A SAS macro to run restricted cubic spline Cox model

https://www.hsph.harvard.edu/donna-spiegelman/software/lgtphcurv9/

http://www.sciencedirect.com/science/article/pii/S0169260797000436

http://epidemiologymatters.org/epidemiology-we-like/methods/spline-regression/

Bayesian regression compare to other regressions

https://stats.stackexchange.com/questions/252577/bayes-regression-how-is-it-done-in-comparison-to-standard-regression

Review some Monte Carlo theorems

1.
Suppose the random variable U has a uniform (0,1) distribution.
Let F be a continuous distribution function. Then the random variable X = F^(-1)(U)
has distribution function F.

Note: F(x)=(x-a)/(b-a), for uniform random variable at [a,b] interval. Here F(u)=u.

2.Monte Carlo Integration

generate x1, x2,...,xn from uniform(a,b), then compute Yi = (b - a)g(Xi). Then mean Y is a consistent estimate of the integral

Note: 1. definite integral is a number.

3. Accept-Reject Generation Algorithm

Difference between indefinite and definite integrals.

 Indefinite integrals are functions while definite integrals are numbers. This is quite useful when we calculate Bayesian estimator

EXPECTATION AND THE INDICATOR FUNCTION

Any distribution that has an inverse can be simulated from Uni(0,1)

Proof:

Highest density region

https://stats.stackexchange.com/a/148452/61705