Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
A. Generation of random samples.
a. Uniform [0,1] random variable.
b. Inverting cumulative distribution.
c. Accept/reject procedure.
d. Normal distribution. Box-Muller procedure.
e. Gibbs sampler.
B. Acceleration of convergence.
C. Longstaff-Schwartz technique.
D. Calculation of sensitivities.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Gibbs sampler.

ibbs sampler produces a sample that converges in distribution to a target multidimensional random variable specified by a distribution known up to a normalization constant. Bayesian analysis ( Bayesian section ) is the main field of application.

Suppose we are given a distribution MATH . The variable $\theta$ is multidimensional: MATH .


The first element $\theta_{t=1}$ of a sample is drawn from some distribution MATH . We proceed to generate the subsequent draws $\theta_{t}$ , $t=2,...$ as follows:

Set MATH .

For every $k=1,...,N$ do

1. form a vector MATH ;

2. generate a draw MATH from the single dimensional conditional distribution MATH ;

3. store the result in the new MATH .

Set MATH .

See [Gelman] for full treatment of the subject.

Notation. Index. Contents.

Copyright 2007