How to derive bernoulli deviance

Question

I found the deviance definition in https://en.wikipedia.org/wiki/Deviance_(statistics) and the one-observation bernoulli deviance in Scikit Binomial Deviance Loss Function. $$ \text{bernoulliDeviance}_i = -2 * (y_i\log(p_i)+(1-y_i)\log(1-p_i)). $$ How can I derive such formula by the wiki definition of deviance?

Answer 1

Definitions:

1. $p \equiv$ your $p_i$, the bernoulli parameter, such that $P(1) = p$

2. $\hat{\theta}_0 \equiv p$, the general parameter vector is, in this case, equal to the bernoulli parameter

With that, we can write the function with the two possible states:

• if $y_i = 0$, $p(y|p) = (1-p)$
• if $y_i = 1$, $p(y|p) = p$

We can combine those using something akin to a delta function by attaching a prefactor that is $1$ when for the corresponding state of $y_i$:

$f(y_i) = (1-y_i) log(1-p) + y_i log(p)$

You can check that this function gives the same values as the conditional function at 0 and 1.