Cosmological likelihoods (likelihoods)

Construct likelihoods from Spherical Fourier coefficients and Cartesian power spectrum multipoles of random fields for cosmological parameter inference.

Probability distributions

complex_normal_pdf(data_vector, ...[, ...])	Compute the complex normal probability density function or its natural logarithm given the zero-centred data vector and its covariance matrix.
multivariate_normal_pdf(data_vector, ...[, ...])	Compute the multivariate normal probability density function or its natural logarithm given the data vector, its mean vector and covariance matrix.
modified_student_pdf(data_vector, ...[, ret_log])	Compute the multivariate modified Student probability density function or its natural logarithm given the data vector, its mean vector and covariance matrix.

Likelihood inference

spherical_covariance(pivot, spherical_model, ...)	Compute the parametrised covariance matrix of spherical Fourier coefficients.
cartesian_moments(pivot, orders, cartesian_model)	Compute the parametrised mean and covariance of Cartesian power spectrum multipoles.
LogLikelihood([spherical_data, ...])	Construct the logarithmic likelihood function from cosmological data.

harmonia.reader.likelihoods.chi_square(data_vector, covariance_matrix)

Calculate chi-square from zero-centred data vector and its covariance matrix.

Parameters:

• data_vector (complex, array_like) – Zero-centred data vector.

• covariance_matrix (complex, array_like) – Covariance matrix.

Returns:

chi_sq – Chi-square value.

Return type:

float numpy.ndarray

harmonia.reader.likelihoods.complex_normal_pdf(data_vector, covariance_matrix, ret_log=True, downscale=None)

Compute the complex normal probability density function or its natural logarithm given the zero-centred data vector and its covariance matrix.

Parameters:

• data_vector (complex, array_like) – Zero-centred data vector.

• covariance_matrix (complex, array_like) – Covariance matrix.

• ret_log (bool, optional) – If True (default), return logarithmic probability density.

• downscale (float or None, optional) – If not None (default), the data vector and covariance matrix are simultaneous downscaled to avoid numerical issue.

Returns:

(Logarithmic) probability density.

Return type:

float, array_like

harmonia.reader.likelihoods.multivariate_normal_pdf(data_vector, expectation_vector, covariance_matrix, ret_log=True)

Compute the multivariate normal probability density function or its natural logarithm given the data vector, its mean vector and covariance matrix.

Parameters:

• data_vector (float, array_like) – Data vector.

• expectation_vector (float, array_like) – Mean vector.

• covariance_matrix (float, array_like) – Covariance matrix.

• ret_log (bool, optional) – If True (default), return logarithmic probability density.

Returns:

density – Logarithmic probability density value.

Return type:

float, array_like

harmonia.reader.likelihoods.modified_student_pdf(data_vector, expectation_vector, covariance_matrix, degree, ret_log=True)

Compute the multivariate modified Student probability density function or its natural logarithm given the data vector, its mean vector and covariance matrix.

Parameters:

• data_vector (float, array_like) – Data vector.

• expectation_vector (float, array_like) – Mean vector.

• covariance_matrix (float, array_like) – Covariance matrix.

• degree (int) – The degree number. This could be the number of empirical covariance matrices used to obtain the estimated covariance_matrix.

• ret_log (bool, optional) – If True (default), return logarithmic probability density.

Returns:

(Logarithmic) probability density value.

Return type:

float, array_like

exception harmonia.reader.likelihoods.LikelihoodWarning

Likelihood evaluation warning.

harmonia.reader.likelihoods.spherical_covariance(pivot, spherical_model, **kwargs)

Compute the parametrised covariance matrix of spherical Fourier coefficients.

Parameters:

• pivot ({‘natural’, ‘spectral’}) – Pivot order for vectorisation.

• spherical_model (SphericalCorrelator) – Spherical correlator base model.

• **kwargs – Parameters (other than pivot) to be passed to correlator_matrix() of spherical_correlator.

Returns:

covariance_matrix – Covariance matrix.

Return type:

complex numpy.ndarray

See also

SphericalCorrelator

harmonia.reader.likelihoods.cartesian_moments(pivot, orders, cartesian_model, covariance_estimator=None, mode_counts=None, **kwargs)

Compute the parametrised mean and covariance of Cartesian power spectrum multipoles.

Parameters:

• pivot ({‘order’, ‘wavenumber’}) – Pivot order for vectorisation.

• orders (list of int) – Orders of the power spectrum multipoles.

• cartesian_model (CartesianMultipoles) – Cartesian power multipoles base model.

• covariance_estimator (CovarianceEstimator or None, optional) – Cartesian power multipole covariance estimator. Its wavenumbers must match wavenumbers associated with cartesian_model. If None, no correlation between power spectrum multipoles is assumed but mode_counts must be provided for calculating the power spectrum variance.

• mode_counts (int, array_like or None, optional) – Number of independent modes for each power spectrum measurement (default is None) used to calculate the power spectrum variance. Ignored if covariance_estimator is provided.

• **kwargs – Parameters (other than orders) to be passed to convolved_power_multipoles() of cartesian_model.

Returns:

• expectation (float numpy.ndarray) – Power spectrum expectation at the specified wavenumbers.

• covariance (float numpy.ndarray) – Power spectrum variance at the specified wavenumbers.

class harmonia.reader.likelihoods.LogLikelihood(spherical_data=None, cartesian_data=None, covariance_estimator=None, mode_counts=None, base_spherical_model=None, base_cartesian_model=None, spherical_pivot='natural', cartesian_pivot='order', nbar=None, contrast=None, tracer_p=1.0, comm=None)

Construct the logarithmic likelihood function from cosmological data.

Parameters:

• spherical_data (SphericalArray or None, optional) – Spherical Fourier coefficient data (default is None).

• cartesian_data (CartesianArray or None, optional) – Spherical Fourier coefficient data (default is None).

• covariance_estimator (CovarianceEstimator or None, optional) – Cartesian multipole covariance estimator (default is None).

• mode_counts (int, array_like or None, optional) – Number of independent modes for each Cartesian data point (default is None) as an alternative to covariance_estimator. Ignored if covariance_estimator is provided.

• base_spherical_model (SphericalCorrelator or None, optional) – Baseline spherical correlator model (default is None).

• base_cartesian_model (CartesianMultipoles or None, optional) – Baseline Cartesian multipole model (default is None).

• spherical_pivot ({‘natural’, ‘spectral’}, optional) – Pivot order for spherical map data vectorisation (default is ‘natural’).

• cartesian_pivot ({‘order’, ‘wavenumber’}, optional) – Pivot order for Cartesian map data vectorisation (default is ‘order’).

• nbar (float or None, optional) – Mean particle number density (in cubic \(h\)/Mpc). If None (default), shot noise is neglected.

• contrast (float or None, optional) – If not None (default), this adds additional shot noise level 1 / (contrast * nbar) due to a FKP-style random catalogue.

• tracer_p (float, optional) – Tracer-dependent parameter for bias modulation by f_nl (default is 1.).

• comm (mpi4py.MPI.Comm or None, optional) – MPI communicator (default is None).

Variables:

• attrs (dict) – Directory holding input parameters not corresponding to any of the following attributes.

• spherical_data (SphericalArray or None) – Spherical Fourier coefficient data.

• cartesian_data (CartesianArray or None) – Spherical Fourier coefficient data.

• covariance_estimator (CovarianceEstimator or None) – Cartesian multipole covariance estimator.

• base_spherical_model (SphericalCorrelator or None) – Baseline spherical correlator model.

• base_cartesian_model (CartesianMultipoles or None) – Baseline Cartesian multipole model.

spherical_map_likelihood(b_1, f_nl, exclude_degrees=(), compression_matrix=None, **kwargs)

Evaluate the spherical map logarithmic likelihood.

Parameters:

• b_1 (float) – Scale-independent linear bias.

• f_nl (float or None) – Local primordial non-Gaussianity.

• exclude_degrees (tuple of int, optional) – If not empty (default), modes whose spherical degree match one of its elements are removed from the likelihood.

• compression_matrix (numpy.ndarray or None, optional) – If not None (default), both the data vector and the model covariance matrix are processed for data compression. This must be compatible with exclude_degrees, i.e. it accounts for elements removed from the data vector and covariance matrix by exclude_degrees.

• **kwargs – Additional parameters to be passed to correlator_matrix() of base_spherical_model.

Returns:

log_likelihood – Logarithmic likelihood.

Return type:

float

See also

generate_compression_matrix

spherical_covariance

correlator_matrix()

cartesian_map_likelihood(b_1, f_nl, orders=None, num_samples=None, **kwargs)

Evaluate the Cartesian map logarithmic likelihood.

Parameters:

• b_1 (float) – Scale-independent linear bias of the tracer particles.

• f_nl (float or None) – Local primordial non-Gaussianity.

• orders (list of int or None, optional) – Orders of the power spectrum multipoles. If None (default), only the monopole is used.

• num_samples (int or None, optional) – If None (default), the normal distribution is used without correction for covariance estimation uncertainty; otherwise it is passed as degree to modified_student_pdf() for covariance estimation uncertainty correction [1].

• **kwargs – Additional parameters to be passed to convolved_power_multipoles() of base_cartesian_model.

Returns:

log_likelihood – Logarithmic likelihood.

Return type:

float

See also

cartesian_moments

convolved_power_multipoles()

[1]

Sellentin E. & Heavens A. F., 2016. MNRAS 456(1), L132–L136. [arXiv: 1511.05969]