# posterior: update the prior distribution with sufficient statistics In bbricks: Bayesian Methods and Graphical Model Structures for Statistical Modeling

## 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

 `1` ```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.

`posterior.LinearGaussianGaussian` for Linear Gaussian and Gaussian conjugate structure, `posterior.GaussianGaussian` for Gaussian-Gaussian conjugate structure, `posterior.GaussianInvWishart` for Gaussian-Inverse-Wishart conjugate structure, `posterior.GaussianNIW` for Gaussian-NIW conjugate structure, `posterior.GaussianNIG` for Gaussian-NIG conjugate structure, `posterior.CatDirichlet` for Categorical-Dirichlet conjugate structure, `posterior.CatDP` for Categorical-DP conjugate structure ...