fit.posterior¶

The posterior of the parameters of a fit: the Laplace approximation.

The Laplace approximation of the posterior is the multivariate normal around the estimate of a fit whose precision is the curvature of the negative log posterior at the estimate, in the space the optimizer searches, the parameter_scale of the fit (Tierney & Kadane 1986; Gelman et al. 2013, ch. 4.1):

precision = J' J / σ² + H_prior,   covariance = precision⁻¹

with the jacobian J of the residuals of the data at the estimate, the Gauss-Newton approximation of the curvature of the negative log-likelihood (exact for normal noise of a known scale and a model linear in the parameters; an approximation for a scale of the noise which is estimated and for Laplace noise), see sbmlsim.fit.fisher, and the Hessian of the negative log density of the priors H_prior, by central differences in the space of the optimizer. What the curvature contains is what the fit minimized, its ObjectiveType:

• posterior: the likelihood and the priors, around the maximum a posteriori estimate;
• likelihood: the likelihood with the uniform priors of the bounds, which have no curvature; priors of the parameters are no part of the objective and are left out;
• least_squares: the weighted least squares as the likelihood of normal noise of the scale σ² = 2 cost / (n - k), the posterior of the uniform priors, whose covariance is the covariance of the Fisher information.

On a logarithmic scale of the optimizer the normal of the logarithms is a log-normal of the values. The priors are the densities of the values, as the fit evaluated them, without the jacobian of the scale.

A direction which neither the data nor the priors constrain has no curvature, the posterior is the uniform prior of the bounds there, which a normal does not describe: posterior_laplace raises and names the parameters of such a direction. It warns where the approximation is questionable: a parameter at a bound, where the posterior is cut off, and a posterior whose interval reaches past a bound. These are the problems of a sampler of the posterior (#374).

A ParameterPosterior is stored as JSON (posterior.json next to the records of a fit, see sbmlsim.run), so an analysis of the posterior does not run the fit, and is the source of the parameters of a propagation (ParameterPosterior.uncertainty_model), of the designs of sbmlsim.simulation.sampling (posterior_parameters, or distributions and rank_correlation for lhs and random) and of a sensitivity analysis.

Tierney L, Kadane JB (1986). Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81:82-86. Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2013). Bayesian Data Analysis, 3^rd ed. CRC Press.

ParameterPosterior(
    opid,
    sid,
    pids,
    values,
    scales,
    covariance,
    units,
    targets,
    lower_bounds,
    upper_bounds,
    objective,
    method="laplace",
    n=0,
    sigma2=1.0,
    step=0.0,
    problems=list(),
)

k

mean

standard_deviations

correlation_matrix

correlation

summary_df

intervals(level=0.95)

distributions(targets=None)

normal_correlation(targets=None)

rank_correlation()

uncertainty_model(targets=None, **kwargs)

to_dict()

from_dict(d)

write(path)

read(path)

prior_hessian(
    log_density, z, to_linear, lower, upper, step=PRIOR_STEP
)

posterior_laplace(
    problem,
    settings,
    parameter_set,
    *,
    fisher=None,
    step=None,
    rank_tolerance=DEFAULT_RANK_TOLERANCE,
)