"In a just world, every article about GPT-4 would nod toward Barthes and Foucault." Downing spoonfuls of mayonnaise as I read Ted's essay, because it is too spicy (link in quote below)
Richard McElreath 🐈⬛
@rmcelreath.bsky.social
19K followers 1.4K following 1.8K posts
Anthropologist - Bayesian modeling - science reform - cat and cooking content too - Director @ MPI for evolutionary anthropology www.eva.mpg.de/ecology/staff/richar…
![Week Date Lecture Reading Practical Study Questions
1 Oct 14 Natural selection Chapter 1 Darwin's theory 1:1,3,5
2 21 Modern synthesis and pop-gen Chapter 3 Basic pop-gen 3:1,2,3
3 28 Speciation Chapter 4 Phylogenetics 4:2,3,8
4 Nov 4 Primate diversity Chapter 5 Primate anatomy 5:1,5,8
5 11 Primate reproduction Chapter 6 Mating and parenting 6:2,7,9
6 25 Primate socioecology Chapter 7 Cooperation 7:1,3,9
7 Dec 2 Primate behavior and cognition Chapter 8 Brains and life history 8:4,5,8
8 9 Hominin fossil record Chapters 9,10 Hominin anatomy 9:7; 10:6,10
9 16 Tools and Homo Chapter 11 Hominin tech 1 11:1,4,5
10 Jan 6 Out of Africa Chapter 12 Neanderthals etc 12:1,7,12
11 13 Origins of modern behavior Chapter 13 Hominin tech 2 13:1,8,10
12 20 Human diversity and adaptation Chapter 14 Genetic diversity 14:5,8,10
13 27 Cultural evolution and technology Chapter 15 Human behavior 15:1,7,9
14 Feb 3 Cooperation and social organization Chapter 16 [none]](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ta2whs6wjtpljvpqgs5leu36/bafkreiaiwk37h2vik76hryefsgcl2xseikqjmxjbyrzgj5jjbx3g46wdgq)
![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)
![\subsection{Occupancy as a function of performance}
Our goal is to model occupancy probability, using the thermal performance function $f$. The function $f$ is not a proper probability function, because it returns negative values above the thermal maximum $T \geq m + n$. We can use a form resembling the Gibbs-Boltzmann distribution to transform $f$ to a probability scale:
\begin{align}
g(T) = \theta \exp \big( \kappa ( f\,(T) - 1 ) \big)
\end{align}
where $\theta$ is the maximum probability of occupancy at the thermal optimum $T=m$, and $\kappa > 0$ determines how peaked the resulting function is around $m$.
% DSolve[{p'[f] == k p[f], p[1] == a}, p[f], f]
One way to motivate this function is to assume that the rate of change in occupancy probability $g$ with thermal performance $f$ is a constant proportion of $g$:
\begin{align}
\frac{dg}{df} = \kappa g
\end{align}
In other words, wherever we are in performance $f$, increasing $f$ produces the same proportional increase in occupancy probability. Solving for $g(f)$:
\begin{align}
g(f\,) = C \exp( \kappa f \,)
\end{align}
To fix the constant $C$, let the occupancy probability at the thermal optimum be $g(f=1)=\theta$. Then $g(1) = \theta = C \exp( \kappa )$. Solving for $C$ gives $C=\theta \exp(-\kappa)$. Substituting, we arrive at our function $g$:
\begin{align}
g(f\,) = \theta \exp(-\kappa) \exp( \kappa f \, ) = \theta \exp( \kappa ( f - 1 ) )
\end{align}
For $\theta=0.7$, $m=20$, and $n=5$, here is $g(T)$ for two values of $\kappa$:
\begin{tikzpicture}
\begin{axis}[axis lines = left, xlabel = $T$, ylabel = $g$,
width=7cm, height=4cm, ymin=0, ymax=1,
tick label style={font=\tiny},
label style={font=\small} ]
\addplot[domain=-3:30, red, thick, samples=100]
{ 0.7*exp( 2*((exp( (x-20)/5 )*( 1 - (x-20)/5 )) - 1 ) ) };
\addplot[domain=-3:30, blue, thick, samples=100]
{ 0.7*exp( 8*((exp( (x-20)/5 )*( 1 - (x-20)/5 )) - 1 ) ) };
\node [text=red] at (80,50) {$\kappa=2$};
\node [text=b…](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ta2whs6wjtpljvpqgs5leu36/bafkreihdkuuohwfyb3djyhrdueswo5qoewtu4hjxjk657edxnp7atwcczq)
