Sentence examples for marginal likelihood is from inspiring English sources
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.
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.
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.
The marginal likelihood was estimated by the stepping stone method78,80 using 50 stones with a chain length of 5,000.
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).
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.
From each model, the marginal likelihood was estimated using the harmonic mean estimator [59] and the Bayes factor found by taking the ratio.
After a run of 1 million iterations in BayesTraits [15] to estimate the correlation between leaf-living ecology and reduced morphology, the following harmonic means of the marginal likelihood were obtained, with virtually no difference between runs: ∼71.50 for the model in which leaf-living ecology and reduced morphology are independent, and ∼56.50 for the model in which they are dependent.
