Update dcp module documentation

lampe/inference/dcp.py

@@ -1,5 +1,26 @@
 r"""Differentiable Coverage Probability components.
 
+This module implements Differentiable Coverage Probability (DCP) regularizer for
+training Neural Ratio Estimator (NRE) and Neural Posterior Estimator (NPE), please refer to
+the documentation of the respective modules to learn about the methods.
+
+DCP regularizer relies on the Expected Coverage Probability (ECP) of Highest Posterior Density Region (HPDR)
+
+.. math:: \text{ECP}(1 - \alpha) = \mathbb{E}_{p(\theta, x)} \left[ \mathds{1}[\theta \in \Theta_{\hat{p}(\theta|x)}(1-\alpha)] \right]
+
+where :math:`\Theta_{\hat{p}(\theta|x)}(1-\alpha)` is the HPDR of the approximate posterior model :math:`\hat{p}(\theta|x)`
+at level :math:`1-\alpha`, and :math:`\mathds{1}[\cdot]` is an indicator function.
+
+It can be shown that
+
+.. math:: \text{ECP}(1 - \alpha) = \mathbb{E}_{p(\theta, x)} \left[\mathds{1}[\alpha(\hat{p},\theta,x) \geqslant \alpha]\right]
+
+where :math:`\alpha(\hat{p},\theta_j,x_j) := \int \hat{p}(\theta|x_j) \mathds{1}[\hat{p}(\theta|x_j) < \hat{p}(\theta_j|x_j)]\textrm{d}\theta`.
+
+A conservative (calibrated) posterior model is one that has `\text{ECP}(1 - \alpha)` above (at) level :math:`1-\alpha` for all :math:`\alpha \in (0,1)`.
+DCP regularizer turns this condition into a differentiable optimization objective minimized in
+parallel to the main optimization objective of NRE or NPE.
+
 References:
     | Calibrating Neural Simulation-Based Inference with Differentiable Coverage Probability (Falkiewicz et al., 2023)
     | https://arxiv.org/abs/2310.13402