Continuous Bernoulli distribution

The continuous Bernoulli can be thought of as a continuous relaxation of the Bernoulli distribution, which is defined on the discrete set ${\displaystyle \{0,1\}}$ by the probability mass function:

${\displaystyle p(x)=p^{x}(1-p)^{1-x},}$

where ${\displaystyle p}$ is a scalar parameter between 0 and 1. Applying this same functional form on the continuous interval ${\displaystyle [0,1]}$ results in the continuous Bernoulli probability density function, up to a normalizing constant.

The Beta distribution has the density function:

${\displaystyle p(x)\propto x^{\alpha -1}(1-x)^{\beta -1},}$

which can be re-written as:

${\displaystyle p(x)\propto x_{1}^{\alpha _{1}-1}x_{2}^{\alpha _{2}-1},}$

where ${\displaystyle \alpha _{1},\alpha _{2}}$ are positive scalar parameters, and ${\displaystyle (x_{1},x_{2})}$ represents an arbitrary point inside the 1-simplex, ${\displaystyle \Delta ^{1}=\{(x_{1},x_{2}):x_{1}>0,x_{2}>0,x_{1}+x_{2}=1\}}$. Switching the role of the parameter and the argument in this density function, we obtain:

${\displaystyle p(x)\propto \alpha _{1}^{x_{1}}\alpha _{2}^{x_{2}}.}$

This family is only identifiable up to the linear constraint ${\displaystyle \alpha _{1}+\alpha _{2}=1}$, whence we obtain:

${\displaystyle p(x)\propto \lambda ^{x_{1}}(1-\lambda )^{x_{2}},}$

corresponding exactly to the continuous Bernoulli density.

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

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