An excellent discussion of the history and issues affiliated with Choice under Risk and Uncertainty is available on the Newschool website. For a detailed description of Savage's axiomatization from his seminal work The Foundations of Statistics, see this article due to Edi Karni. My own research includes two papers aimed at axiomatizing choice when preferences are state dependent.
J.W. Witter has a nice practical guide for the use of Monte Carlo simulation methods in Excel, which is useful when the uncertainty can be reduced to standard probability distributions. When data is present, modeling under uncertainty requires combining prior information with data using Bayes rule. A nice discussion of Bayes' Rule is available at Kevin Murphy's website at The University of British Columbia. For simple models with conjugate priors, the resulting posterior distributions are straightforward to sample.
More complex models may require the use of Markov Chain Monte Carlo techniques. My own research includes papers developing methods for iid sampling for Bayesian Generalized linear models as well as methods for understanding the convergence rates of some Markov Chain Monte Carlo simulation algorithms. Some of the methods have been incorporated into the R-package Baldur, which I have developed. Baldur includes basic documentation for the generalized linear models included. Some of the functions in Baldur are also available as an Excel add in.
Winbugs is an excellent software for learning to build more general Bayesian models. The R-package R2Winbugs is also useful as it allows one to have a useful interface for Winbugs for transfering data and simulation output to and from R for ease of manipulation.
George Dantzig - Inventor of Linear Progamming
My Research
Likelihood Subgradient Paper