# Difference in likelihoods between BM and the stationary distribution of an OU model

I'm calculating the fits of Brownian motion and Ornstein-Uhlenbeck models to data, given a phylogeny.

However, when deriving the likelihood functions, it appears that the Ornstein-Uhlenbeck model is identical to the Brownian motion model except the variance-covariance matrix is scaled by sigma2/(2*alpha) rather than sigma2. (So essentially just adding an extra parameter to do the same thing). Does this seem correct or is there something simple that I am missing?

Likelihood under Brownian motion

xi,...,n ~ MVN(X,sigma2*SIGMA) – where X is a n-by-1 vector with each term being x0 (the ancestral state) – and SIGMA is a covariance matrix (n-by-n) where diag(SIGMA)i = ti, the distance from the ancestral node, and SIGMAi,j = t0 - ti,j , the distance from the ancestral node minus the time since species i and j last shared a common ancestor.

Likelihood under OU process

E[Xi] = E[Xp]exp(-alpha*t(i,p)) + THETA(1-exp(-alpha*t(i,p))) Var[Xi] = (sigma2/(2*alpha))*(1-exp(-2alpha*t(i,p)) + Var[Xp]exp(-2alpha*t(i,p)), where p is an ancestor of i and t(i,p) is the time between i and p.

Assuming X for the ancestral follows the stationary distribution of the OU model s.t. Xp ~ N(THETA,sigma2/(2*alpha)), this then simplifies to a MVN(X,(sigma2/(2*alpha))*SIGMA)

• Use LaTeX code inside dollar signs (e.g., $x$) to write math notation. – Kodiologist Aug 25 '17 at 19:03