  The topic of deep learning has seen a surge of interest in recent years both
within and outside of the field of Statistics. Deep models leverage both
nonlinearity and interaction effects to provide superior predictions in many
cases when compared to linear or generalized linear models. However, one of the
main challenges with deep modeling approaches is quantification of uncertainty.
The use of random weight models, such as the popularized "Extreme Learning
Machine," offer a potential solution in this regard. In addition to uncertainty
quantification, these models are extremely computationally efficient as they do
not require optimization through stochastic gradient descent, which is what is
typically done for deep learning. We show how the use of random weights in a
deep model can fit into a likelihood based framework to allow for uncertainty
quantification of the model parameters and any desired estimates. Furthermore,
we show how this approach can be used to account for informative sampling of
survey data through the use of a pseudo-likelihood. We illustrate the
effectiveness of this methodology through simulation and with a real survey
data application involving American National Election Studies data.

Methodology