loglikelihood(DM::DataModel, θ::AbstractVector) -> Real

Calculates the logarithm of the likelihood $L$, i.e. $\ell(\mathrm{data} \, | \, \theta) \coloneqq \mathrm{ln} \big( L(\mathrm{data} \, | \, \theta) \big)$ given a DataModel and a parameter configuration $\theta$.

MLE(DM::DataModel) -> Vector

Returns the parameter configuration $\theta_\text{MLE} \in \mathcal{M}$ which is estimated to have the highest likelihood of producing the observed data (under the assumption that the specified model captures the true relationship present in the data). For performance reasons, the maximum likelihood estimate is stored as a part of the DataModel type.

MLEuncert(DM::AbstractDataModel, mle::AbstractVector=MLE(DM), F::AbstractMatrix=FisherMetric(DM, mle))

Returns vector of type Measurements.Measurement where the parameter uncertainties are approximated via the diagonal of the inverse Fisher metric. That is, the stated uncertainties are a linearized symmetric approximation of the true parameter uncertainties around the MLE.

LogLikeMLE(DM::DataModel) -> Real

Returns the value of the log-likelihood $\ell$ when evaluated at the maximum likelihood estimate, i.e. $\ell(\mathrm{data} \, | \, \theta_\text{MLE})$. For performance reasons, this value is stored as a part of the DataModel type.

Score(DM::DataModel, θ::AbstractVector{<:Number}; ADmode::Val=Val(:ForwardDiff))

Calculates the gradient of the log-likelihood $\ell$ with respect to a set of parameters $\theta$. ADmode=Val(false) computes the Score using the Jacobian dmodel provided in DM, i.e. by having to separately evaluate both the model as well as dmodel. Other choices of ADmode directly compute the Score by differentiating the formula the log-likelihood, i.e. only one evaluation on a dual variable is performed.

FisherMetric(DM::DataModel, θ::AbstractVector{<:Number})

Computes the Fisher metric $g$ given a DataModel and a parameter configuration $\theta$ under the assumption that the likelihood $L(\mathrm{data} \, | \, \theta)$ is a multivariate normal distribution.

\[g_{ab}(\theta) \coloneqq -\int_{\mathcal{D}} \mathrm{d}^m y_{\mathrm{data}} \, L(y_{\mathrm{data}} \,|\, \theta) \, \frac{\partial^2 \, \mathrm{ln}(L)}{\partial \theta^a \, \partial \theta^b} = -\mathbb{E} \bigg( \frac{\partial^2 \, \mathrm{ln}(L)}{\partial \theta^a \, \partial \theta^b} \bigg)\]

GeometricDensity(DM::AbstractDataModel, θ::AbstractVector) -> Real

Computes the square root of the determinant of the Fisher metric $\sqrt{\mathrm{det}\big(g(\theta)\big)}$ at the point $\theta$.

ChristoffelSymbol(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
ChristoffelSymbol(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(1,2)$ Christoffel symbol $\Gamma$ at a point $\theta$ (i.e. the Christoffel symbol "of the second kind") through finite differencing of the Metric. Accurate to ≈ 3e-11. BigCalc=true increases accuracy through BigFloat calculation.

Riemann(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
Riemann(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(1,3)$ Riemann tensor by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

Ricci(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
Ricci(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(0,2)$ Ricci tensor by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

RicciScalar(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false) -> Real
RicciScalar(Metric::Function, θ::AbstractVector; BigCalc::Bool=false) -> Real

Calculates the Ricci scalar by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

GeodesicDistance(DM::DataModel, P::AbstractVector{<:Number}, Q::AbstractVector{<:Number}; tol::Real=1e-10)
GeodesicDistance(Metric::Function, P::AbstractVector{<:Number}, Q::AbstractVector{<:Number}; tol::Real=1e-10)

Computes the length of a geodesic connecting the points P and Q.

AIC(DM::DataModel, θ::AbstractVector) -> Real

Calculates the Akaike Information Criterion given a parameter configuration $\theta$ defined by $\mathrm{AIC} = 2 \, \mathrm{length}(\theta) -2 \, \ell(\mathrm{data} \, | \, \theta)$. Lower values for the AIC indicate that the associated model function is more likely to be correct. For linearly parametrized models and small sample sizes, it is advisable to instead use the AICc which is more accurate.

AICc(DM::DataModel, θ::AbstractVector) -> Real

Computes Akaike Information Criterion with an added correction term that prevents the AIC from selecting models with too many parameters (i.e. overfitting) in the case of small sample sizes. $\mathrm{AICc} = \mathrm{AIC} + \frac{2\mathrm{length}(\theta)^2 + 2 \mathrm{length}(\theta)}{N - \mathrm{length}(\theta) - 1}$ where $N$ is the number of data points. Whereas AIC constitutes a first order estimate of the information loss, the AICc constitutes a second order estimate. However, this particular correction term assumes that the model is linearly parametrized.

BIC(DM::DataModel, θ::AbstractVector) -> Real

Calculates the Bayesian Information Criterion given a parameter configuration $\theta$ defined by $\mathrm{BIC} = \mathrm{ln}(N) \cdot \mathrm{length}(\theta) -2 \, \ell(\mathrm{data} \, | \, \theta)$ where $N$ is the number of data points.

IsLinearParameter(DM::DataModel, MLE::AbstractVector) -> BitVector

Checks with respect to which parameters the model function model(x,θ) is linear and returns vector of booleans where true indicates linearity. This test is performed by comparing the Jacobians of the model for two random configurations $\theta_1, \theta_2 \in \mathcal{M}$ column by column.

ConfidenceRegionVolume(DM::AbstractDataModel, Confnum::Real; N::Int=Int(1e5), WE::Bool=true, Approx::Bool=false, kwargs...) -> Real

Computes coordinate-invariant volume of confidence region associated with level Confnum via Monte Carlo by integrating the geometric density factor. For likelihoods which are particularly expensive to evaluate, Approx=true can improve the performance by approximating the confidence region via polygons.

Pullback(DM::AbstractDataModel, ω::AbstractVector{<:Number}, θ::AbstractVector) -> Vector

Pull-back of a covector to the parameter manifold $T^*\mathcal{M} \longleftarrow T^*\mathcal{D}$.

Pullback(DM::DataModel, G::AbstractArray{<:Number,2}, θ::AbstractVector) -> Matrix

Pull-back of a (0,2)-tensor G to the parameter manifold.

Pushforward(DM::DataModel, X::AbstractVector, θ::AbstractVector) -> Vector

Calculates the push-forward of a vector X from the parameter manifold to the data space $T\mathcal{M} \longrightarrow T\mathcal{D}$.

ProfileBox(DM::AbstractDataModel, Fs::AbstractVector{<:AbstractInterpolation}, Confnum::Real=2.) -> HyperCube

Constructs HyperCube which bounds the confidence region associated with the confidence level Confnum from the interpolated likelihood profiles.