LikelihoodResult Class

The LikelihoodResult class stores and analyzes the results of likelihood computations, providing methods for best-fit extraction, confidence intervals, and marginalization.

Overview

LikelihoodResult provides:

  • Storage of chi-squared and log-likelihood values across parameter grid

  • Best-fit parameter extraction from likelihood surface

  • Confidence interval computation via profile likelihood

  • Parameter marginalization for 1D posteriors

  • Result persistence and loading

Class Documentation

class picslike.likelihood_result.LikelihoodResult(parameter_grid, chi_squared_values, log_likelihood_values)[source]

Bases: object

Container for pixel-based likelihood computation results.

This class stores and manages results from likelihood analysis, providing methods for result analysis, best-fit parameter extraction, confidence interval computation, and result persistence.

Parameters:

  • parameter_grid (ParameterGrid) – Parameter grid used in the likelihood computation.

  • chi_squared_values (numpy.ndarray) – Chi-squared values for each parameter grid point.

  • log_likelihood_values (numpy.ndarray) – Log-likelihood values for each parameter grid point.

parameter_grid

Parameter grid used in the computation.

Type:

ParameterGrid

chi_squared_values

Array of chi-squared values.

Type:

numpy.ndarray

log_likelihood_values

Array of log-likelihood values.

Type:

numpy.ndarray

likelihood_values

Array of likelihood values (exponential of log-likelihood).

Type:

numpy.ndarray

Examples

Basic result analysis:

>>> result = LikelihoodResult(grid, chi2_values, log_like_values)
>>> best_fit = result.get_best_fit()
>>> confidence_intervals = result.get_confidence_intervals()
>>> result.save("analysis_results.pkl")

Statistical analysis:

>>> # Get 68% confidence intervals
>>> intervals_68 = result.get_confidence_intervals(confidence_level=0.68)
>>> # Get marginalized likelihood for specific parameter
>>> marg_like = result.get_marginalized_likelihood('omega_b')

Notes

The class assumes that likelihood values correspond to parameter grid points in the same order. It provides both chi-squared and likelihood perspectives on the same underlying computation.

__init__(parameter_grid, chi_squared_values, log_likelihood_values)[source]

Initialize likelihood result container.

Parameters:

  • parameter_grid (ParameterGrid) – Parameter grid used in the likelihood computation.

  • chi_squared_values (numpy.ndarray) – Chi-squared values for each parameter grid point.

  • log_likelihood_values (numpy.ndarray) – Log-likelihood values for each parameter grid point.

Raises:

ValueError – If array dimensions are inconsistent with parameter grid.

get_best_fit()[source]

Get best-fit parameter values.

Returns:

best_fit_params – Dictionary mapping parameter names to their best-fit values.

Return type:

dict[str, Any]

Notes

Returns the parameter combination with the highest likelihood (lowest chi-squared value). In case of ties, returns the first occurrence.

get_chi_squared_minimum()[source]

Get minimum chi-squared value.

Returns:

chi_squared_min – Minimum chi-squared value in the analysis.

Return type:

float

get_maximum_likelihood()[source]

Get maximum likelihood value.

Returns:

likelihood_max – Maximum likelihood value in the analysis.

Return type:

float

get_confidence_intervals(confidence_level=0.68)[source]

Compute confidence intervals for each parameter.

Parameters:

confidence_level (float, optional) – Confidence level for interval computation (default: 0.68 for 1σ).

Returns:

confidence_intervals – Dictionary mapping parameter names to (lower, upper) confidence bounds.

Return type:

dict[str, tuple[float, float]]

Notes

Computes confidence intervals by finding parameter ranges that contain the specified fraction of the total likelihood. Uses marginalized likelihood distributions for each parameter.

get_marginalized_likelihood(parameter_name)[source]

Get marginalized likelihood for a specific parameter.

Parameters:

parameter_name (str) – Name of the parameter to marginalize over.

Returns:

marginalized_likelihood – Marginalized likelihood distribution for the parameter.

Return type:

numpy.ndarray

Raises:

ValueError – If parameter name is not found in the grid.

Notes

Computes the marginalized likelihood by integrating (summing) over all other parameters in the grid. The result is normalized to unit area under the curve.

save(output_path)[source]

Save likelihood results to file.

Parameters:

output_path (str or Path) – Path where results should be saved.

Return type:

None

Notes

Saves the complete LikelihoodResult object using pickle format. The saved file can be loaded later for further analysis or visualization.

classmethod load(input_path)[source]

Load likelihood results from file.

Parameters:

input_path (str or Path) – Path to saved results file.

Returns:

result – Loaded likelihood result object.

Return type:

LikelihoodResult

Raises:

FileNotFoundError – If the input file does not exist.

get_summary_statistics()[source]

Get summary statistics for the likelihood analysis.

Returns:

summary – Dictionary containing summary statistics including best-fit values, chi-squared minimum, and confidence intervals.

Return type:

dict[str, Any]

Notes

Provides a comprehensive summary of the analysis results suitable for reporting and quick interpretation of the results.

__repr__()[source]

Return string representation of the result object.

Return type:

str

Key Methods

Initialization

LikelihoodResult.__init__(parameter_grid, chi_squared_values, log_likelihood_values)[source]

Initialize likelihood result container.

Parameters:

  • parameter_grid (ParameterGrid) – Parameter grid used in the likelihood computation.

  • chi_squared_values (numpy.ndarray) – Chi-squared values for each parameter grid point.

  • log_likelihood_values (numpy.ndarray) – Log-likelihood values for each parameter grid point.

Raises:

ValueError – If array dimensions are inconsistent with parameter grid.

Best-Fit Analysis

LikelihoodResult.get_best_fit()[source]

Get best-fit parameter values.

Returns:

best_fit_params – Dictionary mapping parameter names to their best-fit values.

Return type:

dict[str, Any]

Notes

Returns the parameter combination with the highest likelihood (lowest chi-squared value). In case of ties, returns the first occurrence.

LikelihoodResult.get_chi_squared_minimum()[source]

Get minimum chi-squared value.

Returns:

chi_squared_min – Minimum chi-squared value in the analysis.

Return type:

float

LikelihoodResult.get_maximum_likelihood()[source]

Get maximum likelihood value.

Returns:

likelihood_max – Maximum likelihood value in the analysis.

Return type:

float

Statistical Analysis

LikelihoodResult.get_confidence_intervals(confidence_level=0.68)[source]

Compute confidence intervals for each parameter.

Parameters:

confidence_level (float, optional) – Confidence level for interval computation (default: 0.68 for 1σ).

Returns:

confidence_intervals – Dictionary mapping parameter names to (lower, upper) confidence bounds.

Return type:

dict[str, tuple[float, float]]

Notes

Computes confidence intervals by finding parameter ranges that contain the specified fraction of the total likelihood. Uses marginalized likelihood distributions for each parameter.

LikelihoodResult.get_marginalized_likelihood(parameter_name)[source]

Get marginalized likelihood for a specific parameter.

Parameters:

parameter_name (str) – Name of the parameter to marginalize over.

Returns:

marginalized_likelihood – Marginalized likelihood distribution for the parameter.

Return type:

numpy.ndarray

Raises:

ValueError – If parameter name is not found in the grid.

Notes

Computes the marginalized likelihood by integrating (summing) over all other parameters in the grid. The result is normalized to unit area under the curve.

LikelihoodResult.get_summary_statistics()[source]

Get summary statistics for the likelihood analysis.

Returns:

summary – Dictionary containing summary statistics including best-fit values, chi-squared minimum, and confidence intervals.

Return type:

dict[str, Any]

Notes

Provides a comprehensive summary of the analysis results suitable for reporting and quick interpretation of the results.

Persistence

LikelihoodResult.save(output_path)[source]

Save likelihood results to file.

Parameters:

output_path (str or Path) – Path where results should be saved.

Return type:

None

Notes

Saves the complete LikelihoodResult object using pickle format. The saved file can be loaded later for further analysis or visualization.

classmethod LikelihoodResult.load(input_path)[source]

Load likelihood results from file.

Parameters:

input_path (str or Path) – Path to saved results file.

Returns:

result – Loaded likelihood result object.

Return type:

LikelihoodResult

Raises:

FileNotFoundError – If the input file does not exist.

Usage Examples

Creating a LikelihoodResult

import numpy as np
from picslike import LikelihoodResult, ParameterGrid

# Assuming you have a parameter grid and computed chi-squared values
chi2_values = np.array([...])  # Chi-squared at each grid point
log_like_values = -0.5 * chi2_values

result = LikelihoodResult(
    parameter_grid=grid,
    chi_squared_values=chi2_values,
    log_likelihood_values=log_like_values,
)

Extracting Best-Fit Parameters

# Get best-fit parameter values
best_fit = result.get_best_fit()
print(f"Best-fit omega_b: {best_fit['omega_b']:.5f}")
print(f"Best-fit omega_c: {best_fit['omega_c']:.4f}")

# Get minimum chi-squared
chi2_min = result.get_chi_squared_minimum()
print(f"Minimum chi-squared: {chi2_min:.2f}")

Computing Confidence Intervals

# 68% confidence intervals (1-sigma)
intervals_68 = result.get_confidence_intervals(0.68)
for param, (lower, upper) in intervals_68.items():
    print(f"{param}: [{lower:.5f}, {upper:.5f}]")

# 95% confidence intervals (2-sigma)
intervals_95 = result.get_confidence_intervals(0.95)

Marginalizing Over Parameters

# Get 1D marginalized likelihood for omega_b
marg_omega_b = result.get_marginalized_likelihood("omega_b")

# Plot marginalized likelihood
import matplotlib.pyplot as plt

omega_b_values = grid.parameter_ranges["omega_b"]
plt.plot(omega_b_values, marg_omega_b)
plt.xlabel(r"$\Omega_b$")
plt.ylabel("Marginalized Likelihood")
plt.show()

Summary Statistics

# Get comprehensive summary
summary = result.get_summary_statistics()

print(f"Number of parameters: {summary['n_parameters']}")
print(f"Total grid points: {summary['n_grid_points']}")
print(f"Chi-squared minimum: {summary['chi2_min']:.2f}")
print(f"Best-fit values: {summary['best_fit']}")

Saving and Loading Results

# Save results to file
result.save("output/likelihood_results.npz")

# Load results from file
loaded_result = LikelihoodResult.load("output/likelihood_results.npz")

# Verify loaded correctly
assert loaded_result.get_chi_squared_minimum() == result.get_chi_squared_minimum()

Result Format

The LikelihoodResult stores:

  • parameter_grid: The ParameterGrid used for computation

  • chi_squared_values: 1D array of chi-squared values indexed by grid point

  • log_likelihood_values: 1D array of log-likelihood values

File Format

Results are saved in NumPy’s compressed archive format (.npz):

# Contents of saved file
{
    "chi_squared_values": np.ndarray,
    "log_likelihood_values": np.ndarray,
    "parameter_names": list,
    "parameter_ranges": dict,
    # ... additional metadata
}

Statistical Notes

Confidence Intervals

Confidence intervals are computed using the profile likelihood method:

\[\Delta \chi^2 = \chi^2(\theta) - \chi^2_{min}\]

For a single parameter, the confidence levels correspond to:

  • 68% (1σ): \(\Delta \chi^2 < 1.0\)

  • 95% (2σ): \(\Delta \chi^2 < 4.0\)

  • 99% (3σ): \(\Delta \chi^2 < 9.0\)

Marginalization

Marginalized likelihoods are computed by integrating over nuisance parameters:

\[\mathcal{L}_{marg}(\theta_i) = \int \mathcal{L}(\theta_i, \theta_j) d\theta_j\]

For grid-based evaluation, this is approximated as a sum over grid points.

See Also

  • picslike.picslike.PICSLike : Main likelihood computation class

  • picslike.parameter_grid.ParameterGrid : Parameter space management