Topology starts with divergence i.e. $d(model_1, model_2)$. ## What is model divergence? It is categorized as: | - | y | $\theta$ | | ------- | ----------------------------- | ----------------------------------------------- | | preFit | 1. prior predictive check | 2. model distance, simulation-based calibration | | postFit | 3. posterior predictive check | | ## How to compute model divergence? For each 1, 2, 3 above, 1: $d(y, \hat{y_1}), d(y, \hat{y_2})$ - prior predictive check 2: $d(module_1, module_2)$ - structure of parameter - measurement model - e.g. module structure which is similar to network architecture in graphcial neural network. They determine basis function space. 3: $d(\tilde{y_1}, \tilde{y_2})$ - posterior predictive check - $\theta$-based distance: simulation-based distance between edges and vertices (marginalize out y) - $y$-based distance: expected log pointwise predictive density (marginalize out theta) The three category correspond to three types of criticism recommended [this](https://arxiv.org/pdf/2006.02985.pdf) workflow paper for disease transmission. ![[Pasted image 20220622054150.png|800]] ## Why is model divergence needed? This forms model topology which makes "familes of model", "model network" more concrete. Predictions $\tilde{y}$ depend heavily on the model structure (e.g. model 1~ 8 in Birthday problem) and based on this, we assume the existence of `module tree` whose divergence on this [hyperbolic topology](https://en.wikipedia.org/wiki/Hyperbolic_tree) can be the proxy for model divergence. Just as [decision tree is an universal approximator](https://dbertsim.mit.edu/pdfs/papers/2018-sobiesk-optimal-classification-and-regression-trees.pdf), module tree is expressive and forms large enough basis function space. Hence, this structure is eligible to measure model divergence.