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. More information about prod function is present in the Prod Implementation section. 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.AbstractFormConstraintType
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.

See also: CompositeFormConstraint

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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.FormConstraintCheckEachType
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.

See also: FormConstraintCheckLast, default_form_check_strategy, constrain_form, multiply_messages

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ReactiveMP.FormConstraintCheckLastType
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.

See also: FormConstraintCheckLast, default_form_check_strategy, constrain_form, multiply_messages

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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.

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.

Is point mass form constraint (optional)

Every custom functional form may implement a new method for the is_point_mass_form_constraint function that returns either true or false. This is an utility function that simplifes computation of the Bethe Free Energy and is not strictly necessary.

Compatibility with @constraints macro (optional)

To make custom functional form constraint compatible with the @constraints macro, it must implement a new method for the make_form_constraint function.

ReactiveMP.make_form_constraintFunction
make_form_constraint(::Type, args...; kwargs...)

Creates form constraint object based on passed type with given args and kwargs. Used to simplify form constraint specification.

As an example:

make_form_constraint(PointMass)

creates an instance of PointMassFormConstraint and

make_form_constraint(SampleList, 5000, LeftProposal())

should create an instance of SampleListFormConstraint.

See also: AbstractFormConstraint

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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. More information about prod function is present in the Prod Implementation section. 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, GraphPPL

# 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.is_point_mass_form_constraint(::MeanPrecisionFormConstraint) = false
ReactiveMP.default_form_check_strategy(::MeanPrecisionFormConstraint)   = FormConstraintCheckLast()
ReactiveMP.default_prod_constraint(::MeanPrecisionFormConstraint)       = ProdGeneric()

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::DistProduct)
    # DistProduct is the special case, read about this type more in the corresponding documentation section
    # ...
end

At this point we already can use our functional form constraint in the inference backend, however, lets also make our functional form constraint compatible with the @constraints macro from GraphPPL.jl package.

ReactiveMP.make_form_constraint(::Type{ NormalMeanPrecision }, args...; kwargs...) = MeanPrecisionFormConstraint()
@constraints begin
    q(x) :: NormalMeanPrecision
end
Constraints:
  marginals form:
    q(x) :: Main.var"Main".MeanPrecisionFormConstraint() [ prod_constraint = ProdGeneric(fallback = ProdAnalytical()) ]
  messages form:
  factorisation:
Options:
  warn = true