posterior: update the prior distribution with sufficient statistics

Description

This is a generic function that will update the prior distribution of a "BayesianBrick" object by adding information of the observation's sufficient statistics. i.e. for the model structure: $$theta|gamma \sim H(gamma)$$ $$x|theta \sim F(theta)$$ update gamma to gamma_posterior by adding the information of x to gamma. For a given sample set x or it's sufficient statistics ss, and a Bayesian bricks object obj, posterior() will update the posterior parameters in obj for different model structures:

class(obj)="LinearGaussianGaussian"

$$x \sim Gaussian(A z + b, Sigma)$$ $$z \sim Gaussian(m,S)$$ posterior() will update m and S in obj. See ?posterior.LinearGaussianGaussian for details.

class(obj)="GaussianGaussian"

Where $$x \sim Gaussian(mu,Sigma)$$ $$mu \sim Gaussian(m,S)$$ Sigma is known. posterior() will update m and S in obj. See ?posterior.GaussianGaussian for details.

class(obj)="GaussianInvWishart"

Where $$x \sim Gaussian(mu,Sigma)$$ $$Sigma \sim InvWishart(v,S)$$ mu is known. posterior() will update v and S in obj. See ?posterior.GaussianInvWishart for details.

class(obj)="GaussianNIW"

Where $$x \sim Gaussian(mu,Sigma)$$ $$Sigma \sim InvWishart(v,S)$$ $$mu \sim Gaussian(m,Sigma/k)$$ posterior() will update m, k, v and S in obj. See ?posterior.GaussianNIW for details.

class(obj)="GaussianNIG"

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. posterior() will update m, V, a and b in obj. See ?posterior.GaussianNIG for details.

class(obj)="CatDirichlet"

Where $$x \sim Categorical(pi)$$ $$pi \sim Dirichlet(alpha)$$ posterior() will update alpha in obj. See ?posterior.CatDirichlet for details.

class(obj)="CatDP"

Where $$x \sim Categorical(pi)$$ $$pi \sim DirichletProcess(alpha)$$ posterior() will update alpha in obj. See ?posterior.CatDP for details.

class(obj)="DP"

Where $$pi|alpha \sim DP(alpha,U)$$ $$z|pi \sim Categorical(pi)$$ $$theta_z|psi \sim H0(psi)$$ $$x|theta_z,z \sim F(theta_z)$$ posterior() will update alpha and psi in obj. See ?posterior.DP for details.

class(obj)="HDP"

Where $$G|gamma \sim DP(gamma,U)$$ $$pi_j|G,alpha \sim DP(alpha,G), j = 1:J$$ $$z|pi_j \sim Categorical(pi_j)$$ $$k|z,G \sim Categorical(G),\textrm{ if z is a sample from the base measure G}$$ $$theta_k|psi \sim H0(psi)$$ posterior() will update gamma, alpha and psi in obj. See ?posterior.HDP for details.

class(obj)="HDP2"

Where $$G |eta \sim DP(eta,U)$$ $$G_m|gamma,G \sim DP(gamma,G), m = 1:M$$ $$pi_{mj}|G_m,alpha \sim DP(alpha,G_m), j = 1:J_m$$ $$z|pi_{mj} \sim Categorical(pi_{mj})$$ $$k|z,G_m \sim Categorical(G_m),\textrm{ if z is a sample from the base measure } G_m$$ $$u|k,G \sim Categorical(G),\textrm{ if k is a sample from the base measure} G$$ $$theta_u|psi \sim H0(psi)$$ $$x|theta_u,u \sim F(theta_u)$$ posterior() will update eta, gamma, alpha and psi in obj. See ?posterior.HDP2 for details.

Usage

posterior(obj, ...)

Arguments

obj

A "BayesianBrick" object used to select a method.

...

further arguments passed to or from other methods.

Value

None, or an error message if the update fails.