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.
 a. Structure of the model with unknown parameters.
 b. Recursive formula for posterior joint distribution.
 c. Marginal distribution of mean.
 d. Marginal distribution of precision.
 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.

## Structure of the model with unknown parameters.

e have seen in the previous section that the precision is more natural then for Bayesian computations involving normal distribution. We are given a sample from , where the parameters and are unknown. Assume the following prior for the mean and precision :

 (Normal distribution with unknown parameters 1)
where the gamma distribution has the following form:
 (Gamma distribution)
and the constants are known. As usual in Bayesian computation, we disregard multiplicative constants. The expression for the prior distribution takes the form We multiply it with the likelihood and obtain the joint posterior distribution of :
 (Normal distribution with unknown parameters 2)

 Notation. Index. Contents.