Data Likelihoods

fenrir_data_loglik(
    prob::SciMLBase.AbstractODEProblem,
    alg::ProbNumDiffEq.AbstractEK,
    args...;
    observation_matrix,
    observation_noise_cov,
    data,
    kwargs...
)

Compute the Fenrir [3] approximate negative log-likelihood (NLL) of the data.

This is a convenience function that

1. Solves the ODE with a ProbNumDiffEq.EK1 of the specified order and with a diffusion as provided by the diffusion_var argument, and
2. Fits the ODE posterior to the data via Kalman filtering and thereby computes the log-likelihood of the data on the way.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops.

You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

• prob::SciMLBase.AbstractODEProblem: the initial value problem of interest
• alg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.
• data::NamedTuple{(:t, :u)}: the data to be fitted
• observation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix
• observation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance

Reference

• [3] Tronarp et al, "Fenrir: Physics-Enhanced Regression for Initial Value Problems", ICML (2022)

dalton_data_loglik(
    prob::SciMLBase.AbstractODEProblem,
    alg::ProbNumDiffEq.AbstractEK,
    args...;
    observation_matrix,
    observation_noise_cov,
    data,
    kwargs...
)

Compute the DALTON [10] approximate negative log-likelihood (NLL) of the data.

You can control the step-size behaviour of the solver as you would for a standard ODE solve, but additionally the solver always steps through the data.t locations by adding them to tstops. You can also choose steps adaptively by setting adaptive=true, but this is not well-tested so use at your own risk!

Arguments

• prob::SciMLBase.AbstractODEProblem: the initial value problem of interest
• alg::AbstractEK: the probabilistic ODE solver to be used; use EK1 for best results.
• data::NamedTuple{(:t, :u)}: the data to be fitted
• observation_matrix::Union{AbstractMatrix,UniformScaling}: the matrix which maps the ODE state to the measurements; typically a projection matrix
• observation_noise_cov::Union{Number,AbstractMatrix}: the scalar observation noise variance

Reference

• [10] Wu et al, "Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations", arXiv (2023)