add reference

Signed-off-by: Nathaniel <[email protected]>

Author: NathanielF

docs/source/notebooks/inv_prop_latent.ipynb

@@ -1405,7 +1405,7 @@
     "\n",
     "### The Problem of Feedback\n",
     "\n",
-    "The issue here is sometimes called Bayesian feedback or \"collider bias via the likelihood\", and it's a key issue when trying to build joint models for causal inference in the Bayesian paradigm. Because we have fit the outcome and the treatment models simultaneously, and this means that the outcome can influence the posterior distribution of the parameters $\\beta$ in the treatment model and it violates the idea of design-before-analysis. We have here an apparent example of a slight bias due to this effect.  The two stage modular approach seems to better recover the treatment effect reported in the literature and avoids the risk of collider bias i.e. in the modular implementation we are able to use the propensity score to adjust for accuracy and compensate for the missing variables `x2` and `x1`. \n",
+    "The issue here is sometimes called Bayesian feedback or \"collider bias via the likelihood\" {cite:p}`GriffithCollider`, and it's a key issue when trying to build joint models for causal inference in the Bayesian paradigm. Because we have fit the outcome and the treatment models simultaneously, and this means that the outcome can influence the posterior distribution of the parameters $\\beta$ in the treatment model and it violates the idea of design-before-analysis. We have here an apparent example of a slight bias due to this effect.  The two stage modular approach seems to better recover the treatment effect reported in the literature and avoids the risk of collider bias i.e. in the modular implementation we are able to use the propensity score to adjust for accuracy and compensate for the missing variables `x2` and `x1`. \n",
     "\n",
     "\n",
     "**💡 Key Take-away:** With an underspecified outcome model, we may use a well specified propensity score for adjusting the model to retrieve accurate treatment effect estimates. However, this tends to breakdown if we have estimated both propensity score and outcome in a joint bayesian model due to feedback effects. The solution is to use the propensity score in a 2 stage fashion. \n",

docs/source/references.bib

@@ -186,3 +186,11 @@ @book{Hernan2024-HERCIW
 	year = {2024},
   publisher = {CRC Press TBC}
 }
+
+@article{GriffithCollider,
+  title={Collider bias undermines our understanding of COVID-19 disease risk and severity},
+  author={Griffith, G. J. and  Morris, T. T. and Tudball, M. J. and Herbert, A. and Mancano, G. and Pike, L. and Sharp, G. C. and Sterne, J. and Palmer, T. M. and Davey Smith, G. and Tilling, K. and Zuccolo, L. and Davies, N. M. and & Hemani, G.},
+  journal={Nature communications},
+  year={2020},
+  publisher={Nature}
+}