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

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 distributions

Bernoulli distribution

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 distribution

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 distribution

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

Continuous categorical distribution

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. Archived 2020-11-25 at the Wayback Machine
  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).