GNU MCSim Frederic Yves Bois Chair of mathematical modeling and system biology for predictive toxicology

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1 GNU MCSim Frederic Yves Bois Chair of mathematical modeling and system biology for predictive toxicology 20/06/ / 11

2 Talk overview GNU MCSim is... What it can do Statistical modelling with GNU MCSim Specifying distributions Specifying probabilistic (statistical) models 20/06/2011-2

3 GNU MCSim is... A general algebraic / ordinary differential equation solver (stochastic simulations also for algebraic systems). Started around 1990 at UC Berkeley for our simulation needs (compartmental models of the body, for transport and metabolism of chemicals, drugs etc.) It is written in standard C language as a model preprocessor and a library of functions to make a model executable. Models are text files in native language or in SBML. The executable is very fast and very simple: commandline based, no GUI, no bells, no whistles. Under GPL v3, FSF, 20/06/2011-3

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5 What it can do Straigth simulations (set parameter values and initial conditions). Automated serial simulations (called "SetPoint", using pre-specified θ vectors) Monte Carlo simulations (sampling parameters from prior probability distributions) Markov chain Monte Carlo (MCMC) simulations (Bayesian sampling from joint posterior probability distribution, given priors, data, and a multilevel statistical model) Optimal design computing (in a Bayesian framework, finds the design with maximal expected utility given priors and a statistical model) 20/06/2011-5

6 Priors, posteriors, likelihood, multilevel models Posterior = Prior Likelihood Prior Iteration 20/06/

7 Statistical modelling with GNU MCSim Variables are typed as state variables, parameters, outputs and inputs. Input and state variables initial conditions can be parameterized. Parameters have default values. In simulation definition files, defaults can be overridden and distributions or densities can be assigned (if the context is relevant). E.g.: x = 10; # set the initial value if x is a state variable Pi = 3; # to stop worrying about little decimals... Distrib(A, Normal, 0, 1); Distrib(B, Normal, A, 2); All parameters assigned distributions are sampled according to the context: Distrib() defines a prior or a likelihood, depending on the type of variable. 20/06/2011-7

8 Specifying distributions About 20 distributions types are available. They all require at most four parameters. E.g.: Distrib(A, Normal, 0, 1, -10, 10); # mean, SD, LB, UB Arbitrary (stepwise) functions would be useful, but are not available yet. No multivariate distributions (would require matrix notations, not available). Re-using posterior samples is done through SetPoint simulations. Associated random number generators are available (for stochastic simulations). Likelihoods have an extended syntax: Distrib(Data(y), Normal, Prediction(y), 0.01); 20/06/2011-8

9 Specifying probabilistic models for MCMC and OD Statistical models are defined in the simulation definition file. They specify levels, priors, likelihoods, observation times and data. Hierarchy is handled through "cloning". E.g.: Level { # population level Distrib(A, Normal, 0, 1); Likelihood(y, Normal, Prediction(y), 0.05); Level { # sub-population, individual, experiment... level Distrib(A, Normal, A, B); # automatically cloned Simulation { Print (y, 10); Data (y, -0.2);} } } Simulation { Print (y, 15); Data (y, 1.6);} 20/06/2011-9

10 Conclusions Specifying distributions on parameters is relatively easy in the univariate case. The key concept is that of random variable. You need to define a context (or: what to do if not in the Monte Carlo case). The field of application is much increased if you offer the possibility to define distributions on data and statistical models. But basically, most statistical models can be cast as networks of random variables. GNU MCSim has some limitations due to the separation of structural and statistical models, but that has also some advantages (tbd...) 20/06/

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