MCMC Algorithm for Hierarchical Multinomial Logit with Mixture-of-Normals Heterogeneity
rhierMnlRwMixture is a MCMC algorithm for a hierarchical multinomial logit with a mixture of normals heterogeneity distribution. This is a hybrid Gibbs Sampler with a RW Metropolis step for the MNL coefficients for each panel unit.
rhierMnlRwMixture(Data, Prior, Mcmc)
Data |
list(lgtdata, Z, p) |
Prior |
list(a, deltabar, Ad, mubar, Amu, nu, V, a, ncomp, SignRes) |
Mcmc |
list(R, keep, nprint, s, w) |
y_i ~ MNL(X_i,β_i) with i = 1, …, length(lgtdata) and where β_i is nvar x 1
β_i = ZΔ[i,] + u_i
Note: ZΔ is the matrix Z * Δ and [i,] refers to ith row of this product
Delta is an nz x nvar array
u_i ~ N(μ_{ind},Σ_{ind}) with ind ~ multinomial(pvec)
pvec ~ dirichlet(a)
delta = vec(Δ) ~ N(deltabar, A_d^{-1})
μ_j ~ N(mubar, Σ_j (x) Amu^{-1})
Σ_j ~ IW(nu, V)
Note: Z should NOT include an intercept and is centered for ease of interpretation.
The mean of each of the nlgt βs is the mean of the normal mixture.
Use summary() to compute this mean from the compdraw output.
Be careful in assessing prior parameter Amu: 0.01 is too small for many applications.
See chapter 5 of Rossi et al for full discussion.
Data = list(lgtdata, Z, p) [Z optional]
lgtdata: |
list of nlgt=length(lgtdata) lists with each cross-section unit MNL data |
lgtdata[[i]]$y: |
n_i x 1 vector of multinomial outcomes (1, ..., p) |
lgtdata[[i]]$X: |
n_i*p x nvar design matrix for ith unit |
Z: |
nreg x nz matrix of unit chars (def: vector of ones) |
p: |
number of choice alternatives |
Prior = list(a, deltabar, Ad, mubar, Amu, nu, V, a, ncomp, SignRes) [all but ncomp are optional]
a: |
ncomp x 1 vector of Dirichlet prior parameters (def: rep(5,ncomp)) |
deltabar: |
nz*nvar x 1 vector of prior means (def: 0) |
Ad: |
prior precision matrix for vec(D) (def: 0.01*I) |
mubar: |
nvar x 1 prior mean vector for normal component mean (def: 0 if unrestricted; 2 if restricted) |
Amu: |
prior precision for normal component mean (def: 0.01 if unrestricted; 0.1 if restricted) |
nu: |
d.f. parameter for IW prior on normal component Sigma (def: nvar+3 if unrestricted; nvar+15 if restricted) |
V: |
PDS location parameter for IW prior on normal component Sigma (def: nu*I if unrestricted; nu*D if restricted with d_pp = 4 if unrestricted and d_pp = 0.01 if restricted) |
ncomp: |
number of components used in normal mixture |
SignRes: |
nvar x 1 vector of sign restrictions on the coefficient estimates (def: rep(0,nvar))
|
Mcmc = list(R, keep, nprint, s, w) [only R required]
R: |
number of MCMC draws |
keep: |
MCMC thinning parameter -- keep every keepth draw (def: 1) |
nprint: |
print the estimated time remaining for every nprint'th draw (def: 100, set to 0 for no print) |
s: |
scaling parameter for RW Metropolis (def: 2.93/sqrt(nvar)) |
w: |
fractional likelihood weighting parameter (def: 0.1) |
If β_ik has a sign restriction: β_ik = SignRes[k] * exp(β*_ik)
To use sign restrictions on the coefficients, SignRes must be an nvar x 1 vector containing values of either 0, -1, or 1. The value 0 means there is no sign restriction, -1 ensures that the coefficient is negative, and 1 ensures that the coefficient is positive. For example, if SignRes = c(0,1,-1), the first coefficient is unconstrained, the second will be positive, and the third will be negative.
The sign restriction is implemented such that if the the k'th β has a non-zero sign restriction (i.e., it is constrained), we have β_k = SignRes[k] * exp(β*_k).
The sign restrictions (if used) will be reflected in the betadraw output. However, the unconstrained mixture components are available in nmix. Important: Note that draws from nmix are distributed according to the mixture of normals but not the coefficients in betadraw.
Care should be taken when selecting priors on any sign restricted coefficients. See related vignette for additional information.
nmix Detailsnmix is a list with 3 components. Several functions in the bayesm package that involve a Dirichlet Process or mixture-of-normals return nmix. Across these functions, a common structure is used for nmix in order to utilize generic summary and plotting functions.
probdraw: |
ncomp x R/keep matrix that reports the probability that each draw came from a particular component |
zdraw: |
R/keep x nobs matrix that indicates which component each draw is assigned to (here, null) |
compdraw: |
A list of R/keep lists of ncomp lists. Each of the inner-most lists has 2 elemens: a vector of draws for mu and a matrix of draws for the Cholesky root of Sigma.
|
A list containing:
Deltadraw |
R/keep x nz*nvar matrix of draws of Delta, first row is initial value |
betadraw |
nlgt x nvar x R/keep array of beta draws |
nmix |
a list containing: |
loglike |
R/keep x 1 vector of log-likelihood for each kept draw |
SignRes |
nvar x 1 vector of sign restrictions |
Note: as of version 2.0-2 of bayesm, the fractional weight parameter has been changed to a weight between 0 and 1.
w is the fractional weight on the normalized pooled likelihood. This differs from what is in Rossi et al chapter 5, i.e.
like_i^{(1-w)} x like_pooled^{((n_i/N)*w)}
Large R values may be required (>20,000).
Peter Rossi, Anderson School, UCLA, perossichi@gmail.com.
For further discussion, see Chapter 5, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.
http://www.perossi.org/home/bsm-1
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=10000} else {R=10}
set.seed(66)
p = 3 # num of choice alterns
ncoef = 3
nlgt = 300 # num of cross sectional units
nz = 2
Z = matrix(runif(nz*nlgt),ncol=nz)
Z = t(t(Z) - apply(Z,2,mean)) # demean Z
ncomp = 3 # num of mixture components
Delta = matrix(c(1,0,1,0,1,2),ncol=2)
comps=NULL
comps[[1]] = list(mu=c(0,-1,-2), rooti=diag(rep(1,3)))
comps[[2]] = list(mu=c(0,-1,-2)*2, rooti=diag(rep(1,3)))
comps[[3]] = list(mu=c(0,-1,-2)*4, rooti=diag(rep(1,3)))
pvec = c(0.4, 0.2, 0.4)
## simulate from MNL model conditional on X matrix
simmnlwX= function(n,X,beta) {
k = length(beta)
Xbeta = X%*%beta
j = nrow(Xbeta) / n
Xbeta = matrix(Xbeta, byrow=TRUE, ncol=j)
Prob = exp(Xbeta)
iota = c(rep(1,j))
denom = Prob%*%iota
Prob = Prob/as.vector(denom)
y = vector("double",n)
ind = 1:j
for (i in 1:n) {
yvec = rmultinom(1, 1, Prob[i,])
y[i] = ind%*%yvec
}
return(list(y=y, X=X, beta=beta, prob=Prob))
}
## simulate data
simlgtdata = NULL
ni = rep(50, 300)
for (i in 1:nlgt) {
betai = Delta%*%Z[i,] + as.vector(rmixture(1,pvec,comps)$x)
Xa = matrix(runif(ni[i]*p,min=-1.5,max=0), ncol=p)
X = createX(p, na=1, nd=NULL, Xa=Xa, Xd=NULL, base=1)
outa = simmnlwX(ni[i], X, betai)
simlgtdata[[i]] = list(y=outa$y, X=X, beta=betai)
}
## plot betas
if(0){
bmat = matrix(0, nlgt, ncoef)
for(i in 1:nlgt) {bmat[i,] = simlgtdata[[i]]$beta}
par(mfrow = c(ncoef,1))
for(i in 1:ncoef) { hist(bmat[,i], breaks=30, col="magenta") }
}
## set parms for priors and Z
Prior1 = list(ncomp=5)
keep = 5
Mcmc1 = list(R=R, keep=keep)
Data1 = list(p=p, lgtdata=simlgtdata, Z=Z)
## fit model without sign constraints
out1 = rhierMnlRwMixture(Data=Data1, Prior=Prior1, Mcmc=Mcmc1)
cat("Summary of Delta draws", fill=TRUE)
summary(out1$Deltadraw, tvalues=as.vector(Delta))
cat("Summary of Normal Mixture Distribution", fill=TRUE)
summary(out1$nmix)
## plotting examples
if(0) {
plot(out1$betadraw)
plot(out1$nmix)
}
## fit model with constraint that beta_i2 < 0 forall i
Prior2 = list(ncomp=5, SignRes=c(0,-1,0))
out2 = rhierMnlRwMixture(Data=Data1, Prior=Prior2, Mcmc=Mcmc1)
cat("Summary of Delta draws", fill=TRUE)
summary(out2$Deltadraw, tvalues=as.vector(Delta))
cat("Summary of Normal Mixture Distribution", fill=TRUE)
summary(out2$nmix)
## plotting examples
if(0) {
plot(out2$betadraw)
plot(out2$nmix)
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