This function updates the latent utility vector, where (independent across deciders and choice occasions) the utility for each alternative is updated conditional on the other utilities.

## Arguments

- U
The current utility vector of length

`J-1`

.- y
An integer from

`1`

to`J`

, the index of the chosen alternative.- sys
A vector of length

`J-1`

, the systematic utility part.- Sigmainv
The inverted error term covariance matrix of dimension

`J-1`

x`J-1`

.

## Details

The key ingredient to Gibbs sampling for probit models is considering the latent utilities as parameters themselves which can be updated (data augmentation). Independently for all deciders \(n=1,\dots,N\) and choice occasions \(t=1,...,T_n\), the utility vectors \((U_{nt})_{n,t}\) in the linear utility equation \(U_{nt} = X_{nt} \beta + \epsilon_{nt}\) follow a \(J-1\)-dimensional truncated normal distribution, where \(J\) is the number of alternatives, \(X_{nt} \beta\) the systematic (i.e. non-random) part of the utility and \(\epsilon_{nt} \sim N(0,\Sigma)\) the error term. The truncation points are determined by the choices \(y_{nt}\). To draw from a truncated multivariate normal distribution, this function implemented the approach of Geweke (1998) to conditionally draw each component separately from a univariate truncated normal distribution. See Oelschläger (2020) for the concrete formulas.

## References

See Geweke (1998) *Efficient Simulation from the Multivariate Normal and Student-t Distributions Subject
to Linear Constraints and the Evaluation of Constraint Probabilities* for Gibbs sampling
from a truncated multivariate normal distribution. See Oelschläger and Bauer (2020) *Bayes Estimation
of Latent Class Mixed Multinomial Probit Models* for its application to probit utilities.