Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.

Recursive formula for posterior joint distribution.


roposition

With notation of the previous section the posterior distribution takes the form of the prior distribution: MATH MATH MATH with the parameters MATH given by the expressions MATH MATH MATH MATH where the $\bar{x}$ and $s^{2}$ are sample's mean and standard deviation MATH , MATH

Proof.

We replace the dependence on $X$ with the dependence on the statistics $\bar{x}$ and $s^{2}$ : MATH We used the results of the section ( Sufficient statistics for normal sample section ). Therefore, in line with the calculation ( Normal distribution with unknown parameters 1 )-( Normal distribution with unknown parameters 2 ), MATH We extract the MATH term from the above expression: MATH MATH MATH MATH MATH MATH MATH MATH MATH Hence, MATH MATH or MATH and the claim follows.





Notation. Index. Contents.


















Copyright 2007