Conventions

This page defines the mathematical notation and conventions used throughout DerivKit, particularly in ForecastKit. These definitions establish the signs, normalizations, and reference points used in Fisher forecasts, DALI expansions, and systematic bias calculations.

Parameters and Observables

We denote the model parameters by \(\theta\) and the model predictions by \(\nu(\theta)\). The fiducial parameter point is \(\theta_0\), and parameter displacements are defined as

\[\delta^a = \theta^a - \theta_0^a.\]

Parameter indices are denoted by Latin letters \(a,b,c,\ldots\), while observable indices are denoted by \(i,j,k,\ldots\). Repeated indices are summed unless stated otherwise.

Model derivatives are written as

\[\nu_{,a} = \left.\frac{\partial\nu}{\partial\theta^a}\right|_{\theta_0},\]

\[\nu_{,ab} = \left.\frac{\partial^2\nu} {\partial\theta^a\partial\theta^b}\right|_{\theta_0}.\]

Higher-order derivatives follow the same notation.

Likelihood and Posterior

For Gaussian data with parameter-independent covariance \(C\), we define

\[\chi^2(\theta) = [d-\nu(\theta)]^{\mathrm T} C^{-1} [d-\nu(\theta)].\]

The log-likelihood is

\[\log L(\theta) = -\frac{1}{2}\chi^2(\theta) + \mathrm{const}.\]

Given a prior density \(\pi(\theta)\), the log-posterior is

\[\log p(\theta\mid d) = \log L(\theta) + \log\pi(\theta) + \mathrm{const}.\]

Thus, positive contributions to \(\chi^2\) enter the log-likelihood with a negative factor of \(1/2\).

The negative log-posterior is

\[\Phi(\theta) = -\log p(\theta\mid d).\]

Maximizing the log-posterior is equivalent to minimizing \(\Phi\).

Forecast Tensors

ForecastKit constructs Fisher and DALI tensors from derivatives of the model predictions, evaluated at \(\theta_0\).

The Fisher matrix is

\[F_{ab} = \nu_{,a}^{\mathrm T}C^{-1}\nu_{,b}.\]

The doublet DALI tensors are

\[D^{(1)}_{abc} = \nu_{,ab}^{\mathrm T}C^{-1}\nu_{,c},\]

\[D^{(2)}_{abcd} = \nu_{,ab}^{\mathrm T}C^{-1}\nu_{,cd}.\]

The triplet DALI tensors are

\[T^{(1)}_{abcd} = \nu_{,abc}^{\mathrm T}C^{-1}\nu_{,d},\]

\[T^{(2)}_{abcde} = \nu_{,abc}^{\mathrm T}C^{-1}\nu_{,de},\]

\[T^{(3)}_{abcdef} = \nu_{,abc}^{\mathrm T}C^{-1}\nu_{,def}.\]

Tensor definitions do not include Taylor factorial coefficients. These coefficients are applied when evaluating the expansion.

Tensor contractions are understood to be symmetrized over the parameter indices where appropriate.

DALI Expansion

For expected data at the fiducial model, define the change in chi-squared relative to \(\theta_0\) as

\[\Delta\chi^2(\theta) = \chi^2(\theta)-\chi^2(\theta_0).\]

The doublet DALI approximation is

\[\Delta\chi^2_{\mathrm DALI} = F_{ab}\delta^a\delta^b + D^{(1)}_{abc}\delta^a\delta^b\delta^c + \frac{1}{4} D^{(2)}_{abcd}\delta^a\delta^b\delta^c\delta^d.\]

Consequently,

\[\log L_{\mathrm DALI}(\theta) = \mathrm{const} -\frac{1}{2}F_{ab}\delta^a\delta^b -\frac{1}{2}D^{(1)}_{abc}\delta^a\delta^b\delta^c -\frac{1}{8}D^{(2)}_{abcd}\delta^a\delta^b\delta^c\delta^d.\]

The negative signs arise from the definition of the Gaussian log-likelihood, not from the tensor definitions.

DALI truncates the Taylor expansion of the model predictions before constructing the squared residual. It is therefore not equivalent to truncating a Taylor series of the log-likelihood at the same polynomial degree.

The Fisher, doublet, triplet, and quadruplet approximations retain model derivatives through first, second, third, and fourth order, respectively.

Systematic Mismatches and Bias Tensors

A systematic data mismatch is defined as

\[\Delta\nu = \langle d\rangle-\nu(\theta_0).\]

This convention always represents expected data minus the model prediction at the fiducial parameters.

The corresponding bias tensors are

\[b_{a_1\ldots a_n} = \Delta\nu^{\mathrm T} C^{-1} \nu_{,a_1\ldots a_n}.\]

The first-order bias vector is

\[b_a = \Delta\nu^{\mathrm T}C^{-1}\nu_{,a}.\]

With this sign convention, the Fisher parameter bias is

\[\Delta\theta_{\mathrm F}^a = (F^{-1})^{ab}b_b.\]

The mismatch contribution to the log-likelihood is

\[B(\delta) = b_a\delta^a + \frac{1}{2}b_{ab}\delta^a\delta^b + \frac{1}{3!}b_{abc}\delta^a\delta^b\delta^c + \cdots.\]

Reversing the sign of \(\Delta\nu\) reverses the first-order Fisher bias. Higher-order DALI responses need not reverse in the same way.

Parameter Displacements and Systematic Shifts

A parameter displacement is measured relative to the fiducial point:

\[\Delta\theta^{\mathrm{disp}}(A) = \theta(A)-\theta_0.\]

A systematic shift is measured relative to the corresponding estimator in the absence of the mismatch:

\[\Delta\theta^{\mathrm{sys}}(A) = \theta(A)-\theta(0).\]

These quantities are not necessarily equal.

For a posterior mean, the systematic shift is

\[\Delta\bar{\theta}^{\mathrm{sys}}(A) = \bar{\theta}(A)-\bar{\theta}(0).\]

In a non-Gaussian posterior, the unbiased mean need not coincide with the fiducial parameters:

\[\bar{\theta}(0)\neq\theta_0.\]

The same distinction applies to the posterior maximum when the unbiased MAP differs from \(\theta_0\).

MAP and Posterior Mean

The maximum a posteriori estimate is

\[\hat{\theta} = \operatorname*{arg\,max}_{\theta} p(\theta\mid d).\]

The posterior mean is

\[\bar{\theta}^a = \int \theta^a p(\theta\mid d) \,\mathrm{d}^{N_{\mathrm{par}}}\theta.\]

These estimators generally differ for non-Gaussian posteriors. ForecastKit therefore distinguishes MAP shifts from posterior mean shifts.

Unless explicitly stated otherwise, a reported systematic shift refers to the change in the specified estimator relative to its unbiased value.

Scope of the Conventions

The Fisher and DALI tensor expressions above assume a Gaussian likelihood with parameter-independent covariance.

ForecastKit also supports generalized Fisher formalisms for parameter-dependent covariance and other likelihood models. Their additional terms and assumptions are documented in the corresponding method descriptions.

Priors are included separately from the likelihood expansion unless explicitly incorporated into the forecasting calculation.