This is a generic function that will generate the the density value of the posterior predictive distribution. i.e. for the model structure:
$$theta|gamma \sim H(gamma)$$
$$x|theta \sim F(theta)$$
get the probability density/mass of the posterior predictive distribution of a new sample x_new: p(x_new|gamma).
For a given Bayesian bricks object obj and a new sample x, dPosteriorPredictive() will calculate the marginal likelihood for different model structures:
Where
$$x \sim Gaussian(A z + b, Sigma)$$
$$z \sim Gaussian(m,S)$$
dPosteriorPredictive() will return p(x|m,S,A,b,Sigma)
See ?dPosteriorPredictive.LinearGaussianGaussian for details.
Where
$$x \sim Gaussian(mu,Sigma)$$
$$mu \sim Gaussian(m,S)$$
Sigma is known.
dPosteriorPredictive() will return p(x|m,S,Sigma)
See ?dPosteriorPredictive.GaussianGaussian for details.
Where
$$x \sim Gaussian(mu,Sigma)$$
$$Sigma \sim InvWishart(v,S)$$
mu is known.
dPosteriorPredictive() will return p(x|mu,v,S)
See ?dPosteriorPredictive.GaussianInvWishart for details.
Where
$$x \sim Gaussian(mu,Sigma)$$
$$Sigma \sim InvWishart(v,S)$$
$$mu \sim Gaussian(m,Sigma/k)$$
dPosteriorPredictive() will return p(x|m,k,v,S)
See ?dPosteriorPredictive.GaussianNIW for details.
Where
$$x \sim Gaussian(X beta,sigma^2)$$
$$sigma^2 \sim InvGamma(a,b)$$
$$beta \sim Gaussian(m,sigma^2 V)$$
X is a row vector, or a design matrix where each row is an obervation.
dPosteriorPredictive() will return p(x,X|m,V,a,b)
See ?dPosteriorPredictive.GaussianNIG for details.
Where
$$x \sim Categorical(pi)$$
$$pi \sim Dirichlet(alpha)$$
dPosteriorPredictive() will return p(x|alpha)
See ?dPosteriorPredictive.CatDirichlet for details.
Where
$$x \sim Categorical(pi)$$
$$pi \sim DirichletProcess(alpha)$$
dPosteriorPredictive() will return p(x|alpha)
See ?dPosteriorPredictive.CatDP for details.
dPosteriorPredictive(obj, ...)A "BayesianBrick" object used to select a method.
further arguments passed to or from other methods.
numeric, the density value
dPosteriorPredictive.LinearGaussianGaussian for Linear Gaussian and Gaussian conjugate structure, dPosteriorPredictive.GaussianGaussian for Gaussian-Gaussian conjugate structure, dPosteriorPredictive.GaussianInvWishart for Gaussian-Inverse-Wishart conjugate structure, dPosteriorPredictive.GaussianNIW for Gaussian-NIW conjugate structure, dPosteriorPredictive.GaussianNIG for Gaussian-NIG conjugate structure, dPosteriorPredictive.CatDirichlet for Categorical-Dirichlet conjugate structure, dPosteriorPredictive.CatDP for Categorical-DP conjugate structure ...