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.
 a. Joint posterior distribution of mean and hyperparameters.
 b. Posterior distribution of mean conditionally on hyperparameters.
 c. Marginal posterior distribution of hyperparameters.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Joint posterior distribution of mean and hyperparameters.

n agreement with the general Bayesian idea ( Bayesian technique ) we write The observable data depends directly only on , hence Also, in agreement with the structure of the model, Hence, We replace the dependence on the sample with dependence on the statistics according to the formula and drop the multiplicative constants. We obtain

 (Hierarchical1)
where we are using the notation .

 Notation. Index. Contents.