2. Functional Causal Graphical Models

Pywhy-graphs provides a layer to convert imbue causal graphs with a data-generating model. Currently, we only support linear models, but we plan to support non-linear and we also do not support latent confounders yet.

To add a latent confounder, one can add a confounder explicitly, generate the data and then drop the confounder varialble in the final dataset. In the roadmap of this submodule, the plan is to represent any bidirected edge as a uniformly randomly distributed variable that has an additive noise effect on both variables simulatanously.

2.1. Linear

In order to represent linear functions, we imbue nodes with a set of node attributes:

• parent_functions: a mapping of functions that map each node to a nested dictionary

  of parents and their corresponding weight and function that map parent values to values that are input to the node value with the weight.

• gaussian_noise_function: a dictionary with keys mean and std that

  encodes the data-generating function for the Gaussian noise.

  For example, if the node is \(X\) and its parents are \(Y\) and \(Z\), then parent_functions and gaussian_noise_function for node \(X\) is:

  {
      'X': {
          'parent_functions': {
              'Y': {
                  'weight': <weight of Y added to X>,
                  'func': <function that takes input Y>,
              },
              'Z': {
                  'weight': <weight of Z added to X>,
                  'func': <function that takes input Z>,
              },
          },
          'gaussian_noise_function': {
              'mean': <mean of gaussian noise added to X>,
              'std': <std of gaussian noise added to X>,
          }
      }
  }
  

2.2. Linear functional graphs

make_graph_linear_gaussian(G[, ...])	Convert an existing DAG to a linear Gaussian graphical model.
apply_linear_soft_intervention(G, targets[, ...])	Applies a soft intervention to a linear Gaussian graph.

2.3. Multidomain

Currently, this submodule only supports linear functions.

Multiple-domain causal graphs are represented by selection diagrams [1], or augmented selection diagrams (TODO: CITATION FOR LEARNING SEL DIAGRAMS).

In order to represent multidomain functions, we imbue nodes with a set of node attributes in addition to the ones for linear functions. The nodes that are imbued with extra attributes are the direct children of an S-node.

• invariant_domains: a list of domain IDs that are invariant for this node.

• domain_gaussian_noise_function: a dictionary with keys mean and std that

  encodes the data-generating function for the Gaussian noise for each non-invariant domain.

  {
      'X': {
          'domain_gaussian_noise_function': {
              <domain_id>: {
                  'mean': <mean of gaussian noise added to X>,
                  'std': <std of gaussian noise added to X>,
              },
          'invariant_domains': [<domain_id>, ...],
          }
      }
  }
  

2.4. Linear functional selection diagrams

make_graph_multidomain(G[, n_domains, ...])

Convert an existing linear Gaussian DAG to a multi-domain selection diagram model.