental model: Baysian SIHR model with latent Markov structure, which treated the numbers of infected and infectious individuals without isolation to be the latent variables and allowed the effective reproduction number to change over time, thus the effects of policies could be reasonably estimated. By using the epidemic data of Wuhan, Wenzhou and Shenzhen, we migrated the corresponding estimated policy modes to South Korea, Italy, and the United States and simulated the potential outcomes for these countries when they adopted similar policy strategies of three cities in China. We found that the mild interventions implemented in Shenzhen were effective to control the epidemic in the early stage, while more stringent policies which were issued in Wuhan and Wenzhou were necessary if the epidemic was more severe and needed to be controlled in a short time.

Simplified integrated nested Laplace approximation

Summary Integrated nested Laplace approximation provides accurate and efficient approximations for marginal distributions in latent Gaussian random field models. Computational feasibility of the original Rue et al. (2009) methods relies on efficient approximation of Laplace approximations for the marginal distributions of the coefficients of the latent field, conditional on the data and hyperparameters. The computational efficiency of these approximations depends on the Gaussian field having a Markov structure. This note provides equivalent efficiency without requiring the Markov property, which allows for straightforward use of latent Gaussian fields without a sparse structure, such as reduced rank multi-dimensional smoothing splines. The method avoids the approximation for conditional modes used in Rue et al. (2009), and uses a log determinant approximation based on a simple quasi-Newton update. The latter has a desirable property not shared by the most commonly used variant of the original method.

Lower bounds for the decay of correlations in non-uniformly expanding maps

We give conditions under which non-uniformly expanding maps exhibit lower bounds of polynomial type for the decay of correlations and for a large class of observables. We show that if the Lasota–Yorke-type inequality for the transfer operator of a first return map is satisfied in a Banach space ${\mathcal{B}}$, and the absolutely continuous invariant measure obtained is weak mixing, in terms of aperiodicity, then, under some renewal condition, the maps have polynomial decay of correlations for observables in ${\mathcal{B}}.$ We also provide some general conditions that give aperiodicity for expanding maps in higher dimensional spaces. As applications, we obtain lower bounds for piecewise expanding maps with an indifferent fixed point and for which we also allow non-Markov structure and unbounded distortion. The observables are functions that have bounded variation or satisfy quasi-Hölder conditions and have their support bounded away from the neutral fixed points.