# MAGI posterior density Computes the MAGI log-posterior value and gradient for an ODE model with the given inputs: the observations $Y$, the latent system trajectories $X$, the parameters $\theta$, the noise standard deviations $\sigma$, and covariance kernels. ```r MagiPosterior( y, xlatent, theta, sigma, covAllDimInput, odeModel, priorTemperatureInput = 1, useBand = FALSE ) ``` ## Arguments - `y`: data matrix of observations - `xlatent`: matrix of system trajectory values - `theta`: vector of parameter values $\theta$ - `sigma`: vector of observation noise for each system component - `covAllDimInput`: list of covariance kernel objects for each system component. Covariance calculations may be carried out with `calCov`. - `odeModel`: list of ODE functions and inputs. See details. - `priorTemperatureInput`: vector of tempering factors for the GP prior, derivatives, and observations, in that order. Controls the influence of the GP prior relative to the likelihood. Recommended values: the total number of observations divided by the total number of discretization points for the GP prior and derivatives, and 1 for the observations. - `useBand`: logical: should the band matrix approximation be used? If `TRUE`, `covAllDimInput` must include CinvBand, mphiBand, and KinvBand as computed by `calCov`. ## Returns A list with elements `value` for the value of the log-posterior density and `grad` for its gradient. ## Examples ```r # Trajectories from the Fitzhugh-Nagumo equations tvec <- seq(0, 20, 2) Vtrue <- c(-1, 1.91, 1.38, -1.32, -1.5, 1.73, 1.66, 0.89, -1.82, -0.93, 1.89) Rtrue <- c(1, 0.33, -0.62, -0.82, 0.5, 0.94, -0.22, -0.9, -0.08, 0.95, 0.3) # Noisy observations Vobs <- Vtrue + rnorm(length(tvec), sd = 0.05) Robs <- Rtrue + rnorm(length(tvec), sd = 0.1) # Prepare distance matrix for covariance kernel calculation foo <- outer(tvec, t(tvec), '-')[, 1, ] r <- abs(foo) r2 <- r^2 signr <- -sign(foo) # Choose some hyperparameter values to illustrate rphi <- c(0.95, 3.27) vphi <- c(1.98, 1.12) phiTest <- cbind(vphi, rphi) # Covariance computations curCovV <- calCov(phiTest[,1], r, signr, kerneltype = "generalMatern") curCovR <- calCov(phiTest[,2], r, signr, kerneltype = "generalMatern") # Y and X inputs to MagiPosterior yInput <- data.matrix(cbind(Vobs, Robs)) xlatentTest <- data.matrix(cbind(Vtrue, Rtrue)) # Create odeModel list for FN equations fnmodel <- list( fOde = fnmodelODE, fOdeDx = fnmodelDx, fOdeDtheta = fnmodelDtheta, thetaLowerBound = c(0, 0, 0), thetaUpperBound = c(Inf, Inf, Inf) ) MagiPosterior(yInput, xlatentTest, theta = c(0.2, 0.2, 3), sigma = c(0.05, 0.1), list(curCovV, curCovR), fnmodel) ```

MagiPosterior function