First hour: why should we talk about estimation?<br>
<br>
Basic structure: Markov chain f(\theta) --- \theta --- X^n --- T(X^n).  Parameter estimation given (possibly dependent) data X^n.
<br>
<br>
Example: \theta is a visual scene, f(\theta) indicates the presence of a given object, X^n is some collection of cortical cells<br>
<br>
Basic questions:<br>
1) What do we mean by "encode?"  Collection of conditional distributions P(X^n ; \theta)<br>
2) How much information does the population encode?  I(X^n ; \theta), or some Bayes error.<br>
3) How much information can the population encode?  Constrained maximization problem again.<br>
4) Data analysis: how do we measure how much info is encoded?<br>
5) How is info "read out?"<br>
<br>
Special cases:<br>
1) Detection/discrimination<br>
2) continuous/perhaps high-dimensional estimation<br>
<br>
<br>
Second hour: outline of mathematical approaches to the above questions.<br>
<br>
Asymptotic approaches are usually required; we'll talk about four.<br>
<br>
1) Asymptotics of linear Gaussian estimation<br>
2) Max likelihood; more generally, M-estimation and associated empirical processes: where K-L divergence, Fisher information come from<br>
3) Asymptotics of a posteriori distributions, I(\theta ; X^n)<br>
4) Hypothesis testing and Chernoff information<br>