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BIC

Bayesian Information Criterion (BIC).

In statistics, the Bayesian Information Criterion (BIC) 1, or Schwarz Information Criterion (SIC), is a criterion for model selection among a finite set of models; the model with the highest BIC is preferred. It is based, in part, on the likelihood function and is closely related to the Akaike Information Criterion (AIC).

Let

  • k being the number of clusters,

  • \(n_i\) being the number of points within each cluster, \(n_1 + n_2 + ... + n_k = n\),

  • \(d\) being the dimension of the clustering problem.

Then, the variance of the clustering solution will be calculated as

\[ \hat{\sigma}^2 = \frac{1}{(n - m) \times d} \sum_{i = 1}^n \lVert x_i - c_j \rVert^2. \]

The maximum likelihood function, used in the BIC version of River, would be

\[ \hat{l}(D) = \sum_{i = 1}^k n_i \log(n_i) - n \log n - \frac{n_i \times d}{2} \times \log(2 \pi \hat{\sigma}^2) - \frac{(n_i - 1) \times d}{2}, \]

and the BIC will then be calculated as

\[ BIC = \hat{l}(D) - 0.5 \times k \times log(n) \times (d+1). \]

Using the previously mentioned maximum likelihood function, the higher the BIC value, the better the clustering solution is. Moreover, the BIC calculated will always be less than 0 2.

Attributes

  • bigger_is_better

    Indicates if a high value is better than a low one or not.

Examples

>>> from river import cluster
>>> from river import stream
>>> from river import metrics

>>> X = [
...     [1, 2],
...     [1, 4],
...     [1, 0],
...     [4, 2],
...     [4, 4],
...     [4, 0],
...     [-2, 2],
...     [-2, 4],
...     [-2, 0]
... ]

>>> k_means = cluster.KMeans(n_clusters=3, halflife=0.4, sigma=3, seed=0)
>>> metric = metrics.cluster.BIC()

>>> for x, _ in stream.iter_array(X):
...     k_means = k_means.learn_one(x)
...     y_pred = k_means.predict_one(x)
...     metric = metric.update(x, y_pred, k_means.centers)

>>> metric
BIC: -30.060416

Methods

get

Return the current value of the metric.

revert

Revert the metric.

Parameters

  • x
  • y_pred
  • centers
  • sample_weight – defaults to 1.0
update

Update the metric.

Parameters

  • x
  • y_pred
  • centers
  • sample_weight – defaults to 1.0
works_with

Indicates whether or not a metric can work with a given model.

Parameters

  • model (river.base.estimator.Estimator)

References

  1. Wikipedia contributors. (2020, December 14). Bayesian information criterion. In Wikipedia, The Free Encyclopedia, from https://en.wikipedia.org/w/index.php?title=Bayesian_information_criterion&oldid=994127616 ↩

  2. BIC Notes, https://github.com/bobhancock/goxmeans/blob/master/doc/BIC_notes.pdf ↩