Likelihood Theory ================= LiLit implements multiple likelihood approximations for CMB analysis. This section provides detailed formulations and implementation notes. Overview -------- LiLit supports both single-field and multi-field likelihood calculations with the following approximations: - **Exact likelihood**: Based on Hamimeche & Lewis (2008) - **Gaussian likelihood**: Standard Gaussian approximation with proper multi-field covariance - **Correlated Gaussian likelihood**: Accounts for multipole correlations (single-field only, under development) Each approximation handles field-specific multipole ranges (:math:`\ell_{\rm min}`, :math:`\ell_{\rm max}`) and sky fractions (:math:`f_{\rm sky}`) appropriately. Exact Likelihood ---------------- The exact likelihood approximation follows Hamimeche & Lewis (2008). Single-field Case ~~~~~~~~~~~~~~~~~ For a single field, the log-likelihood is: .. math:: \log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}\left[\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}-\log\left(\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}\right)-1\right] Multi-field Case ~~~~~~~~~~~~~~~~ For :math:`N` fields, the formula becomes: .. math:: \log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}^{\rm eff}\left[\text{Tr}\left(\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right) - \log\left|\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right| - N\right] where the covariance matrix has the structure: .. math:: \mathcal{C}_{\rm obs} = \left(\begin{array}{ccc} C_{\ell}^{XX} & C_{\ell}^{XY} & C_{\ell}^{XZ}\\ C_{\ell}^{YX} & C_{\ell}^{YY} & C_{\ell}^{YZ}\\ C_{\ell}^{ZX} & C_{\ell}^{ZY} & C_{\ell}^{ZZ} \end{array}\right) Each entry includes signal and noise contributions: .. math:: C_{\ell}^{XX} = C_{\ell}^{\rm CMB} + C_{\ell}^{\rm FGs} + N_{\ell}^{X} + \ldots Implementation Details ~~~~~~~~~~~~~~~~~~~~~~ **Multipole Ranges**: The multipole range for cross-correlations between two fields is determined by the intersection of their individual ranges. **Sky Fraction**: For multiple :math:`f_{\rm sky}` values, an effective value is computed as the geometric mean: :math:`f_{\rm sky}^{\rm eff} = \sqrt[N]{\prod_i f_{\rm sky}^i}`. **Singular Matrices**: When multipole cuts or excluded probes make the covariance matrix singular, LiLit automatically identifies and removes null diagonal entries along with their corresponding rows and columns. **Excluded Probes**: Specific cross-correlations can be excluded (e.g., ``excluded_probes = ["xz"]``) by setting the corresponding covariance entries to zero. Gaussian Likelihood ------------------- The Gaussian approximation assumes normally distributed power spectra with known covariance. Single-field Case ~~~~~~~~~~~~~~~~~ .. math:: \log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)f_{\rm sky}\left[\frac{\left(C_{\ell}^{\rm obs} - C_{\ell}^{\rm th}\right)^2}{\sigma^{2}_{\ell}}\right] where the variance is: .. math:: \sigma^{2}_{\ell} = \frac{2}{(2\ell+1)f_{\rm sky}}\left(C_{\ell}^{\rm obs}\right)^2 Multi-field Case ~~~~~~~~~~~~~~~~ The data vector is formed from the upper triangular part of the covariance matrix: .. math:: X_\ell = \left(C_{\ell}^{XX}, C_{\ell}^{XY}, C_{\ell}^{XZ}, C_{\ell}^{YY}, C_{\ell}^{YZ}, C_{\ell}^{ZZ}\right) The covariance matrix for this data vector is: .. math:: \text{Cov}^{\rm ABCD}_{\ell} = \frac{1}{(2\ell+1)f_{\rm sky}^{AB}f_{\rm sky}^{CD}}\left( \sqrt{f_{\rm sky}^{AC}f_{\rm sky}^{BD}}C_\ell^{AC}C_\ell^{BD} + \sqrt{f_{\rm sky}^{AD}f_{\rm sky}^{BC}}C_\ell^{AD}C_\ell^{BC} \right) where cross-correlation sky fractions are: :math:`f_{\rm sky}^{XY} = \sqrt{f_{\rm sky}^{XX}f_{\rm sky}^{YY}}`. The likelihood is then: .. math:: \log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)\left[X_\ell \cdot \text{Cov}^{-1}_{\ell} \cdot X_\ell^{\rm T}\right] Implementation Differences from Exact Case ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ **Multipole Ranges**: Cross-correlation ranges use geometric mean: :math:`\ell_{\rm max}^{XY} = \sqrt{\ell_{\rm max}^{XX}\ell_{\rm max}^{YY}}`. **Masking**: The covariance matrix is computed for the full range, then masked entries are removed before inversion. The data vector is masked consistently. **Sky Fractions**: Each field retains its individual :math:`f_{\rm sky}` factor, unlike the exact case which uses an effective value. Correlated Gaussian Likelihood ------------------------------ This approximation accounts for correlations between different multipoles (currently single-field only). Formulation ~~~~~~~~~~~ .. math:: \log\mathcal{L} = -\frac{1}{2}\left[\left(\vec{C^{\rm obs}} - \vec{C^{\rm th}}\right) \cdot \text{Ext}^{-1} \cdot \left(\vec{C^{\rm obs}} - \vec{C^{\rm th}}\right)^{\rm T}\right] where :math:`\vec{C^{\rm obs}}` and :math:`\vec{C^{\rm th}}` are vectors over the multipole range, and :math:`\text{Ext}` is an externally provided covariance matrix. Requirements ~~~~~~~~~~~~ - External covariance matrix must exclude :math:`\ell = 0, 1` - Covariance matrix should already account for :math:`f_{\rm sky}` effects - Multi-field extension is under development Sky Cut Approximations ---------------------- All likelihood implementations make approximations regarding sky cuts: 1. **Mode Counting**: The factor :math:`(2\ell+1)f_{\rm sky}` approximates the reduction in available modes due to masking 2. **No Mode Coupling**: Correlations between different multipoles induced by the mask are neglected 3. **Effective Sky Fraction**: Multi-field cases use geometric averaging of individual sky fractions These approximations are valid when: - Sky cuts are not too severe (:math:`f_{\rm sky} \gtrsim 0.3`) - Mask boundaries are not too complex - Cross-correlation regions have substantial overlap For more accurate treatment of sky cuts, external covariance matrices accounting for mode coupling should be used with the correlated Gaussian likelihood. References ---------- - Hamimeche, S. & Lewis, A., 2008, *Likelihood analysis of CMB temperature and polarization power spectra*, `arXiv:0801.0554 `_ - Campeti, P. et al., 2020, *Measuring the spectrum of primordial gravitational waves with CMB, PTA and Laser Interferometers*, `arXiv:2007.04241 `_ - Allys, E. et al., 2022, *Probing cosmic inflation with the LiteBIRD cosmic microwave background polarization survey*, `arXiv:2202.02773 `_