TABLE 3.

Extensions of our proposed models from homogeneous classification process studies done by Spiers et al. (2022) and Wright et al. (2020).

Author	Model framework	Link to our model
Wright et al. (2020) and the models their proposed framework generalises such as Chambert et al. (2015), Chambert, Grant, et al. (2018) and Chambert, Waddle, et al. (2018)	Ecological process: absolute counts $z_{\mathit{js}}\sim \text{Bernoulli}\left({\psi }_{\mathit{js}}\right)$ $\left[V_{\mathit{jks}},|,z_{\mathit{js}}=1\right]\sim \text{Poisson}\left({\lambda }_{\mathit{jks}}\right)$	Ecological process: relative abundance Assume no occupancy sub‐model and for each individual, then $\left[V_{\mathit{jks}}\right]\sim \text{Categorical}\left({\lambda }_{\mathit{jks}}/{\sum }_j{\lambda }_{\mathit{jks}}\right)$
	Observation process: $\left[Y_{\mathit{jks}},|,V_{\mathit{jks}}=v_{\mathit{jks}},,z_{\mathit{js}}=1\right]\sim \text{Multinomial}\left(v_{\mathit{jks}},|,{\mathrm{\Omega }}_{\mathit{jk}}\right)$, where ${\mathrm{\Omega }}_{\mathit{jk}}\sim \text{Dirichlet}\left(\alpha \right)$	Observation process: $\left[Y_{\mathit{ijs}},|,V_{\mathit{ijs}}=v_{\mathit{jks}}\right]\sim \text{Categorical}\left({\mathrm{\Omega }}_{\mathit{jks}}\right)$, where ${\mathrm{\Omega }}_{\mathit{jks}}$ can be chosen as any of the homogeneous models described in Table 2
Spiers et al. (2022)	Ecological process: occupancy dynamics and encounter rates $z_{\mathit{jst}}\sim \text{Bernoulli}\left({\psi }_{\mathit{jst}}\right)$ $\left[V_{\mathit{\text{jist}}}\right]\sim \text{Categorical}\left(\frac{{\lambda }_{\mathit{\text{jist}}}{z}_{\mathit{jst}}}{{\sum }_j{\lambda }_{\mathit{\text{jist}}}{z}_{\mathit{jst}}}\right);$	Ecological process: relative abundance Choose $t=1$ and ignore the occupancy sub‐model. $\left[V_{\mathit{jis}}\right]\sim \text{Categorical}\left(\frac{{\lambda }_{\mathit{jis}}}{{\sum }_s{\lambda }_{\mathit{jis}}}\right);$
	Classification process: $\left[Y_{\mathit{jis}},|,V_{\mathit{jis}}\right]\sim \text{Categorical}\left({\mathrm{\Omega }}_{\mathit{jk}}\right)$, where ${\mathrm{\Omega }}_{\mathit{jk}}\sim \text{Dirichlet}\left(\alpha \right)$	Classification process: $\left[Y_{\mathit{jis}},|,V_{\mathit{jis}}=1\right]\sim \text{Categorical}\left({\mathrm{\Omega }}_{\mathit{jks}}\right)$, where ${\mathrm{\Omega }}_{\mathit{jks}}$ can be chosen as any of the homogeneous models described in Table 2

Note: The table specifies the ecological process model for Wright et al. (absolute abundance model), Spiers et al. (occupancy dynamics and encounter rate model) and ours (relative abundance model); and also the observation model for Wright et al. and Spiers et al. (homogeneous classification process with classification probabilities simulated from Dirichlet distribution) and ours from heterogeneous models described in Table 2. Since our framework extends the work done by (Wright et al., 2020), it is safe to say that the classification component of our proposed framework are also generalised forms of Chambert et al. (2015), Chambert, Grant, et al. (2018) and Chambert, Waddle, et al. (2018). The index j refers to the true state identity, k refers to the reported state identity, s refers to the location in Spiers et al. (2022) and Wright et al. (2020) but refers to individuals in this study, i refers to the visit, and t refers to the year. In addition, the random variable Y refers to the reported observations, V to the verified observations and z to the occupancy state of the individuals.