I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 2 Classical statistics.
 3 Bayesian statistics.
 A. Basic idea of Bayesian analysis.
 B. Estimating the mean of normal distribution with known variance.
 C. Estimating unknown parameters of normal distribution.
 D. Hierarchical analysis of normal model with known variance.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Basic idea of Bayesian analysis.

here is a sample and there is an unknown parameter . However, unlike the classical approach, the is regarded as a random variable. The nature selects from some "prior" population , then it selects the sample according to . The goal is to recover the distribution . Such recovery is accomplished by the repeated application of the ( Bayes formula ):

 (Bayesian technique)
The is the likelihood function, is called the "prior" distribution and the is a normalization constant: The may hold some prior knowledge about the There are at least three general strategies to choose the prior distribution: non-informative (diffuse) prior, invariant prior (Jeffrey's principle) and hierarchical modelling. For analytical convenience one should try and choose prior so that the posterior distribution , likelihood and the prior would belong to the same class of functions (normal, binomial, exponential and so fourth). Such prior distribution is called "conjugate".

The principal technical tool is to drop normalization constants from all calculations and track only the essential part. The normalization constant is then recovered from the final solution according to . In particular, we write ( Bayesian technique ) as

 Notation. Index. Contents.