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Custom Functional Form Specification

In a nutshell, functional form constraints defines a function that approximates the product of colliding messages and computes posterior marginal that can be used later on during the inference procedure. An important part of the functional forms constraint implementation is the prod function in the BayesBase package. For example, if we refer to our CustomFunctionalForm as to f we can see the whole functional form constraints pipeline as:

\[q(x) = f\left(\frac{\overrightarrow{\mu}(x)\overleftarrow{\mu}(x)}{\int \overrightarrow{\mu}(x)\overleftarrow{\mu}(x) \mathrm{d}x}\right)\]

Interface

ReactiveMP.jl, however, uses some extra utility functions to define functional form constraint behaviour. Here we briefly describe all utility function. If you are only interested in the concrete example, you may directly head to the Custom Functional Form example at the end of this section.

Abstract super type

ReactiveMP.AbstractFormConstraint — Type
AbstractFormConstraint

Every functional form constraint is a subtype of AbstractFormConstraint abstract type.

Note: this is not strictly necessary, but it makes automatic dispatch easier and compatible with the CompositeFormConstraint.

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ReactiveMP.UnspecifiedFormConstraint — Type
UnspecifiedFormConstraint

One of the form constraint objects. Does not imply any form constraints and simply returns the same object as receives. However it does not allow DistProduct to be a valid functional form in the inference backend.

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ReactiveMP.CompositeFormConstraint — Type
CompositeFormConstraint

Creates a composite form constraint that applies form constraints in order. The composed form constraints must be compatible and have the exact same form_check_strategy.

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Form check strategy

Every custom functional form must implement a new method for the default_form_check_strategy function that returns either FormConstraintCheckEach or FormConstraintCheckLast.

  • FormConstraintCheckLast: q(x) = f(μ1(x) * μ2(x) * μ3(x))
  • FormConstraintCheckEach: q(x) = f(f(μ1(x) * μ2(x)) * μ3(x))
ReactiveMP.default_form_check_strategy — Function
default_form_check_strategy(form_constraint)

Returns a default check strategy (e.g. FormConstraintCheckEach or FormConstraintCheckEach) for a given form constraint object.

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ReactiveMP.FormConstraintCheckEach — Type
FormConstraintCheckEach

This form constraint check strategy checks functional form of the messages product after each product in an equality chain. Usually if a variable has been connected to multiple nodes we want to perform multiple prod to obtain a posterior marginal. With this form check strategy constrain_form function will be executed after each subsequent prod function.

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ReactiveMP.FormConstraintCheckLast — Type
FormConstraintCheckEach

This form constraint check strategy checks functional form of the last messages product in the equality chain. Usually if a variable has been connected to multiple nodes we want to perform multiple prod to obtain a posterior marginal. With this form check strategy constrain_form function will be executed only once after all subsequenct prod functions have been executed.

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Prod constraint

Every custom functional form must implement a new method for the default_prod_constraint function that returns a proper prod_constraint object.

ReactiveMP.default_prod_constraint — Function
default_prod_constraint(form_constraint)

Returns a default prod constraint needed to apply a given form_constraint. For most form constraints this function returns ProdGeneric.

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Constrain form, a.k.a f

The main function that a custom functional form must implement, which we referred to as f in the beginning of this section, is the constrain_form function.

ReactiveMP.constrain_form — Function
constrain_form(form_constraint, distribution)

This function must approximate distribution object in a form that satisfies form_constraint.

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Custom Functional Form Example

In this demo we show how to build a custom functional form constraint that is compatible with the ReactiveMP.jl inference backend. An important part of the functional forms constraint implementation is the prod function in the BayesBase package. We show a relatively simple use-case, which might not be very useful in practice, but serves as a simple step-by-step guide. Assume that we want a specific posterior marginal of some random variable in our model to have a specific Gaussian parametrisation, for example mean-precision. We can use built-in NormalMeanPrecision distribution, but we still need to define our custom functional form constraint:

using ReactiveMP, BayesBase

# First we define our functional form structure with no fields
struct MeanPrecisionFormConstraint <: AbstractFormConstraint end

Next we define the behaviour of our functional form constraint:

ReactiveMP.default_form_check_strategy(::MeanPrecisionFormConstraint)   = FormConstraintCheckLast()
ReactiveMP.default_prod_constraint(::MeanPrecisionFormConstraint)       = GenericProd()

function ReactiveMP.constrain_form(::MeanPrecisionFormConstraint, distribution)
    # This is quite a naive assumption, that a given `dsitribution` object has `mean` and `precision` defined
    # However this quantities might be approximated with some other external method, e.g. Laplace approximation
    m = mean(distribution)      # or approximate with some other method
    p = precision(distribution) # or approximate with some other method
    return NormalMeanPrecision(m, p)
end

function ReactiveMP.constrain_form(::MeanPrecisionFormConstraint, distribution::BayesBase.ProductOf)
    # ProductOf is the special case, read about this type more in the corresponding documentation section
    # of the `BayesBase` package
    # ...
end