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pathfinder(ℓ; kwargs...)
pathfinder(fun::SciMLBase::OptimizationFunction; kwargs...)
pathfinder(prob::SciMLBase::OptimizationProblem; kwargs...)

Find the best multivariate normal approximation encountered while maximizing a log density.

From an optimization trajectory, Pathfinder constructs a sequence of (multivariate normal) approximations to the distribution specified by a log density function. The approximation that maximizes the evidence lower bound (ELBO), or equivalently, minimizes the KL divergence between the approximation and the true distribution, is returned.

The covariance of the multivariate normal distribution is an inverse Hessian approximation constructed using at most the previous history_length steps.

Arguments

• ℓ: an object, representing the log-density of the target distribution and its gradient, that implements the LogDensityProblems interface.
• fun::SciMLBase.OptimizationFunction: an optimization function that represents the negative log density with its gradient. It must have the necessary features (e.g. a Hessian function, if applicable) for the chosen optimization algorithm. For details, see Optimization.jl: OptimizationFunction.
• prob::SciMLBase.OptimizationProblem: an optimization problem containing a function with the same properties as fun, as well as an initial point, in which case init and dim are ignored.

Keywords

• dim::Int: dimension of the target distribution, needed only if fun is provided and init is not.
• init::AbstractVector{<:Real}: initial point of length dim from which to begin optimization. If not provided, an initial point of type Vector{Float64} and length dim is created and filled using init_sampler.
• init_scale::Real: scale factor $s$ such that the default init_sampler samples entries uniformly in the range $[-s, s]$
• init_sampler: function with the signature (rng, x) -> x that modifies a vector of length dims in-place to generate an initial point
• ndraws_elbo::Int=5: Number of draws used to estimate the ELBO
• ndraws::Int=ndraws_elbo: number of approximate draws to return
• rng::Random.AbstractRNG: The random number generator to be used for drawing samples
• executor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to perform ELBO computation in parallel. The default (SequentialEx()) performs no parallelization. If rng is known to be thread-safe, and the log-density function is known to have no internal state, then Transducers.PreferParallel() may be used to parallelize log-density evaluation. This is generally only faster for expensive log density functions.
• history_length::Int=6: Size of the history used to approximate the inverse Hessian.
• optimizer: Optimizer to be used for constructing trajectory. Can be any optimizer compatible with Optimization.jl, so long as it supports callbacks. Defaults to Optim.LBFGS(; m=history_length, linesearch=LineSearches.HagerZhang(), alphaguess=LineSearches.InitialHagerZhang()). See the Optimization.jl documentation for details.
• ntries::Int=1_000: Number of times to try the optimization, restarting if it fails. Before every restart, a new initial point is drawn using init_sampler.
• fail_on_nonfinite::Bool=true: If true, optimization fails if the log-density is a NaN or Inf or if the gradient is ever non-finite. If nretries > 0, then optimization will be retried after reinitialization.
• kwargs... : Remaining keywords are forwarded to Optimization.solve.

Returns

• PathfinderResult

PathfinderResult

Container for results of single-path Pathfinder.

Fields

• input: User-provided input object, e.g. a LogDensityProblem, optim_fun, optim_prob, or another object.
• optimizer: Optimizer used for maximizing the log-density
• rng: Pseudorandom number generator that was used for sampling
• optim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization
• logp: Log-density function
• fit_distribution::Distributions.MvNormal: ELBO-maximizing multivariate normal distribution
• draws::AbstractMatrix{<:Real}: draws from multivariate normal with size (dim, ndraws)
• fit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.
• draws_transformed: draws transformed to be draws from fit_distribution_transformed.
• fit_iteration::Int: Iteration at which ELBO estimate was maximized
• num_tries::Int: Number of tries until Pathfinder succeeded
• optim_solution::SciMLBase.OptimizationSolution: Solution object of optimization.
• optim_trace::Pathfinder.OptimizationTrace: container for optimization trace of points, log-density, and gradient. The first point is the initial point.
• fit_distributions::AbstractVector{Distributions.MvNormal}: Multivariate normal distributions for each point in optim_trace, where fit_distributions[fit_iteration + 1] == fit_distribution
• elbo_estimates::AbstractVector{<:Pathfinder.ELBOEstimate}: ELBO estimates for all but the first distribution in fit_distributions.
• num_bfgs_updates_rejected::Int: Number of times a BFGS update to the reconstructed inverse Hessian was rejected to keep the inverse Hessian positive definite.

Returns

• PathfinderResult

multipathfinder(ℓ, ndraws; kwargs...)
multipathfinder(fun::SciMLBase.OptimizationFunction, ndraws; kwargs...)

Run pathfinder multiple times to fit a multivariate normal mixture model.

For nruns=length(init), nruns parallel runs of pathfinder produce nruns multivariate normal approximations $q_k = q(\phi | \mu_k, \Sigma_k)$ of the posterior. These are combined to a mixture model $q$ with uniform weights.

$q$ is augmented with the component index to generate random samples, that is, elements $(k, \phi)$ are drawn from the augmented mixture model

\[\tilde{q}(\phi, k | \mu, \Sigma) = K^{-1} q(\phi | \mu_k, \Sigma_k),\]

where $k$ is a component index, and $K=$ nruns. These draws are then resampled with replacement. Discarding $k$ from the samples would reproduce draws from $q$.

If importance=true, then Pareto smoothed importance resampling is used, so that the resulting draws better approximate draws from the target distribution $p$ instead of $q$. This also prints a warning message if the importance weighted draws are unsuitable for approximating expectations with respect to $p$.

Arguments

• ℓ: an object, representing the log-density of the target distribution and its gradient, that implements the LogDensityProblems interface.
• fun::SciMLBase.OptimizationFunction: an optimization function that represents a negative log density with its gradient. It must have the necessary features (e.g. a Hessian function) for the chosen optimization algorithm. For details, see Optimization.jl: OptimizationFunction.
• ndraws::Int: number of approximate draws to return

Keywords

• init: iterator of length nruns of initial points of length dim from which each single-path Pathfinder run will begin. length(init) must be implemented. If init is not provided, nruns must be, and dim must be if fun provided.
• nruns::Int: number of runs of Pathfinder to perform. Ignored if init is provided.
• ndraws_per_run::Int: The number of draws to take for each component before resampling. Defaults to a number such that ndraws_per_run * nruns > ndraws.
• importance::Bool=true: Perform Pareto smoothed importance resampling of draws.
• rng::AbstractRNG=Random.GLOBAL_RNG: Pseudorandom number generator. It is recommended to use a parallelization-friendly PRNG like the default PRNG on Julia 1.7 and up.
• executor::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor that determines if and how to run the single-path runs in parallel. If a transducer for multi-threaded computation is selected, you must first verify that rng and the log density function are thread-safe.
• executor_per_run::Transducers.Executor=Transducers.SequentialEx(): Transducers.jl executor used within each run to parallelize PRNG calls. Defaults to no parallelization. See pathfinder for a description.
• kwargs... : Remaining keywords are forwarded to pathfinder.

Returns

• MultiPathfinderResult

MultiPathfinderResult

Container for results of multi-path Pathfinder.

Fields

• input: User-provided input object, e.g. either logp, (logp, ∇logp), optim_fun, optim_prob, or another object.
• optimizer: Optimizer used for maximizing the log-density
• rng: Pseudorandom number generator that was used for sampling
• optim_prob::SciMLBase.OptimizationProblem: Otimization problem used for optimization
• logp: Log-density function
• fit_distribution::Distributions.MixtureModel: uniformly-weighted mixture of ELBO- maximizing multivariate normal distributions from each run.
• draws::AbstractMatrix{<:Real}: draws from fit_distribution with size (dim, ndraws), potentially resampled using importance resampling to be closer to the target distribution.
• draw_component_ids::Vector{Int}: component id of each draw in draws.
• fit_distribution_transformed: fit_distribution transformed to the same space as the user-supplied target distribution. This is only different from fit_distribution when integrating with other packages, and its type depends on the type of input.
• draws_transformed: draws transformed to be draws from fit_distribution_transformed.
• pathfinder_results::Vector{<:PathfinderResult}: results of each single-path Pathfinder run.
• psis_result::Union{Nothing,<:PSIS.PSISResult}: If importance resampling was used, the result of Pareto-smoothed importance resampling. psis_result.pareto_shape also diagnoses whether draws can be used to compute estimates from the target distribution. See PSIS.PSISResult for details