wilmott.com

Hi guys, I am trying to understand the differences between Bayesian MCMC methods and maximum likelihood techniques. I am looking at applying this to a customized binary response model and have heard that while MCMC methods take longer to find the parameters, they are more robust then MLE techniques (despite my fond memories in gradschool of playing with a dozen starting values deep into the night...). I learn best by example as it is harder for me to apply theories (I can understand them mathematically but applying the arguments is a different story). lets say that I am trying to estimate the relationship (in the form of a parameter) between the temperature outside and rate of rainfall. my proposed model looks like this:where y is the amount of rainfall, x is the temperature and epsilon approximates some distribution \phi with a mean of \mu and a standard deviation of \sigma.A MLE technique would involve the following:1: state some starting values for alpha, beta, mu2: find out the log (sum of the errors) epsilon that are created from the equation3: state other values for alpha beta until the log of errors is minimized 4: find the inverse hessian to find a quasi-standard error for the parametersWhat I am trouble understanding is how an MCMC approaches this issue. from what I tentatively understand I would need to do the following:1) given a prior distribution of alpha, beta, mu, find a sequence of candidates proposed my an MCMC algorithim.2) weight each proposed combination of parameters proportionally to the log error that they generate in the model.3) eventually, as the number of sequences goes to infinity, I imagine that I will have a distribution of the entire parameter space whose expected value is the relationship between rainfall and temperature. the reason that I am having trouble with this topic is that most of the MCMC literature concerns how do most efficiently draw candidates for the various supported distributions rather than how to actually implement MCMC methods. If I am totally off, please correct me here thanks, honeyoak.