Box 2.

Estimating relative abundances

Consider a survey designed to quantify the abundances $N_l$ at locations $l=1,\cdots ,L$ from abundances reported by observers $j=1,\cdots ,J$ from a total of $V$ visits. Let $d_v$ denote the reported abundance during visit $v=1,\cdots ,V$ conducted at location $l_v$ by observer $o_v$. Here, $d_v$ is affected by both the abundance $N_{l_v}$ at location $l_v$ as well as by the detection probability $p_{o_v}$ of observer $o_v$ such that $P\left(d_v,|,N_{l_v},,,p_{o_v}\right)=\left(\genfrac{}{}{0.0pt}{}{N_{l_v}}{d_v}\right){p_{o_v}}^{d_v}{(1-p_{o_v})}^{N_{l_v}-d_v}$ is given by binomial sampling. Since $N_{l_v}$ and $p_{o_v}$ are confounded, estimating them individually is difficult (DasGupta & Rubin 2005). To illustrate this, consider a case with two locations with $N_1=100$ and $N_2=200$ surveyed $m=5$ times each by a single observer with detection probability $p=0.2$. As shown in Fig. 5A, the uncertainty associated with abundance estimated from that data under mild priors $N_1,N_2\sim \text{Exp}(0.001)$ spans about two orders of magnitude. This is because the data is well explained by pretty much any abundance if paired with a corresponding detection probability and more informative priors would be required to constrain the range of possible values. However, there is considerable evidence that $N_2$ is about twice $N_1$ (Fig. 5B), illustrating that relative abundances may be learned accurately from such surveys. To benefit from this in a realistic setting, we here generalize the inference of relative abundances to many locations. Let us assume that the abundances $N_l=N_0{e}^{{\rho }_l}$ are scaled by location-specific factors ${\rho }_l\sim N(0,{\sigma }_{\rho }^2)$ that are themselves normally distributed with mean zero and variance ${\sigma }_{\rho }^2$. Similarly, we assume that the detection probabilities $p_j=\text{logistic}({\pi }_0+{\pi }_j)$ are scaled by observer-specific effects ${\pi }_j\sim N\left(0,,,{\sigma }_{\pi }^2\right)$ that are also normally distributed with mean zero and variance ${\sigma }_{\pi }^2$. Here, the logistic transformation ensures $0\le p_j\le 1$. We further enforce the conditions $\frac{1}{L}{\sum }_i{\rho }_l=0$ and $\frac{1}{J}{\sum }_j{\pi }_j=0$ by scaling $N_0$ and $p_0$ accordingly. If observers do not visit multiple locations, the ${\pi }_j$ need to be modelled using informative covariates. We conducted simulations with $N_0=100$, ${\sigma }_{\rho }^2=0.2$, ${\pi }_0=-1$ and ${\sigma }_{\pi }^2=0.5$, corresponding to an average detection probability $p_0=\text{logistic}\left({\pi }_0\right)=0.27$. As shown in Figs. 5C and 5D, neither $N_0$ nor $p_0$ can be inferred accurately, regardless of whether $L=20$ or $L=100$ locations were surveyed by $J=20$ or $J=100$ observers visiting $m=5$ different locations each, corresponding to $V=100$ and $V=500$ visits, respectively. In contrast, the relative abundances are estimated well, and easily distinguish locations with high from those with low abundances (Figs. 5E and 5F).