Probabilistic machine learning for local volatility

Bayesian inference of polymerase dynamics over the exclusion process

“…The difficulty of sampling covariance hyperparameters is also addressed, e.g. in [ 52 ]. The predicted transient density profiles at t = 30, 60, 300 also are in very good agreement with the input data ( figure 2 d ; in fact, all sampled rate profiles yield similar time-course density profiles despite wide CIs in certain regions (see also electronic supplementary material, figure S3).…”

“…The general problem addressed here is the identification of the PDE parameters that best describe data as a subset of the true PDE solution (see, e.g. [ 52 , 56 – 58 ] and references therein). We focused on the hydrodynamic TASEP with smoothly varying jump rates (which are the parameters to be inferred) as a paradigmatic and well-characterized model of transport.…”

“…Prior distributions are chosen to pull Markov chain Monte Carlo (MCMC) samples away from inappropriate results that are consistent with the likelihood but would not be consistent with domain knowledge [51]. By using scaled sigmoid Gaussian prior probability bounded by θ min and θ max , we only search for solutions constrained in an appropriate interval [52]. By virtue of the Bayes theorem the joint posterior probability for θ and f satisfies Pfalse(f,m,σϵ,κ,σf,lfalsefalse|yfalse)∝Pfalse(yfalsefalse|f,σϵ,κfalse)Pfalse(ffalsefalse|m,σf,lfalse)Pfalse(mfalse)Pfalse(σϵfalse)Pfalse(κfalse)Pfalse(σffalse)Pfalse(lfalse),which we draw random samples from by MCMC sampling, more specifically block Gibbs sampling with elliptical slice sampling at each block [52,53] (electronic supplementary material, appendix C).…”

“…By using scaled sigmoid Gaussian prior probability bounded by θ min and θ max , we only search for solutions constrained in an appropriate interval [52]. By virtue of the Bayes theorem the joint posterior probability for θ and f satisfies Pfalse(f,m,σϵ,κ,σf,lfalsefalse|yfalse)∝Pfalse(yfalsefalse|f,σϵ,κfalse)Pfalse(ffalsefalse|m,σf,lfalse)Pfalse(mfalse)Pfalse(σϵfalse)Pfalse(κfalse)Pfalse(σffalse)Pfalse(lfalse),which we draw random samples from by MCMC sampling, more specifically block Gibbs sampling with elliptical slice sampling at each block [52,53] (electronic supplementary material, appendix C). Equation (2.15) expresses the distribution of parameters given the observed data y and completes the definition of the model.…”

“…Prior distributions are chosen to pull Markov chain Monte Carlo (MCMC) samples away from inappropriate results that are consistent with the likelihood but would not be consistent with domain knowledge [51]. By using scaled sigmoid Gaussian prior probability bounded by θ min and θ max , we only search for solutions constrained in an appropriate interval [52].…”