# Importance sampling MNL This function samples from the posterior distribution using importance sampling, assuming a multivariate (truncated) normal prior distribution and a MNL likelihood. ```r 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`. ## Returns - **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. ## 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. ## Examples ```r ## 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)) ``` ## References \insertRef juidefix

ImpsampMNL function