When X'X is invertible

X'X is always positive semidefinite, because for any nonzero a, a'X'Xa = (Xa)'(Xa) = ||Xa||² >= 0. Moreover, Xa = 0 (and hence ||Xa||² = 0) if and only if the columns of X are linearly dependent, so if X has full column rank then X'X is positive definite.

Every positive definite matrix is invertible, because if Ax=0 for x =/= 0 then x'Ax = dot(x, 0) = 0 which means A is not positive definite.

Therefore, if X has full column rank then X'X is invertible

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: