# Built-in Functional Forms

This section describes built-in functional forms that can be used for posterior marginal and/or messages form constraints specification. Read more information about constraints specification syntax in the Constraints Specification section. 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.

## Custom functional forms

See the Custom functional form specification section for more information about defining novel custom functional forms that are compatible with `ReactiveMP`

inference backend.

## UnspecifiedFormConstraint

Unspecified functional form constraint is used by default and uses only analytical update rules for computing posterior marginals. Throws an error if a product of two colliding messages cannot be computed analytically.

```
@constraints begin
q(x) :: Nothing # This is the default setting
end
```

`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.

**Traits**

`is_point_mass_form_constraint`

=`false`

`default_form_check_strategy`

=`FormConstraintCheckLast()`

`default_prod_constraint`

=`ProdAnalytical()`

`make_form_constraint`

=`Nothing`

(for use in`@constraints`

macro)

See also: `constrain_form`

, `DistProduct`

## PointMassFormConstraint

The most basic form of posterior marginal approximation is the `PointMass`

function. In a few words `PointMass`

represents delta function. In the context of functional form constraints `PointMass`

approximation corresponds to the MAP estimate. For a given distribution `d`

- `PointMass`

functional form simply finds the `argmax`

of the `logpdf(d, x)`

by default.

```
@constraints begin
q(x) :: PointMass # Materializes to the `PointMassFormConstraint` object
end
```

`ReactiveMP.PointMassFormConstraint`

— Type`PointMassFormConstraint`

One of the form constraint objects. Constraint a message to be in a form of dirac's delta point mass. By default uses `Optim.jl`

package to find argmin of -logpdf(x). Accepts custom `optimizer`

callback which might be used to customise optimisation procedure with different packages or different arguments for `Optim.jl`

package.

**Keyword arguments**

`optimizer`

: specifies a callback function for logpdf optimisation. See also:`ReactiveMP.default_point_mass_form_constraint_optimizer`

`starting_point`

: specifies a callback function for initial optimisation point: See also:`ReactiveMP.default_point_mass_form_constraint_starting_point`

`boundaries`

: specifies a callback function for determining optimisation boundaries: See also:`ReactiveMP.default_point_mass_form_constraint_boundaries`

**Custom optimizer callback interface**

```
# This is an example of the `custom_optimizer` interface
function custom_optimizer(::Type{ Univariate }, ::Type{ Continuous }, constraint::PointMassFormConstraint, distribution)
# should return argmin of the -logpdf(distribution)
end
```

**Custom starting point callback interface**

```
# This is an example of the `custom_starting_point` interface
function custom_starting_point(::Type{ Univariate }, ::Type{ Continuous }, constraint::PointMassFormConstraint, distribution)
# built-in optimizer expects an array, even for a univariate distribution
return [ 0.0 ]
end
```

**Custom boundaries callback interface**

```
# This is an example of the `custom_boundaries` interface
function custom_boundaries(::Type{ Univariate }, ::Type{ Continuous }, constraint::PointMassFormConstraint, distribution)
# returns a tuple of `lower` and `upper` boundaries
return (-Inf, Inf)
end
```

**Traits**

`is_point_mass_form_constraint`

=`true`

`default_form_check_strategy`

=`FormConstraintCheckLast()`

`default_prod_constraint`

=`ProdGeneric()`

`make_form_constraint`

=`PointMass`

(for use in`@constraints`

macro)

See also: `constrain_form`

, `DistProduct`

## SampleListFormConstraint

`SampleListFormConstraints`

approximates the resulting posterior marginal (product of two colliding messages) as a list of weighted samples. Hence, it requires one of the arguments to be a proper distribution (or at least be able to sample from it). This setting is controlled with `LeftProposal()`

, `RightProposal()`

or `AutoProposal()`

objects. It is worth to note that `SampleListFormConstraints`

approximates only `DistProduct`

object and leaves any other distribution untouched. It also accepts an optional `method`

object, but the only one available sampling method currently is the `BootstrapImportanceSampling`

.

```
@constraints begin
q(x) :: SampleList(1000)
# or
q(y) :: SampleList(1000, LeftProposal())
end
```

`ReactiveMP.SampleListFormConstraint`

— Type`SampleListFormConstraint(rng, strategy, method)`

One of the form constraint objects. Approximates `DistProduct`

with a SampleList object.

**Traits**

`is_point_mass_form_constraint`

=`false`

`default_form_check_strategy`

=`FormConstraintCheckLast()`

`default_prod_constraint`

=`ProdGeneric()`

`make_form_constraint`

=`SampleList`

(for use in`@constraints`

macro)

See also: `constrain_form`

, `DistProduct`

## FixedMarginalFormConstraint

Fixed marginal form constraint replaces the resulting posterior marginal obtained during the inference procedure with the prespecified one. Worth to note that the inference backend still tries to compute real posterior marginal and may fail during this process. Might be useful for debugging purposes. If `nothing`

is passed then the computed posterior marginal is returned.

```
@constraints function block_updates(x_posterior = nothing)
# `nothing` returns the computed posterior marginal
q(x) :: Marginal(x_posterior)
end
```

`ReactiveMP.FixedMarginalFormConstraint`

— Type`FixedMarginalFormConstraint`

One of the form constraint objects. Provides a constraint on the marginal distribution such that it remains fixed during inference. Can be viewed as blocking of updates of a specific edge associated with the marginal. If `nothing`

is passed then the computed posterior marginal is returned.

**Traits**

`is_point_mass_form_constraint`

=`false`

`default_form_check_strategy`

=`FormConstraintCheckLast()`

`default_prod_constraint`

=`ProdAnalytical()`

`make_form_constraint`

=`Marginal`

(for use in`@constraints`

macro)

See also: `constrain_form`

, `DistProduct`

## CompositeFormConstraint

It is possible to create a composite functional form constraint with either `+`

operator or using `@constraints`

macro, e.g:

`form_constraint = SampleListFormConstraint(1000) + PointMassFormConstraint()`

```
@constraints begin
q(x) :: SampleList(1000) :: PointMass()
end
```

`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`

. Any functional form constraint that defines `is_point_mass_form_constraint() = true`

may be used only as the last element of the composition.