Likelihood Approximations

LiLit implements two likelihood approximations for CMB power spectrum analysis: the exact likelihood and the Gaussian likelihood. This section details how they work and when to use each.

Exact Likelihood

The exact likelihood is based on the Hamimeche & Lewis (2008) formulation. For a single field, the log-likelihood is:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)\left[\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}-\log\left(\frac{C_{\ell}^{\rm obs}}{C_{\ell}^{\rm th}}\right)-1\right]\]

For multiple fields (N fields), the formula generalizes to:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)\left[\text{Tr}\left(\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right) - \log\left|\mathcal{C}_{\rm obs}\mathcal{C}^{-1}_{\rm th}\right| - N\right]\]

where \(\mathcal{C}_{\rm obs}\) and \(\mathcal{C}_{\rm th}\) are the covariance matrices containing the observed and theoretical power spectra.

Using the Exact Likelihood

from lilit import LiLit

# Create likelihood with exact approximation
likelihood = LiLit(
    name="example_exact",
    fields=["t", "e", "b"],
    like="exact",
    nl_file="/path/to/noise.pkl",
    lmax=[2500, 2000, 500],
    lmin=[2, 2, 2],
    fsky=[0.7, 0.7, 0.6],
    debug=False
)

Gaussian Likelihood

The Gaussian approximation assumes that the power spectra are Gaussian distributed around their theoretical values:

\[\log\mathcal{L} = -\frac{1}{2}\sum_{\ell}(2\ell+1)\left[\frac{(C_{\ell}^{\rm obs} - C_{\ell}^{\rm th})^2}{\sigma^{2}_{\ell}}\right]\]

For a single field, the variance is:

\[\sigma^{2}_{\ell} = \frac{2}{(2\ell+1)f_{\rm sky}}(C_{\ell}^{\rm obs})^2\]

For multiple fields, the data vector is formed from the upper triangular part of the covariance matrix, and the covariance between different power spectrum elements is:

\[\text{Cov}^{\rm ABCD}_{\ell} = \frac{1}{(2\ell+1)f_{\rm sky}^{AB}f_{\rm sky}^{CD}}\left( \sqrt{f_{\rm sky}^{AC}f_{\rm sky}^{BD}}C_\ell^{AC}C_\ell^{BD} + \sqrt{f_{\rm sky}^{AD}f_{\rm sky}^{BC}}C_\ell^{AD}C_\ell^{BC} \right)\]

Using the Gaussian Likelihood

from lilit import LiLit

# Create likelihood with Gaussian approximation
likelihood = LiLit(
    name="example_gaussian",
    fields=["t", "e", "b"],
    like="gaussian",
    nl_file="/path/to/noise.pkl",
    lmax=[2500, 2000, 500],
    lmin=[2, 2, 2],
    fsky=[0.7, 0.7, 0.6],
    debug=False
)

Field-Specific Multipole Ranges

LiLit supports different multipole ranges for different fields. The handling differs between the two likelihood approximations:

Exact Likelihood: - Cross-correlation ranges are determined by the intersection of individual field ranges - Entries outside the valid range for each field are set to zero in the covariance matrix - Singular matrices are handled by removing null diagonal entries and corresponding rows/columns

Gaussian Likelihood: - Cross-correlation ranges use the geometric mean: \(\ell_{\rm max}^{XY} = \sqrt{\ell_{\rm max}^{XX}\ell_{\rm max}^{YY}}\) - A mask is applied to exclude invalid multipoles before matrix inversion

Excluding Specific Probes

You can exclude specific cross-correlations from the analysis:

# Exclude T-B cross-correlation
likelihood = LiLit(
    name="example_no_tb",
    fields=["t", "e", "b"],
    like="exact",
    excluded_probes=["tb"],  # Exclude T-B cross-correlation
    lmax=[2500, 2000, 500],
    fsky=[0.7, 0.7, 0.6]
)

Sky Coverage Effects

LiLit approximates partial sky coverage effects by rescaling the effective number of modes:

Single \(f_{\rm sky}\) value: Applied uniformly to all fields
Multiple values: Geometric mean is computed and applied
Cross-correlations: \(f_{\rm sky}^{XY} = \sqrt{f_{\rm sky}^{XX}f_{\rm sky}^{YY}}\)

Note: This approach does not account for mode coupling introduced by the sky cut.

Debug Mode

Enable debug mode to check the likelihood setup:

likelihood = LiLit(
    name="debug_example",
    fields=["t", "e"],
    like="exact",
    debug=True,  # Enable debugging output
    lmax=[2500, 2000],
    fsky=[0.7, 0.7]
)

When to Use Each Approximation

Use Exact Likelihood when: - You need the most accurate likelihood for parameter inference - Working with low signal-to-noise data - Precise error estimation is critical

Use Gaussian Likelihood when: - Computational speed is important - Signal-to-noise is high (Gaussian approximation is more accurate) - Working with large-scale surveys or many multipoles

Performance Considerations: The exact likelihood requires matrix operations at each multipole, making it computationally more expensive than the Gaussian approximation, especially for many fields or high \(\ell_{\rm max}\) values.