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# bayesian_info_criterion_lsq

astropy.stats.bayesian_info_criterion_lsq(ssr, n_params, n_samples)

  Computes the Bayesian Information Criterion (BIC) assuming that the observations come from a Gaussian distribution.

  In this case, BIC is given as

  \[\mathrm{BIC} = n\ln\left(\dfrac{\mathrm{SSR}}{n}\right) + k\ln(n)\]

  in which \(n\) is the sample size, \(k\) is the number of free parameters and \(\mathrm{SSR}\) stands for the sum of squared residuals between model and data.

  This is applicable, for instance, when the parameters of a model are estimated using the least squares statistic. See [1] and [2].

  Parameters:
    ssr `python:float`

      Sum of squared residuals (SSR) between model and data.

    n_params `python:int`

      Number of free parameters of the model, i.e., dimension of the parameter space.

    n_samples `python:int`

      Number of observations.

  Returns:
    bic `python:float`

  References

  [1]

  Wikipedia. Bayesian Information Criterion. <https://en.wikipedia.org/wiki/Bayesian_information_criterion>

  [2]

  Origin Lab. Comparing Two Fitting Functions. <https://www.originlab.com/doc/Origin-Help/PostFit-CompareFitFunc>

  [3]

  Astropy Models and Fitting <https://docs.astropy.org/en/stable/modeling>

  Examples

  Consider the simple 1-D fitting example presented in the Astropy modeling webpage [3]. There, two models (Box and Gaussian) were fitted to a source flux using the least squares statistic. However, the fittings themselves do not tell much about which model better represents this hypothetical source. Therefore, we are going to apply to BIC in order to decide in favor of a model.

      >>> import numpy as np
      >>> from astropy.modeling import models, fitting
      >>> from astropy.stats.info_theory import bayesian_info_criterion_lsq
      >>> # Generate fake data
      >>> np.random.seed(0)
      >>> x = np.linspace(-5., 5., 200)
      >>> y = 3 * np.exp(-0.5 * (x - 1.3)**2 / 0.8**2)
      >>> y += np.random.normal(0., 0.2, x.shape)
      >>> # Fit the data using a Box model.
      >>> # Bounds are not really needed but included here to demonstrate usage.
      >>> t_init = models.Trapezoid1D(amplitude=1., x_0=0., width=1., slope=0.5,
      ...                             bounds={"x_0": (-5., 5.)})
      >>> fit_t = fitting.LevMarLSQFitter()
      >>> t = fit_t(t_init, x, y)
      >>> # Fit the data using a Gaussian
      >>> g_init = models.Gaussian1D(amplitude=1., mean=0, stddev=1.)
      >>> fit_g = fitting.LevMarLSQFitter()
      >>> g = fit_g(g_init, x, y)
      >>> # Compute the mean squared errors
      >>> ssr_t = np.sum((t(x) - y)*(t(x) - y))
      >>> ssr_g = np.sum((g(x) - y)*(g(x) - y))
      >>> # Compute the bics
      >>> bic_t = bayesian_info_criterion_lsq(ssr_t, 4, x.shape[0])
      >>> bic_g = bayesian_info_criterion_lsq(ssr_g, 3, x.shape[0])
      >>> bic_t - bic_g
      30.644474706065466

  Hence, there is a very strong evidence that the Gaussian model has a significantly better representation of the data than the Box model. This is, obviously, expected since the true model is Gaussian.