The β k and (sigma _{k}^{2}) which maximize the log marginal likelihood are then found iteratively by setting β and σ 2 to initial values and then finding values for (boldsymbol {mu }^{n}_{c^,c^,i}) and (boldsymbol {Sigma }^{n}_{c^,c^,i}) from (12) and (13).
For each treatment (column 1), the two runs of highest harmonic-mean marginal likelihood are reported and compared using Bayes Factors as calculated using 1000 bootstrap pseudoreplicates in Tracer v.1.5, which employs a weighted likelihood bootstrap estimator.
The marginal likelihood was estimated by the stepping stone method78,80 using 50 stones with a chain length of 5,000.
A Bayesian approach to model comparison based on the integrated or marginal likelihood is considered, and applications to linear regression models and nonlinear ordinary differential equation (ODE) models are used as the setting in which to elucidate and further develop existing statistical methodology.
With these transformations, the marginal likelihood is given by Eq. (16).
Since GPR is a form of Bayesian regression, the marginal likelihood is equal to the integral over the product of the prior and the likelihood function.
A useful transformation of the marginal likelihood is the so-called energy function, which is sometimes more convenient to deal with.
More formally, if for each unit the response variables are assumed to be independent conditional on the random effects and if one maintains the assumption that the units are independent, the marginal likelihood is L boldsymbol{theta})=prod_{i=1}^{n}left[int~_{i}pilambda y_{i1} left(prod_{t=2}^{T}boldsymbol{gamma}_{it}lambda y_{it})right)mathbf{1^{prime}}f xi_{i1},xi_{i2})dxi_{i1}dxi_{i2}right].
The marginal likelihood is given as: (pleft( Y|X,theta right) =prod _{n=1}^{N}pleft( y_{n}mid x_{n},theta right) ), where (X=left[ x_{n}right] _{n=1}^{N}in mathbb {R}^{dtimes N}), (Y=left[ y_{n}right] _{n=1}^{N}in mathbb {R}^{N}), and (theta =left( theta _{1},theta _{2},ldots,theta _{m}right) ) encompass the parameters of all experts.
Generally speaking, for simplicity, the log marginal likelihood is maximised [13]: begin{array}{*{20}l} log pleft(boldsymbol{Y}|boldsymbol{X}, thetaright) =& - frac{1}{2} boldsymbol{Y}^{T} (K + {sigma_{n}^{2}}boldsymbol{I})^{-1}boldsymbol{Y}& - frac{1}{2} log|K + {sigma_{n}^{2}}boldsymbol{I}| - frac{n}{2} log2pi.
In a Bayesian framework, the estimated marginal likelihood was significantly higher for clock-like model in nearly all cases, and not significantly different for the remaining cases.
