BayesLogit (version 0.3)

CUBS: Conjugate Update Backward Sample

Description

Draw an approximation of the states in a dynamic generalized linear model using conjugate updating and a backwards sampling.

THIS IS A BETA VERSION. AN ERROR CAN CRASH R. BE CAREFUL.

Usage

CUBS.C(z, X, n, mu, phi, W, m0, C0, obs=c("binom", "nbinom", "norm"), eps.rel=1e-8, max.iter=100)
CUBS.R(z, X, n, mu, phi, W, m0, C0, obs=c("binom", "nbinom", "norm"), max.iter=100)

Arguments

z

An N dimensional vector; $z_i$ is the response at $x_i$.

X

An N x P dimensional design matrix. X must be a _matrix_.

n

An N dimensional vector; the interpretation of $n_i$ changes with obs.

mu

A P.b dim. vector; the mean of beta.

phi

A P.b dim. vector; the autocorrelation of beta.

W

A P.b x P.b dim matrix; the innovation variance of beta.

m0

A P.b dimensional prior mean.

C0

A P.b x P.b dimensional prior precision. C0 must be a _matrix_.

obs

The type of response.

eps.rel

The relative error condition used when calculating the conjugate update parameters.

max.iter

The maximum number of iterations used when calculating the conjugate update parameters.

Value

• alpha
  A P.a array; the posterior sample of the static regression coefficient. When there is no static component this returns 0.
• beta
  A (N+1) x P.b array; the posterior sample of the dynamic regression coefficients.
• log.dens
  The log density (up to a shift) of the sample.

Details

In a generalized linear model, the observation $y_i$ is related, non-linearly, to some parameter $\psi_i$, which is modeled linearly: $x_i \beta$. In a dynamic model, $beta$ is allowed to change in time so that $\psi_i = x_i \beta_i$ where $beta_i$ is governed by some pre-specified dynamics. For the purposes of this routine we assume that $beta_i \sim AR(1)$. One may approximately draw $\beta | y$ by using linear Bayes with a conjugate updating procedure and the using backward sampling formulas.

The parameter n has different interpretations depending on the type of observation. For "binom", n is the number of trials at each observation. For "nbinom", n is the over-dispersion parameter at each response. For "norm", n is the observation variance.

References

Mike West, Jeff Harrison, and Helio S. Migon. Dynamics Generalized Linear Models and Bayesian Forecasting. Journal of the American Statistical Association, 1985.

Examples

data(rain)

  y = rain$y
  X = rain$X
  n = rain$n
  N = length(y+1)

  mu  = 0.0
  phi = 1.0
  W   = 0.1
  m0  = -1.0
  C0  = matrix(1.0)

  out = CUBS.C(y, X, n, mu, phi, W, m0, C0, "binom")		

  samp = 1
  beta = matrix(0, nrow=samp, ncol=N+1)

  for (i in 1:samp) {
    ## if (i%%100==0) {cat("Iteration", i, "\n")}
    out = CUBS.C(y, X, n, mu, phi, W, m0, C0, "binom")
    beta[i,] = out$beta
  }	

  qt = apply(beta, 2, function(x){quantile(x, c(0.25, 0.5, 0.75))})

  ## plot (qt[2,], col=1, type="l")
  ## lines(qt[1,], col=3)
  ## lines(qt[3,], col=3)