Fisher and DALI forecasts

DSF provides the model predictions and covariance needed for derivative-based forecasts of the excess surface density, \(\Delta\Sigma(R)\). Fisher and DALI calculations are performed using DerivKit.

Fisher forecasts

Let \(\boldsymbol{\theta}\) denote the model parameters, \(\boldsymbol{\nu}(\boldsymbol{\theta})\) the stacked \(\Delta\Sigma\) data vector, and \(C\) its covariance. For a Gaussian likelihood with parameter-independent covariance, the Fisher matrix is

\[F_{ab} = \frac{\partial \boldsymbol{\nu}^{\mathrm T}}{\partial \theta_a} C^{-1} \frac{\partial \boldsymbol{\nu}}{\partial \theta_b}.\]

The inverse Fisher matrix approximates the parameter covariance in the local Gaussian forecast, with marginalized uncertainties

\[\sigma(\theta_a) = \sqrt{(F^{-1})_{aa}}.\]

Independent Gaussian prior information can be included by adding its precision matrix to the likelihood Fisher matrix.

DALI forecasts

The Fisher approximation describes a Gaussian parameter distribution near a fiducial point. DALI (Derivative Approximation for LIkelihoods) extends this approach by retaining higher-order derivatives of the model predictions. Writing \(\boldsymbol{\delta}=\boldsymbol{\theta}-\boldsymbol{\theta}_0\), the model is expanded as

\[\boldsymbol{\nu}(\boldsymbol{\theta}) \simeq \boldsymbol{\nu}(\boldsymbol{\theta}_0) + \boldsymbol{\nu}_{,a}\,\delta^a + \frac{1}{2}\boldsymbol{\nu}_{,ab}\,\delta^a\delta^b + \cdots.\]

Inserting this expansion into the Gaussian likelihood yields higher-order corrections to the Fisher approximation, allowing curved and non-elliptical parameter contours. Like Fisher, DALI is a local approximation whose accuracy depends on the model and the region of parameter space considered.

Using DSF

DeltaSigmaForecastBuilder prepares the tomographic data vector and matching covariance. Once configured, the builder provides the inputs for both methods:

forecast = builder.prepare()
kit = builder.forecastkit()

fisher = kit.fisher()
dali = kit.dali(forecast_order=3)

The current DSF forecast builder uses a fiducial, fixed gm covariance; these calculations therefore use derivatives of the data vector rather than covariance derivatives.

For the full mathematical formalism, derivative options, priors, posterior sampling, and worked examples, see the DerivKit forecasting documentation.