ImpsampMNL {idefix}

Importance sampling MNL

Description

This function samples from the posterior distribution using importance sampling, assuming a multivariate (truncated) normal prior distribution and a MNL likelihood.

Usage

ImpsampMNL(
  n.draws,
  prior.mean,
  prior.covar,
  des,
  n.alts,
  y,
  alt.cte = NULL,
  lower = NULL,
  upper = NULL
)

Arguments

n.draws

Numeric value indicating the number of draws.

prior.mean

Numeric vector indicating the mean of the multivariate normal distribution (prior).

prior.covar

Covariance matrix of the prior distribution.

des

A design matrix in which each row is a profile. If alternative specific constants are present, those should be included as the first column(s) of the design. Can be generated with Modfed or CEA.

n.alts

Numeric value indicating the number of alternatives per choice set.

y

A binary response vector. RespondMNL can be used to simulate response data.

alt.cte

A binary vector indicating for each alternative whether an alternative specific constant is desired. The default is NULL.

lower

Numeric vector of lower truncation points, the default is NULL.

upper

Numeric vector of upper truncation points, the default is NULL.

Details

For the proposal distribution a t-distribution with degrees of freedom equal to the number of parameters is used. The posterior mode is estimated using optim, and the covariance matrix is calculated as the negative inverse of the generalized Fisher information matrix. See reference for more information.

From this distribution a lattice grid of draws is generated.

If truncation is present, incorrect draws are rejected and new ones are generated untill n.draws is reached. The covariance matrix is in this case still calculated as if no truncation was present.

Value

sample

Numeric vector with the (unweigthted) draws from the posterior distribution.

weights

Numeric vector with the associated weights of the draws.

max

Numeric vector with the estimated mode of the posterior distribution.

covar

Matrix representing the estimated variance covariance matrix.

References

Yu J, Goos P, Vandebroek M (2011). “Individually adapted sequential Bayesian conjoint-choice designs in the presence of consumer heterogeneity.” https://www.sciencedirect.com/science/article/pii/S0167811611000668.

Examples

## Example 1: sample from posterior, no constraints, no alternative specific constants 
# choice design  
design <- example_design
# Respons.
truePar <- c(0.7, 0.6, 0.5, -0.5, -0.7, 1.7) # some values
set.seed(123)
resp <- RespondMNL(par = truePar, des = design, n.alts = 2)
#prior
pm <- c(1, 1, 1, -1, -1, 1) # mean vector 
pc <- diag(1, ncol(design)) # covariance matrix 
# draws from posterior.
ImpsampMNL(n.draws = 100, prior.mean =  pm, prior.covar = pc,
           des = design, n.alts = 2, y = resp)

## example 2:  sample from posterior with constraints 
# and alternative specific constants
# choice design. 
design <- example_design2
# Respons.
truePar <- c(0.2, 0.8, 0.7, 0.6, 0.5, -0.5, -0.7, 1.7) # some values
set.seed(123)
resp <- RespondMNL(par = truePar, des = design, n.alts = 3)
# prior
pm <- c(1, 1, 1, 1, 1, -1, -1, 1) # mean vector 
pc <- diag(1, ncol(design)) # covariance matrix
low = c(-Inf, -Inf, 0, 0, 0, -Inf, -Inf, 0)
up = c(Inf, Inf, Inf, Inf, Inf, 0, 0, Inf)
# draws from posterior.
ImpsampMNL(n.draws = 100, prior.mean =  pm, prior.covar = pc, des = design,
           n.alts = 3, y = resp, lower = low, upper = up, alt.cte = c(1, 1, 0))