### DAGs (Directed Acyclic Graphs)

#### 1. Description

DAG (Directed Acyclic Graph) is a graphical structure where nodes are connected through directed edges. It relies on conditional independences which allow decomposition of information on strength of associations in BR models into distinct probability distributions.[1][2][3][4]

#### 2. Evaluation

##### 2.1 Principle

- The principles of DAGs are mathematically exhaustive and require technical knowledge of probability theory.

- Whenever a DAG is consistent with the conditional independences among variables, the joint probability distribution can be computed as the product of Conditional Probabilities Tables (CPTs).

- The joint probability distribution can address the uncertainty of any subset of variables in the model, and conditionally on an observed subset of other variables (belief updating).

##### 2.2 Features

- The graphical structure of Bayesian Networks (BNs) may provide large domains with a compact probabilistic representation.

- The CPTs are often difficult to elicit from the parameters reported in the literature or from the knowledge of experts.

- The DAG structure can be exploited to facilitate the estimate of the quantitative BN component from data.

- There are a few extensions to allow continuous random variables to be treated without discretisation and the DAG representing one unit of observation being meant as replicate for all available observations.

- MCMC methods can be applied to learn the distribution of the quantitative parameters in the model.

- BNs may also support personalised medical decisions on several drug options, taking into account of a possibly large set of patient characteristics through simulations.

- It can summarise the balance between risk and benefits in one single measure.

- Influence diagrams extend BNs to incorporate decision nodes and utility nodes in order to avoid the need to iterate a simulation for each decision option.

- Variables representing risk and benefits have to be linked to utility nodes after having been preventively defined on the same quantitative scale.[4]

##### 2.3 Visualisation

- The visualisation associated with DAGs is the network diagram.

- Some expertise may be needed in the interpretation of arrows, as their direction represents how the joint probability distribution is marginalised over the variables, not necessarily information about causal relations.

##### 2.4 Assessability and accessibility

- Conditional independence among variables may be assessed by means of statistical tests applied on data, e.g. using log-linear regression analysis.

- Conditional independencies could be also anticipated from knowledge on how the variables are causally related, e.g. from experimental study designs or from information about the temporal order of event.

- Software packages available to support DAGs implementation are:
##### Genie (http://genie.sis.pitt.edu/)

##### WinBUGS (http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml)

##### R-JAGs (http://www-ice.iarc.fr/~martyn/software/jags)

##### Analytica (http://www.lumina.com/why-analytica)

##### Hugin (http://developer.hugin.com)

##### Netica (https://www.norsys.com)

#### 3. References

[1] Pearl J. Probabilistic reasoning in intelligent systems: networks of plausible inference. San Francisco, CA: Morgan Kaufmann Publishers, Inc.; 1988.[2] Darwiche A. Bayesian Networks. http://reasoning.cs.ucla.edu/. 2009.

[3] Jensen FV. An introduction to Bayesian Networks. 1 ed. Secaucus, NJ: Springer-Verlag New York, Inc.; 1996.