Approximate probability propagation with mixtures of truncated exponentials

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

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Making high-dimensional molecular distribution functions tractable through Belief Propagation on Factor Graphs

10.1101/2021.06.28.450193 ◽

2021 ◽

Author(s):

Zachary Smith ◽

Pratyush Tiwary

Keyword(s):

Belief Propagation ◽

Conformational Dynamics ◽

Joint Probability ◽

Md Simulations ◽

Distribution Functions ◽

Factor Graph ◽

High Dimensional ◽

Enhanced Sampling ◽

Small Peptides ◽

Approximate Probability

Molecular dynamics (MD) simulations provide a wealth of high-dimensional data at all-atom and femtosecond resolution but deciphering mechanistic information from this data is an ongoing challenge in physical chemistry and biophysics. Theoretically speaking, joint probabilities of the equilibrium distribution contain all thermodynamic information, but they prove increasingly difficult to compute and interpret as the dimensionality increases. Here, inspired by tools in probabilistic graphical modeling, we develop a factor graph trained through belief propagation that helps factorize the joint probability into an approximate tractable form that can be easily visualized and used. We validate the study through the analysis of the conformational dynamics of two small peptides with 5 and 9 residues. Our validations include testing the conditional dependency predictions through an intervention scheme inspired by Judea Pearl. Secondly we directly use the belief propagation based approximate probability distribution as a high-dimensional static bias for enhanced sampling, where we achieve spontaneous back-and-forth motion between metastable states that is up to 350 times faster than unbiased MD. We believe this work opens up useful ways to thinking about and dealing with high-dimensional molecular simulations.

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INCREMENTAL COMPILATION OF BAYESIAN NETWORKS BASED ON MAXIMAL PRIME SUBGRAPHS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488511006952 ◽

2011 ◽

Vol 19 (02) ◽

pp. 155-191 ◽

Cited By ~ 5

Author(s):

M. JULIA FLORES ◽

JOSE A. GÁMEZ ◽

KRISTIAN G. OLESEN

Keyword(s):

Bayesian Networks ◽

Bayesian Network ◽

Great Improvement ◽

Experimental Evaluation ◽

Iterative Process ◽

Dynamic Models ◽

Probability Distributions ◽

Probability Propagation ◽

Incremental Compilation

When a Bayesian network (BN) is modified, for example adding or deleting a node, or changing the probability distributions, we usually will need a total recompilation of the model, despite feeling that a partial (re)compilation could have been enough. Especially when considering dynamic models, in which variables are added and removed very frequently, these recompilations are quite resource consuming. But even further, for the task of building a model, which is in many occasions an iterative process, there is a clear lack of flexibility. When we use the term Incremental Compilation or IC we refer to the possibility of modifying a network and avoiding a complete recompilation to obtain the new (and different) join tree (JT). The main point we intend to study in this work is JT-based inference in Bayesian networks. Apart from undertaking the triangulation problem itself, we have achieved a great improvement for the compilation in BNs. We do not develop a new architecture for BNs inference, but taking some already existing framework JT-based for probability propagation such as Hugin or Shenoy and Shafer, we have designed a method that can be successfully applied to get better performance, as the experimental evaluation will show.

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