Here is a link to Problem Set #1, due on September 27.

I remarked in class on Nate Silver’s election blog in the New York Times. He’s using Bayesian methods and simulation to make predictions about the upcoming elections.

In class we first looked at the difference between Bayesian credible intervals and frequentist confidence intervals. The key to understanding this is that in the Bayesian case, the data are considered fixed and known, and the unknown parameter is a random variable that has a probability distribution that is described by the (fixed and known) credible interval computed from the data. On the other hand, in the frequentist case the unknown parameter is considered fixed, and the probability model generates hypothetical data from which we compute hypothetical confidence intervals. A certain fraction of these hypothetical intervals (for example, 95%) will contain the unknown parameter; the remainder (5% in this example) will not. The data we actually observed will generate one of those confidence intervals, the one we observe. But we cannot conclude that the parameter has a 95% probability of lying within *this particular confidence interval*. The confidence interval definition only refers to the statistical properties of confidence intervals in general, not to the particular one we calculate from the one data set we happen to have observed.

I did not mention this in class, but later in the course I will give an example of a valid frequentist 90% confidence interval that has the property that we know for certain that the parameter in question does not lie in that interval. This cannot happen with Bayesian credible intervals. Sorry, it will take some time to get to this example, as we have some other machinery to develop first.

We talked about estimating functions of a parameter, with examples by simulation. We talked about robustness under prior variation and locally uniform priors, and looked at our voting example under a triangular prior. We gave an example in the voting example about how resampling can quickly let us examine the effect of changing the prior. We finished by looking at continuous versus discrete parameters.

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September 18, 2012 at 4:10 pm |

Anyone want to form a homework group?

September 19, 2012 at 3:00 pm |

When were you thinking of starting the homework?

September 19, 2012 at 3:45 pm

I’ve started on it a bit already – please e-mail me and we can arrange a time to meet if you’d like to work together (mwagy@uvm.edu)