The simulations were run on R : 
1. 100,000 datasets, 25 variables, 10,000 data points
2. Randomly generated DAGs with 25 nodes, each with probability 0.1
3. Randomly selected target variable.

University of Michigan - Ann Arbor

More application based, but I enjoyed reading about the approach : 
[arxiv.org/abs/2002.09414](https://arxiv.org/abs/2002.09414) 

Directed Acyclic Graphs and causal thinking in clinical risk prediction modeling

Use Directed Acyclic Graphs to model causal assumptions and answer queries.
Trying to determine the usefulness of a DAG in clinical risk predictions. 

Key Idea

Reasons to keep a minimal number of predictors in a risk prediction model : 
1. Increase performance
2. Reduce time/cost/effort of maintaining and collecting a lot of data
3. To optimize adaptability of model to different settings
4. Increase interpretability.

Reasons to limit number of predictors

Methodology

8 prediction tools were developed to estimate the probability of the target variable being 1. 
These were : 
1. Logistic regression using Markov-Blanket set of outcome as predictors
2. Logistic regression with all 24 variables as predictors
3. Logistic regression with only variables that had a path leading to target node
4. Logistic regression including only output of parent nodes to the target node.
5. Logistic lasso regression model with all 24 variables as predictors
6. Logistic Ridge Regression with all 24 variables
7. Logistic Elastic Net Regression (alpha 0.5) 24 variables.
8. Random Forest algorithm with 24 variables.

Testing Method

Is defined as the smallest subset of variables such that, no variable NOT contained in this set contributes any additional information about the target variable.
Let the Markov Blanket be denoted by $MB(Y)$ where $Y$ is the target variable. Let $\bar{MB(Y)}$  be the complement of $MB(Y)$ in the universe of variables in the model. Then, 
$$P(Y|MB(Y), \bar{MB(Y)}) = P(Y|MB(Y))$$


Markov Blanket

In clinical risk prediction modeling, utilizing the Markov blanket of an outcome variable can optimize predictions. Evaluations of different logistic regression tools yield the following results:

1. Lasso regression selects exactly the Markov blanket at least 10% of the time.
2. Markov blanket logistic regression demonstrates better performance than other models while consistently requiring fewer predictor variables.
3. The Markov blanket model achieves the lowest Integrated Calibration Index, indicating superior model calibration.

Performance of Markov Blanket Models in Clinical Risk Prediction

In clinical risk management, understanding and utilizing underlying causal structures, such as those modeled by Directed Acyclic Graphs (DAGs), can significantly improve prediction model performance and optimize the selection of predictors.

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