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A Python package implementing Structural Causal Models (SCM).

The library uses the CAS library SymPy to allow the user to state arbitrary assignment functions and noise distributions as supported by SymPy and builds the DAG with networkx.

It supports the features: - Sampling - Intervening - Plotting - Printing

and by extension all methods on a DAG provided by networkx after accessing the member variable dag

Installation

Either install via pip pip install scmodels or via cloning the repository and running the installation locally git clone https://github.com/maichmueller/scmodels cd scmodels pip install .

Building an SCM

In order to declare the SCM

$X ~ \leftarrow ~ Z ~ \rightarrow ~ V ~ \rightarrow ~ W$

$\downarrow ~ ~ \swarrow$

$Y$

with the assignments

$Z \sim \text{LogLogistic} (\alpha=1,\beta=1)$

$X=3Z^2 \cdot N \quad [N \sim \text{LogNormal}(\mu=1,\sigma=1)]$

$Y = 2Z + \sqrt{X} + N \quad [N \sim \text{Normal}(\mu=2,\sigma=1)]$

$V = Z + N^P \quad [N \sim \text{Normal}(\mu=0,\sigma=1), P \sim \text{Ber}(0.5)]$

$W = V + \exp(T) - \log(M) * N \quad [N \sim \text{Normal}(\mu=0,\sigma=1), T \sim \text{StudentT}(\nu = 0.5), M \sim \text{Exp}(\lambda = 1)]$

there are 2 different ways implemented.

1. List Of Strings

Describe the assignments as strings of the form:

'VAR = FUNC(Noise, parent1, parent2, ...), Noise ~ DistributionXYZ'

Note that in this case, one must not restate the noise symbol string in the distribution (as would otherwise be necessary in constructing sympy distributions).

```python from scmodels import SCM

assignment_seq = [ "Z = M; M ~ LogLogistic(alpha=1, beta=1)", "X = N * 3 * Z ** 2; N ~ LogNormal(mean=1, std=1)", "Y = P + 2 * Z + sqrt(X); P ~ Normal(mean=2, std=1)", "V = N**P + Z; N ~ Normal(0,1) / P ~ Bernoulli(0.5)", "W = exp(T) - log(M) * N + V; M ~ Exponential(1) / T ~ StudentT(0.5) / N ~ Normal(0, 2)", ]

myscm = SCM(assignment_seq) ```

Agreements: - The name of the noise variable in the distribution specification (e.g. P ~ Normal(mean=2, std=1)) has to align with the noise variable name (P) of the assignment string. - Multiple noise models can be composited as done for variable W in the above example. The noise model string segment must specify all individual noise models separated by a '/'

2. Assignment Map

One can construct the SCM via an assignment map with the variables as keys and a tuple defining the assignment and the noise.

2-Tuple: Assignments via SymPy parsing

To refer to SymPy's string parsing capability (this includes numpy functions) provide a dict entry with 2-tuples as values of the form:

'var': ('assignment string', noise)

```python from sympy.stats import LogLogistic, LogNormal, Normal, Bernoulli, StudentT, Exponential

assignment_map = { "Z": ( "M", LogLogistic("M", alpha=1, beta=1) ), "X": ( "N * 3 * Z ** 2", LogNormal("N", mean=1, std=1), ), "Y": ( "P + 2 * Z + sqrt(X)", Normal("P", mean=2, std=1), ), "V": ( "N**P + Z", [Normal("N", 0, 1), Bernoulli("P", 0.5)] ), "W": ( "exp(T) - log(M) * N + V", [Normal("N", 0, 1), StudentT("T", 0.5), Exponential("M", 1)] ) }

myscm2 = SCM(assignment_map) ```

Agreements: - the name of the noise distribution provided in its constructor (e.g. Normal("N", mean=2, std=1)) must align with the noise variable name (N) in the assignment string. - Multiple noise models in the assignment must be wrapped in an iterable (e.g. a list [], or tuple ())

3-Tuple: Assignments with arbitrary callables

One can also declare the SCM via specifying the variable assignment in a dictionary with the variables as keys and as values a sequence of length 3 of the form:

'var': (['parent1', 'parent2', ...], Callable, Noise)

This allows the user to supply functions that are not limited to predefined function sets of scmodels' dependencies.

```python import numpy as np

def Y_assignment(p, z, x): return p + 2 * z + np.sqrt(x)

functionalmap = { "Z": ( [], lambda m: m, LogLogistic("M", alpha=1, beta=1) ), "X": ( ["Z"], lambda n, z: n * 3 * z ** 2, LogNormal("N", mean=1, std=1), ), "Y": ( ["Z", "X"], Yassignment, Normal("P", mean=2, std=1), ), "V": ( ["Z"], lambda n, p, z: n ** p + z, [Normal("N", mean=0, std=1), Bernoulli("P", p=0.5)] ), "W": ( ["V"], lambda n, t, m, v: np.exp(m) - np.log(m) * n + v, [Normal("N", mean=0, std=1), StudentT("T", nu=0.5), Exponential("M", rate=1)] ) }

myscm3 = SCM(functional_map) ```

Agreements: - The callable's first parameter MUST be the noise inputs in the order that they are given, e.g. see variable $W$. If the noise distribution list contains only None, then they have to be omitted from the parameter list of the callable. - The order of variables in the parents list determines the order of input for parameters in the functional past the noise parameters (left to right).

Features

Prettyprint

The SCM supports a form of informative printing of its current setup, which includes mentioning active interventions and the assignments.

python print(myscm)

Structural Causal Model of 5 variables: Z, X, V, Y, W
Variables with active interventions: []
Assignments:
Z := f(M) = M    [ M ~ LogLogistic(alpha=1, beta=1) ]
X := f(N, Z) = N * 3 * Z ** 2    [ N ~ LogNormal(mean=1, std=1) ]
Y := f(P, Z, X) = P + 2 * Z + sqrt(X)    [ P ~ Normal(mean=2, std=1) ]
V := f(N, P, Z) = N**P + Z   [ N ~ Normal(mean=0, std=1), P ~ Bernoulli(p=0.5, succ=1, fail=0) ]
W := f(M, T, N, V) = exp(T) - log(M) * N + V     [ M ~ Exponential(rate=1), T ~ StudentT(nu=0.5), N ~ Normal(mean=0, std=2) ]

In the case of custom callable assignments, the output is less informative

python print(myscm3)

Structural Causal Model of 5 variables: Z, X, V, Y, W
Variables with active interventions: []
Assignments:
Z := f(M) = __unknown__  [ M ~ LogLogistic(alpha=1, beta=1) ]
X := f(N, Z) = __unknown__   [ N ~ LogNormal(mean=1, std=1) ]
Y := f(P, Z, X) = __unknown__    [ P ~ Normal(mean=2, std=1) ]
V := f(N, P, Z) = __unknown__    [ N ~ Normal(mean=0, std=1), P ~ Bernoulli(p=0.5, succ=1, fail=0) ]
W := f(N, T, M, V) = __unknown__     [ N ~ Normal(mean=0, std=1), T ~ StudentT(nu=0.5), M ~ Exponential(rate=1) ]

Interventions

One can easily perform interventions on the variables, e.g. a Do-intervention or also general interventions, which remodel the connections, assignments, and noise distributions. For general interventions, the passing structure is dict of the following form:

{var: (New Parents (Optional), New Assignment (optional), New Noise (optional))}

Any part of the original variable state, that is meant to be left unchanged, has to be passed as None. E.g. to assign a new callable assignment to variable X without changing parents or noise, one would call:

```python mynewcallable = lambda n, z: n + z

myscm.intervention({"X": (None, mynewcallable, None)}) ```

For the example of a do-intervention $\text{do}(X=1)$, the helper method do_intervention is provided. Its counterpart for noise interventions is soft_intervention:

```python myscm.do_intervention([("X", 1)])

from sympy.stats import FiniteRV

myscm.soft_intervention([("X", FiniteRV(myscm["X"].noise[0].name, density={-1: .5, 1: .5}))]) ```

Refer to the sympy docs for more information on building random variables.

Calling undo_intervention restores the original state of all variables. If variables are specified (variables=['X', 'Y']), any interventions on only those variables are undone.

python myscm.undo_intervention(variables=["X"])

Sampling

The SCM allows drawing as many samples as needed through the method myscm.sample(n).

```python n = 5 samples = myscm.sample(n)

display_data(samples) ```

| | Z | V | X | W | Y | |----|----------|----------|-----------|---------|----------| | 0 | 0.854202 | 0.193336 | 15.9145 | 1.60315 | 7.55694 | | 1 | 0.634422 | 0.951119 | 2.43486 | 5.59541 | 3.6478 | | 2 | 1.02354 | 2.02354 | 4.11357 | 2.12862 | 6.23834 | | 3 | 0.376173 | 2.2962 | 0.252316 | 2.83333 | 3.64392 | | 4 | 4.19568 | 2.894 | 36.2844 | 3.07636 | 16.4083 |

If infinite sampling is desired, one can also receive a sampling generator through

python container = {var: [] for var in myscm} sampler = myscm.sample_iter(container)

container is an optional target dictionary to store the computed samples in.

```python for i in range(n): next(sampler)

display_data(container) ```

| | Z | V | X | W | Y | |----|----------|-----------|------------|--------------|----------| | 0 | 0.245579 | 0.814717 | 1.59715 | 2.10267 | 2.26984 | | 1 | 7.88344 | 8.88344 | 122.197 | 1.90523e+13 | 29.1743 | | 2 | 1.26534 | 1.29397 | 12.5345 | 0.523827 | 8.77073 | | 3 | 1.37904 | 2.37904 | 13.327 | 2.08063 | 9.14935 | | 4 | 0.230563 | -0.966714 | 0.354393 | -1.22201 | 3.09432 |

If the target container is not provided, the generator returns a new dict for every sample.

```python sample = next(myscm.sample_iter())

display_data(sample) ```

| | Z | V | X | W | Y | |----|---------|--------|---------|--------|---------| | 0 | 2.02308 | 3.1259 | 59.8718 | 4.2209 | 13.7312 |

Plotting

If you have graphviz installed, you can plot the DAG by calling

python myscm.plot(node_size=1000, alpha=1)