Continuous Bernoulli distribution

In probability theory, statistics, and machine learning, the continuous Bernoulli distribution[1][2][3] is a family of continuous probability distributions parameterized by a single shape parameter , defined on the unit interval , by:

Continuous Bernoulli distribution
Probability density function
Probability density function of the continuous Bernoulli distribution

The continuous Bernoulli distribution arises in deep learning and computer vision, specifically in the context of variational autoencoders,[4][5] for modeling the pixel intensities of natural images. As such, it defines a proper probabilistic counterpart for the commonly used binary cross entropy loss, which is often applied to continuous, -valued data.[6][7][8][9] This practice amounts to ignoring the normalizing constant of the continuous Bernoulli distribution, since the binary cross entropy loss only defines a true log-likelihood for discrete, -valued data.

The continuous Bernoulli also defines an exponential family of distributions. Writing for the natural parameter, the density can be rewritten in canonical form: .

Related distributionsEdit

Bernoulli distributionEdit

The continuous Bernoulli can be thought of as a continuous relaxation of the Bernoulli distribution, which is defined on the discrete set   by the probability mass function:


where   is a scalar parameter between 0 and 1. Applying this same functional form on the continuous interval   results in the continuous Bernoulli probability density function, up to a normalizing constant.

Beta distributionEdit

The Beta distribution has the density function:


which can be re-written as:


where   are positive scalar parameters, and   represents an arbitrary point inside the 1-simplex,  . Switching the role of the parameter and the argument in this density function, we obtain:


This family is only identifiable up to the linear constraint  , whence we obtain:


corresponding exactly to the continuous Bernoulli density.

Exponential distributionEdit

An exponential distribution restricted to the unit interval is equivalent to a continuous Bernoulli distribution with appropriate parameter.

Continuous categorical distributionEdit

The multivariate generalization of the continuous Bernoulli is called the continuous-categorical.[10]


  1. ^ Loaiza-Ganem, G., & Cunningham, J. P. (2019). The continuous Bernoulli: fixing a pervasive error in variational autoencoders. In Advances in Neural Information Processing Systems (pp. 13266-13276).
  2. ^ PyTorch Distributions.
  3. ^ Tensorflow Probability.
  4. ^ Kingma, D. P., & Welling, M. (2013). Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114.
  5. ^ Kingma, D. P., & Welling, M. (2014, April). Stochastic gradient VB and the variational auto-encoder. In Second International Conference on Learning Representations, ICLR (Vol. 19).
  6. ^ Larsen, A. B. L., Sønderby, S. K., Larochelle, H., & Winther, O. (2016, June). Autoencoding beyond pixels using a learned similarity metric. In International conference on machine learning (pp. 1558-1566).
  7. ^ Jiang, Z., Zheng, Y., Tan, H., Tang, B., & Zhou, H. (2017, August). Variational deep embedding: an unsupervised and generative approach to clustering. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (pp. 1965-1972).
  8. ^ PyTorch VAE tutorial:
  9. ^ Keras VAE tutorial:
  10. ^ Gordon-Rodriguez, E., Loaiza-Ganem, G., & Cunningham, J. P. (2020). The continuous categorical: a novel simplex-valued exponential family. In 36th International Conference on Machine Learning, ICML 2020. International Machine Learning Society (IMLS).