Low-level usage - Code Overview

This is the code overview for the Halerium core subpackage. With the core subpackage the user can create and modify Halerium structures. The objects in the high-level Halerium package, e.g. the CausalStructure, can be viewed as factories for Halerium core code.


Halerium structures are built out of graphs, entities and variables. The building blocks can be nested to build deep and hierarchical structures in a convenient way with the scoping mechanism.


A scope is the context in a Halerium structure. Scopes are managed via entering and exiting python with blocks.

A scopetor is an object that can provide such a scope. The Halerium scopetor classes are Graph, Entity, Variable, and StaticVariable.

A scopee is an object that a becomes child of the scope in which it is created. All scopetors are also scopees as well as all operators.

For example:

>>> with Entity("s") as s:
>>>     e = Entity("e")
>>>     print(e.scope)
<halerium.Entity 's'>

More details can be found in


A Graph instance can contain other Graph instances as well as Entity, Variable, and StaticVariable instances as children.

Additionally it has the special children inputs and outputs that can only contain Entity instances.

In the Halerium platform graphs can be displayed for interactive inspection with the show() method, see


A Entity instance can contain other Entity instances as well as Variable, and StaticVariable instances as children.


Dynamic Variables

The shape of dynamic variables scales with the amount of data. A dynamic variable with a shape of (3, 5) will correspond to an array of shape (11, 3, 5) in a model with n_data=11.

A Variable instance can contain other Variable instances as well as StaticVariable instances as children.

Static Variables

The shape of static variables does not scale with the amount of data. Their value is therefore universal, which makes them suitable for parameters that are to be learned from a set of training data.

A StaticVariable instance can only contain other StaticVariable instances as children.

Printing child trees

The function print_child_tree() can be applied to any scopetor to see the scopetors child tree.

>>> e = Entity("e")
>>> with e:
>>>     Entity("ee")
>>>     with ee:
>>>         Variable("v")
>>>     Variable("w")
>>> print_child_tree(e)
│ └─v


Mathematical operations can be conveniently created with the functions in the halerium.core.math module. All math functions are available at the top-module level, e.g.

# equivalent to

For basic arithmetic use the overloaded python operators +, -, *, /, **. abs() and numpy-stype slicing is also supported. The math functions are designed to mimic their numpy counterparts as closely as possible.

Floats and numpy arrays are automatically casted to Const operators when included in a Halerium operation, e.g.

# equivalent to

Mathematical operations create operators, which are scopees. See their documentation in the halerium.core.operator module. All operators that may be used when defining Halerium structures are accessible therein, e.g.

halerium.core.operator.Add(1., 1.)

Printing operand trees

The function print_operand_tree() can be applied to any operator to see the operators that lead to it.

>>> a = hal.constant(0.)
>>> b = hal.constant(1.)
>>> c = a + b
>>> d = c * a
>>> print_operand_tree(d)
<halerium.Mul 'Mul'>
├─<halerium.Add 'Add'>
│ ├─<halerium.Const 'Const'>
│ └─<halerium.Const 'Const'>
└─<halerium.Const 'Const'>


Data can be linked to Variable or StaticVariable instances in a Halerium structure.

To link data you provide a dictionary with the variables as keys and numpy arrays as values as the data argument of e.g. a model factory.

data={graph.var1: np.zeros((10, 4)),
      graph.var2: np.zeros((10, 2, 3)),
      graph.static_var_1, np.zeros((3,))}
model = get_generative_model(graph=graph,

Alternatively a DataLinker instance can be created from a data dictionary with get_data_linker()

dl = get_data_linker(data={graph.var1: np.zeros((10, 4)),
                           graph.var2: np.zeros((10, 2, 3)),
                           graph.static_var_1, np.zeros((3,))})

The DataLinker instance is the explicit representation of the data links. It too can be provided as the data argument.

model = get_generative_model(graph=graph,


Creating Models

Models can only be created from Graph instances.

Models are created by combining a Halerium graph with data. Models implement a specific algorithm/solution strategy in order to do actual numerical calculations.

The common way to get a model instance is by calling either get_generative_model(), get_posterior_model(), or get_optimizer_model().

get_generative_model() will return an instance of ForwardModel. This model is purely for generating data in a feed-forward fashion. Information from data only flows forwards along the dependencies of the structure.

get_posterior_model() will return an instance of either MAPFisherModel, MAPModel, ADVIModel, or MGVIModel. These models calculate estimates for the variables in the structure that take all data and all possible directions of information flow into account. Which model class (and therefore solution strategy) is used depend on the keyword argument method.

get_optimizer_model() will return an instance of ForwardOptimizerModel. This model is used to calculate the optimal values for a set of variables that minimize a cost function that depends on a set of (different) variables in the graph.

Evaluating Models

Models need to be solved with their solve method, before they can be evaluated. By default models created using the get_..._model functions come in a trained state.

A trained model can
  • generate a single sample of variables with the get_example method

  • generate samples of variables with the get_samples method

  • calculate the mean of variables with the get_means method

  • calculate the standard deviation get_standard_deviations method

  • calculate the variance of variables with the get_variances method.

From posterior models you can also extract a posterior graph by calling the get_posterior_graph method. The posterior graph will contain updated probability distribution for all StaticVariable instances in its graph.

Further explanations on creating and solving/training models can be found in

Training Graphs

To directly get a posterior graph from the combination of a Graph instance and data the Trainer class can be used. Upon instantiation the class will create and solve a model. When called the Trainer returns the posterior graph.

>>> trainer = Trainer(graph=graph, data=train_data)
>>> trained_graph = trainer()

For a more detailed example see

To understand the mathematical background of the training process see


The objective classes introduces in the main package overview can also be used on the core level. Here the objective class is instantiated with a (trained) graph instance and additional arguments like data. When the instantiated objective is called the objective result is returned for the provided scopetors.

>>> predictor = Predictor(graph=graph, data=prediction_input_data)
>>> predictor(graph.y)
array([1., 2., 3., 4.])
The available objectives are

They are explained in detail in


Every StaticVariable or Variable in a Halerium structure has an underlying probability distribution. By default this is the NormalDistribution.

The user can create variables with different types of distributions by providing the distribution class as the distribution argument when creating a variable.

v = Variable(name="v", distribution=LogNormalDistribution)

Each distribution class supports different defining parameters. For the NormalDistribution these are mean and variance, which are commonly used with Halerium variables with the default distribution. For the LogNormalDistribution used in the example above they are mean_log and variance_log.

v.mean_log = 0.
v.variance_log = 1.

The Halerium distributions are explained in more detail in

Below is a short overview of the available distributions and their parameters.

Supported distributions

Currently, Halerium supports the following distributions:

Furthermore there is the possibility of a variable having the NoDistribution in which case it is not a random variable at all, but rather acts as a placeholder for data. Consequently, a variable with the NoDistribution has to be completely determined by data when creating a model.

Regression factories

Regression factories help the user to connect Variable instances by parametrized mathematical formulas with the parameters being StaticVariable instances.

Currently, Halerium provides the following factories for creating the static variables and creating the result of the formula applied to the inputs:

For even more convenience connect_via_regression() and connect_via_gaussian_process() directly set distribution parameters of desired output variables to the regression results.

For further explanations see

Causal Calculus

Causal calculus is realized via the Do operation, do_operation(). With this operation a Graph or other scopetor can be modified to make variables of choice independent of all other variables. This is required to model interventions and to distinguish the effect of interventions from observations. The modified Graph can then be utilized further, e.g. to make predictions. The combination of a do operation and prediction is conveniently accessible in the InterventionPredictor

For a basic introduction of the do operation see

Time Series

Halerium Variable instances are expanded with the data dimension at model creation time. This way a Graph can be formulated irrespective of the amount of data and the conditional independence along the data axis is ensured by construction.

If we use the data axis as a time axis the conditional independence is not desirable. The data values do not represent independent and identically distributed samples, but a time-series. In this time series a particular value is conditionally independent of its corresponding future values, but it can depend on the past.

To access the past values of a Variable or any other dynamic operator, we can utilize the TimeShift operator and the TimeIndex singleton.

v = Variable("v")
past_v = v[TimeIndex-3]
# does the same as
past_v = TimeShift(operand=v, shift=-3, initial_values=0.)

Only past values of a dynamic operator are available. A positive shift would lead to an error.

>>> future_v = v[TimeIndex+1]
RuntimeError: Positive shifts are not supported.

A shift of 0 has the same effect as the Identity operator. For further explanations see