Introduction
Some problems, especially in numerical integration and Markov Chain Monte Carlo, benefit from transformation of variables: for example, if $σ > 0$ is a standard deviation parameter, it is usually better to work with log(σ)
which can take any value on the real line. However, in general such transformations require correcting density functions by the determinant of their Jacobian matrix, usually referred to as "the Jacobian".
Also, is usually easier to code MCMC algorithms to work with vectors of real numbers, which may represent a "flattened" version of parameters, and would need to be decomposed into individual parameters, which themselves may be arrays, tuples, or special objects like lower triangular matrices.
This package is designed to help with both of these use cases. For example, consider the "8 schools" problem from Chapter 5.5 of Gelman et al (2013), in which SAT scores $y_{ij}$ in $J=8$ schools are modeled using a conditional normal
and the $θⱼ$ are assume to have a hierarchical prior distribution
For this problem, one could define a transformation
using TransformVariables
t = as((μ = asℝ, σ = asℝ₊, τ = asℝ₊, θs = as(Array, 8)))
dimension(t)
11
which would then yield a NamedTuple
with the given names, with one of them being a Vector
:
julia> x = randn(dimension(t))
11-element Array{Float64,1}:
0.2972879845354616
0.3823959677906078
-0.5976344767282311
-0.01044524463737564
-0.839026854388764
0.31111133849833383
2.2950878238373105
-2.2670863488005306
0.5299655761667461
0.43142152642291204
0.5837082875687786
julia> y = transform(t, x)
(μ = 0.2972879845354616, σ = 1.4657923768059056, τ = 0.5501113994952206, θs = [-0.01044524463737564, -0.839026854388764, 0.31111133849833383, 2.2950878238373105, -2.2670863488005306, 0.5299655761667461, 0.43142152642291204, 0.5837082875687786])
julia> keys(y)
(:μ, :σ, :τ, :θs)
julia> y.θs
8-element Array{Float64,1}:
-0.01044524463737564
-0.839026854388764
0.31111133849833383
2.2950878238373105
-2.2670863488005306
0.5299655761667461
0.43142152642291204
0.5837082875687786
Further worked examples of using this package can be found in the DynamicHMCExamples.jl repository. It is recommended that the user reads those first, and treats this documentation as a reference.
General interface
TransformVariables.dimension
— Functiondimension(t::AbstractTransform)
The dimension (number of elements) that t
transforms.
Types should implement this method.
TransformVariables.transform
— Functiontransform(t, x)
Transform x
using t
.
TransformVariables.transform_and_logjac
— Functiontransform_and_logjac(t, x)
Transform x
using t
; calculating the log Jacobian determinant, returned as the second value.
TransformVariables.inverse
— Functioninverse(t::AbstractTransform, y)
Return x
so that transform(t, x) ≈ y
.
inverse(t)
Return a callable equivalen to y -> inverse(t, y)
.
TransformVariables.inverse!
— Functioninverse!(x, transformation, y)
Put inverse(t, y)
into a preallocated vector x
, returning x
.
Generalized indexing should be assumed on x
.
See inverse_eltype
for determining the type of x
.
TransformVariables.inverse_eltype
— Functioninverse_eltype(t::AbstractTransform, y)
The element type for vector x
so that inverse!(x, t, y)
works.
TransformVariables.transform_logdensity
— Functiontransform_logdensity(t, f, x)
Let $y = t(x)$, and $f(y)$ a log density at y
. This function evaluates f ∘ t
as a log density, taking care of the log Jacobian correction.
TransformVariables.random_arg
— Functionrandom_arg(x; kwargs...)
A random argument for a transformation.
Keyword arguments
A standard multivaritate normal or Cauchy is used, depending on cauchy
, then scaled with scale
. rng
is the random number generator used.
TransformVariables.random_value
— Functionrandom_value(t; kwargs...)
Random value from a transformation.
Keyword arguments
A standard multivaritate normal or Cauchy is used, depending on cauchy
, then scaled with scale
. rng
is the random number generator used.
Defining transformations
The as
constructor and aggregations
Some transformations, particularly aggregations use the function as
as the constructor. Aggregating transformations are built from other transformations to transform consecutive (blocks of) real numbers into the desired domain.
It is recommended that you use as(Array, ...)
and friends (as(Vector, ...)
, as(Matrix, ...)
) for repeating the same transformation, and named tuples such as as((μ = ..., σ = ...))
for transforming into named parameters. For extracting parameters in log likelihoods, consider Parameters.jl.
See methods(as)
for all the constructors, ?as
for their documentation.
TransformVariables.as
— Functionas(T, args...)
Shorthand for constructing transformations with image in T
. args
determines or modifies behavior, details depend on T
.
Not all transformations have an as
method, some just have direct constructors. See methods(as)
for a list.
Examples
as(Real, -∞, 1) # transform a real number to (-∞, 1)
as(Array, 10, 2) # reshape 20 real numbers to a 10x2 matrix
as(Array, as𝕀, 10) # transform 10 real numbers to (0, 1)
as((a = asℝ₊, b = as𝕀)) # transform 2 real numbers a NamedTuple, with a > 0, 0 < b < 1
Scalar transforms
The symbol ∞
is a placeholder for infinity. It does not correspond to Inf
, but acts as a placeholder for the correct dispatch. -∞
is valid.
TransformVariables.∞
— ConstantPlaceholder representing of infinity for specifing interval boundaries. Supports the -
operator, ie -∞
.
as(Real, a, b)
defines transformations to finite and (semi-)infinite subsets of the real line, where a
and b
can be -∞
and ∞
, respectively. The following constants are defined for common cases.
TransformVariables.asℝ
— ConstantTransform to the real line (identity).
asℝ
and as_real
are equivalent alternatives.
TransformVariables.asℝ₊
— ConstantTransform to a positive real number.
asℝ₊
and as_positive_real
are equivalent alternatives.
TransformVariables.asℝ₋
— ConstantTransform to a negative real number.
asℝ₋
and as_negative_real
are equivalent alternatives.
TransformVariables.as𝕀
— ConstantTransform to the unit interval (0, 1)
.
as𝕀
and as_unit_interval
are equivalent alternatives.
Special arrays
TransformVariables.UnitVector
— TypeUnitVector(n)
Transform n-1
real numbers to a unit vector of length n
, under the Euclidean norm.
TransformVariables.UnitSimplex
— TypeUnitSimplex(n)
Transform n-1
real numbers to a vector of length n
whose elements are non-negative and sum to one.
TransformVariables.CorrCholeskyFactor
— TypeCorrCholeskyFactor(n)
Cholesky factor of a correlation matrix of size n
.
Transforms $n×(n-1)/2$ real numbers to an $n×n$ upper-triangular matrix U
, such that U'*U
is a correlation matrix (positive definite, with unit diagonal).
Notes
If
z
is a vector ofn
IID standard normal variates,σ
is ann
-element vector of standard deviations,U
is obtained fromCorrCholeskyFactor(n)
,
then Diagonal(σ) * U' * z
will be a multivariate normal with the given variances and correlation matrix U' * U
.
Defining custom transformations
TransformVariables.logjac_forwarddiff
— Functionlogjac_forwarddiff(f, x; handleNaN, chunk, cfg)
Calculate the log Jacobian determinant of f
at x
using `ForwardDiff.
Note
f
should be a bijection, mapping from vectors of real numbers to vectors of equal length.
When handleNaN = true
(the default), NaN log Jacobians are converted to -Inf.
TransformVariables.value_and_logjac_forwarddiff
— Functionvalue_and_logjac_forwarddiff(f, x; flatten, handleNaN, chunk, cfg)
Calculate the value and the log Jacobian determinant of f
at x
. flatten
is used to get a vector out of the result that makes f
a bijection.
TransformVariables.CustomTransform
— TypeCustomTransform(g, f, flatten; chunk, cfg)
Wrap a custom transform y = f(transform(g, x))
in a type that calculates the log Jacobian of
∂y/∂x
using
ForwardDiff` when necessary.
Usually, g::TransformReals
, but when an integer is used, it amounts to the identity transformation with that dimension.
flatten
should take the result from f
, and return a flat vector with no redundant elements, so that $x ↦ y$ is a bijection. For example, for a covariance matrix the elements below the diagonal should be removed.
chunk
and cfg
can be used to configure ForwardDiff.JacobianConfig
. cfg
is used directly, while chunk = ForwardDiff.Chunk{N}()
can be used to obtain a type-stable configuration.