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Monte Carlo Markov Chain Algorithm Convergence
Published in Richard M. Golden, Statistical Machine Learning, 2020
Currently, “probabilistic programming languages” have been developed that allow users to specify customized probabilistic models and then use Monte Carlo Markov Chain methods to sample from those models and generate inferences (e.g., Davidson-Pilon 2015; Pfeffer 2016).
Investigating long-term performance of flexible pavement using Bayesian multilevel models
Published in Road Materials and Pavement Design, 2023
Haimei Liang, Hongren Gong, Yiren Sun, Jiachen Shi, Lin Cong, Wenyang Han, Peng Guo
We used a type of Monte Carlo sampling algorithm, Hamiltonian Monte Carlo (HMC), to simulate the posterior distributions (Neal et al., 2011). We coded our Bayesian models using the Stan probabilistic programming language (Gelman et al., 2015). The concerned posterior distributions were simulated using five parallel Markov chains, with each containing 10,000 samples, in which 5000 samples were used for ‘warming up’ the chains. For details of training Bayesian models through HMC and Stan, please refer to McElreath (2020) and Chen et al. (2020). All parameters (including the intercept terms) were assigned a prior to a weakly informative normal distribution N(0,2.5), while the standard deviation term (σ) was assigned a non-negative distribution cauchy(0,1). Using weakly informative priors instead of diffusive (flat) prior distributions has been recommended by van de Schoot et al. (2021) and McElreath (2020).
Bayesian estimation of discrete choice models: a comparative analysis using effective sample size
Published in Transportation Letters, 2022
Jason Hawkins, Khandker Nurul Habib
Most applications of Bayesian inference in the transportation literature rely on an implementation of the AT method in an existing software package (i.e., Apollo (Hess and Palma 2019), bayesm (Rossi 2019), or code adapted from that provided by Kenneth Train (2006)). Ding et al. (2016) specify an ordered logit model for auto ownership choice using custom-built GAUSS code. They do not comment on the runtime of the model. In some instances, researchers have developed VB methods that are specific to a model specification (Bansal et al. 2020). In the case of using an existing implementation of AT, the researcher is constrained in the available model structure to MMNL. In the second instance, while the development of VB methods for a particular model is a worthy research objective, it can be time-consuming and require specific technical knowledge. The solution we propose to these two challenges is the adoption of probabilistic programming. Probabilistic programming packages aim to provide a generalized tool for Bayesian inference. They include BUGS (Lunn et al. 2012), PyMC (Salvatier, Wiecki, and Fonnesbeck 2016), Stan (Carpenter et al. 2017), and Tensorflow Probability (Dillon et al. 2017).
A Comprehensive Review on Stochastic Optimal Power Flow Problems and Solution Methodologies
Published in IETE Technical Review, 2023
Ankur Maheshwari, Yog Raj Sood, Supriya Jaiswal
In power system operations, uncertainties are inevitable and can arise from various sources. Integrating RESs and dynamic loads into power systems has increased uncertainty in power system operation. The most common techniques that have been used to model uncertainties and stochastic behavior of RESs and dynamic loads are as follows: Probabilistic programming: This method incorporates PDFs, such as Weibull, Lognormal, Beta, Normal and others, to model the uncertainty of RESs. Probabilistic programming models rely on data to develop accurate predictions and estimations of uncertainty. While probabilistic programming may be computationally intensive, it may provide a fairer presentation of the uncertainty.Stochastic optimization: In this method, rather than relying on a single, fixed model, stochastic optimization employs a set of scenarios to capture the range of possible behaviors that RESs may exhibit over time. By analyzing these scenarios and their associated probabilities, stochastic optimization can minimize the expected value of the objective function, which measures the system’s performance or cost. Developing accurate stochastic optimization models can be computationally intensive, particularly for large and complex systems requiring significant computing resources.Robust optimization: Robust optimization is a powerful technique for handling uncertainties in power systems. This approach involves identifying worst-case scenarios for uncertain parameters and developing solutions that perform well under all possible scenarios. Rather than optimizing for a single point estimate of the uncertain parameters, robust optimization considers a set of possible realizations and performs the optimization over this set. This approach provides a robust solution that can perform well under all possible realizations of the uncertain parameters, even those not explicitly considered in the modeling. However, robust optimization can also result in a conservative and computationally expensive solution, as it seeks to address all possible scenarios, including those that may be unlikely or rare. Additionally, robust optimization may not always result in the optimal solution if the worst-case scenarios do not occur.