Introduction

This package serves two purposes:

1. Introduce a common API for packages that operate on log densities, which for these purposes are black box $\mathbb{R}^n \to \mathbb{R}$ mappings. Using the interface of introduced in this package, you can query $n$, evaluate the log density and optionally its gradient, and determine if a particular object supports these methods using traits. This usage is relevant primarily for package developers who write generic algorithms that use (log) densities that correspond to posteriors and likelihoods, eg MCMC, MAP, ML. An example is DynamicHMC.jl. This is documented in the API section.

2. Make it easier for users who want to perform inference using the above methods (and packages) to

   • define their own log densities, either taking a vector of real numbers as input, or extracting and transforming parameters using the TransformVariables.jl package,
   • obtain gradients of these log densities using one of the supported automatic differentiation packages of Julia.

This is documented in the next section, with a worked example.

For the purposes of this package, log densities are still valid when shifted by a constant that may be unknown, but is consistent within calls. This is necessary for Bayesian inference, where log posteriors are usually calculated up to a constant. See LogDensityProblems.logdensity for details.

Working with log density problems

Consider an inference problem where IID draws are obtained from a normal distribution,

\[x_i \sim \mathcal{N}(\mu, \sigma)\]

for $i = 1, \dots, N$. It can be shown that the log likelihood conditional on $\mu$ and $\sigma$ is

\[\ell = -N\log \sigma - \sum_{i = 1}^N \frac{(x_i-\mu)^2}{2\sigma^2} = -N\left( \log \sigma + \frac{S + (\bar{x} - \mu)^2}{2\sigma^2} \right)\]

where we have dropped constant terms, and defined the sufficient statistics

\[\bar{x} = \frac{1}{N} \sum_{i = 1}^N x_i\]

and

\[S = \frac{1}{N} \sum_{i = 1}^N (x_i - \bar{x})^2\]

Finally, we use priors

\[\mu \sim \mathcal{N}(0, 5), \sigma \sim \mathcal{N}(0, 2)\]

which yield the log prior

\[-\sigma^2/8 - \mu^2/50\]

which is added to the log likelihood to obtain the log posterior.

It is useful to define a callable that implements this, taking some vector x as an input and calculating the summary statistics, then, when called with a NamedTuple containing the parameters, evaluating to the log posterior.

using Statistics, SimpleUnPack # imported for our implementation

struct NormalPosterior{T} # contains the summary statistics
    N::Int
    x̄::T
    S::T
end

# calculate summary statistics from a data vector
function NormalPosterior(x::AbstractVector)
    NormalPosterior(length(x), mean(x), var(x; corrected = false))
end

# define a callable that unpacks parameters, and evaluates the log likelihood
function (problem::NormalPosterior)(θ)
    @unpack μ, σ = θ
    @unpack N, x̄, S = problem
    loglikelihood = -N * (log(σ) + (S + abs2(μ - x̄)) / (2 * abs2(σ)))
    logprior = - abs2(σ)/8 - abs2(μ)/50
    loglikelihood + logprior
end

problem = NormalPosterior(randn(100))

Let's try out the posterior calculation:

julia> problem((μ = 0.0, σ = 1.0))-48.60297471151442

Using the TransformVariables package

In our example, we require $\sigma > 0$, otherwise the problem is meaningless. However, many MCMC samplers prefer to operate on unconstrained spaces $\mathbb{R}^n$. The TransformVariables package was written to transform unconstrained to constrained spaces, and help with the log Jacobian correction (more on that later). That package has detailed documentation, now we just define a transformation from a length 2 vector to a NamedTuple with fields μ (unconstrained) and σ > 0.

julia> using LogDensityProblems, TransformVariables, TransformedLogDensities
julia> ℓ = TransformedLogDensity(as((μ = asℝ, σ = asℝ₊)), problem)TransformedLogDensity of dimension 2

Then we can query the dimension of this problem, and evaluate the log density:

julia> LogDensityProblems.dimension(ℓ)2
julia> LogDensityProblems.logdensity(ℓ, zeros(2))-48.60297471151442

Manual unpacking and transformation

If you prefer to implement the transformation yourself, you just have to define the following three methods for your problem: declare that it can evaluate log densities (but not their gradient, hence the 0 order), allow the dimension of the problem to be queried, and then finally code the density calculation with the transformation. (Note that using TransformedLogDensities.TransformedLogDensity takes care of all of these for you, as shown above).

function LogDensityProblems.capabilities(::Type{<:NormalPosterior})
    LogDensityProblems.LogDensityOrder{0}()
end

LogDensityProblems.dimension(::NormalPosterior) = 2

function LogDensityProblems.logdensity(problem::NormalPosterior, x)
    μ, logσ = x
    σ = exp(logσ)
    problem((μ = μ, σ = σ)) + logσ
end

julia> LogDensityProblems.logdensity(problem, zeros(2))-48.60297471151442

Here we use the exponential function to transform from $\mathbb{R}$ to the positive reals, but this requires that we correct the log density with the logarithm of the Jacobian, which here happens to be $\log(\sigma)$.

Automatic differentiation

Using either definition, you can transform to another object which is capable of evaluating the gradient, using automatic differentiation. For this, you need the LogDensityProblemsAD.jl package.

Now observe that we can obtain gradients, too:

julia> import ForwardDiff
julia> using LogDensityProblemsAD
julia> ∇ℓ = ADgradient(:ForwardDiff, ℓ)ForwardDiff AD wrapper for TransformedLogDensity of dimension 2, w/ chunk size 2
julia> LogDensityProblems.capabilities(∇ℓ)LogDensityProblems.LogDensityOrder{1}()
julia> LogDensityProblems.logdensity_and_gradient(∇ℓ, zeros(2))(-48.60297471151442, [2.8381559225185673, -2.294050576971158])

Manually calculated derivatives

If you prefer not to use automatic differentiation, you can wrap your own derivatives following the template

function LogDensityProblems.capabilities(::Type{<:NormalPosterior})
    LogDensityProblems.LogDensityOrder{1}() # can do gradient
end

LogDensityProblems.dimension(::NormalPosterior) = 2 # for this problem

function LogDensityProblems.logdensity_and_gradient(problem::NormalPosterior, x)
    logdens = ...
    grad = ...
    logdens, grad
end

Various utilities

You may find these utilities useful for debugging and optimization.

stresstest(f, ℓ; N, rng, scale)

Log densities API

Use the functions below for evaluating gradients and querying their dimension and other information. These symbols are not exported, as they are mostly used by package developers and in any case would need to be imported or qualified to add methods to.

capabilities(T)

struct LogDensityOrder{K}

dimension(ℓ)

logdensity(ℓ, x)

logdensity(::StandardMultivariateNormal) = -0.5 * sum(abs2, x)

logdensity_and_gradient(ℓ, x)

logdensity_gradient_and_hessian(ℓ, x)