Richard McElreath 🐈⬛
@rmcelreath.bsky.social
18K followers 1.4K following 1.7K posts
Anthropologist - Bayesian modeling - science reform - cat and cooking content too - Director @ MPI for evolutionary anthropology www.eva.mpg.de/ecology/staff/richar…
![If we needed just one parameter for each species, we would be done there. But we want to use the covariance matrix $\mathbf K$ to draw a vector of 4 parameters $\phi_s = [m_s,\log n_s,\mathrm{logit} \theta_s, \log \kappa_s ]$ for each species $s$ in $N$ total species. This requires that we also specify a covariance matrix $\Omega$ for the four parameters in $\phi$. The goal is to have the covariance between $\phi_{s,i}$ and $\phi_{t,j}$ be:
\begin{align}
\mathrm{Cov}( \phi_{s,i} , \phi_{t,j} ) = K_{s,t} \, \Omega_{i,j}
\end{align}
which is just the product of the GP covariance between the species $s$ and $t$ and the covariance for the parameter types $i$ and $j$. The joint probability of every $\phi_s$ is then defined by a Kronecker product covariance matrix:
\begin{align}
\mathrm{vec}(\Phi) \sim \mathrm{MVNormal}\left( 0 , \mathbf K \otimes \Omega \right)
\end{align}](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ta2whs6wjtpljvpqgs5leu36/bafkreigggwvh2ll4c4ta5um4whlk7g3liw577r6zgfgr6s7ggwrxy6uoly)
