Information theory


Information relates to uncertainty. The Shannon information content of an outcome \(x\) is \(h(x)=-log_{2}P(x)\). The rare event has larger information than a common event. The unit of information is a bit (binary digit). Coding is a mapping from an outcome of an ensemble to binary digits \(\{0,1\}^+\). A symbol code is a code for a single ensemble. A block code is a code for a sequence ensemble. A set of sequences of the ensemble has a typical subset.


This is a note for Elements of information theory of Thomas M. Cover. The entropy (\(H\)) is a measure of uncertainty of a variable which is the answer to what is the ultimate data compression. Is the conditional probability \(p(x|y)\) considered as a probability of the “conditional variable” \((X|Y=y)\)? Yes, it is the subset of \(X\) given \(Y=y\). If you sum all of the subset probabilities, it becomes the cardinality of \(X\).