rxmc.likelihood_model.UnknownNormalizationModel

class rxmc.likelihood_model.UnknownNormalizationModel

Bases: ParametricLikelihoodModel

A ParametricLikelihoodModel in which the (log of the) multiplicative factor of the model output is a free parameter. In this case, the covariance does not include systematic errors due to the normalization, as the data are explicitly re-normalized.

This corresponds to a statistical model:

\[y_i + \epsilon_i = \rho y_m(x_i, \alpha) + \epsilon_i + \dots\]

where $rho$ is a free parameter corresponding to the multiplicative normalization factor. This corresponds to the Kennedy & O’Hagan (2001) treatment of normalization of model output in in ‘Bayesian calibration of computer models’. Here the normalization is not associated with the data, but is a latent parameter which scales the model output to best fit the data.

This should not be confused with UnknownDataNormalizationModel, in which the normalization of the data is a free parameter.

In fact, the confusion between them is related to Peelle’s Pertinent Puzzle, or D’Agostin bias.

__init__()

Initializes the LikelihoodModel, optionally with a fractional uncorrelated error.

Methods

__init__()	Initializes the LikelihoodModel, optionally with a fractional uncorrelated error.
chi2(observation, ym, log_rho)	Calculate the generalised chi-squared statistic.
covariance(observation, ym, log_rho)	Returns the statistical covariance matrix:
log_likelihood(observation, ym, log_rho)	Returns the log_likelihood that ym reproduces y, given the covariance
residual(observation, ym, log_rho)	Returns the residual between the renormalized model prediction ym and observation.y

covariance(observation: Observation, ym: ndarray, log_rho: float)

Returns the statistical covariance matrix:

\[\Sigma_{ij}^{stat} = \sigma^2_{i}^{stat} \delta_{ij}\]

where $sigma^2_{i}^{stat}$ is the statistical variance of the i-th observation, (observation.statistical_covariance).

When the normalization is a free parameter, the systematic normalization contribution to the covariance is not included, as the data are explicitly re-normalized.

Parameters:

• observation (Observation) – The observation object containing the observed data.

• ym (np.ndarray) – Model prediction for the observation

• log_rho (float) – natural log of the multiplicative normalization factor.

residual(observation: Observation, ym: ndarray, log_rho: float)

Returns the residual between the renormalized model prediction ym and observation.y

Parameters:

• observation (Observation) – The observation object containing the observed data.

• ym (np.ndarray) – Model prediction for the observation.

• log_rho (float) – natural log of the multiplicative normalization factor.

Returns:

np.ndarray – Residual vector.

chi2(observation: Observation, ym: ndarray, log_rho: float)

Calculate the generalised chi-squared statistic. This is the square of the Mahalanobis distance between y and ym

Parameters:

• observation (Observation) – The observation object containing the observed data.

• ym (np.ndarray) – Model prediction for the observation.

• log_rho (float) – natural log of the multiplicative normalization factor.

Returns:

float – Chi-squared statistic.

log_likelihood(observation: Observation, ym: ndarray, log_rho: float)

Returns the log_likelihood that ym reproduces y, given the covariance

Parameters:

• ym (np.ndarray) – Model prediction for the observation.

• observation (Observation) – The observation object containing the observed data.

• log_rho (float) – natural log of the multiplicative normalization factor.

Returns:

float