# Recursive Bayesian Estimation, $p(C_k|x)$ as likelihood

I''ve been struggeling with this problem for the last couple of days.

The main goal is to use the probabilistic classification output $p(C_k|x)$, from for example a logistic regression, to enhance and make a overall classification performance more robust, over a given set of measurements.

In other words: I want to make a recursive bayesian estimation. Now, I'm wondering if there is any change to do this.

Assuming I would just count $p(C_k|x) > 0.5$ as a success, $y=1$ (interpreting as a Bernoulli distribution). I could easily set up a $Beta(\alpha_0,\beta_0)$ and start updating, as the $Beta$ is a conjugate prior, with \begin{align*} \alpha_1 &= \alpha_0 + y, \\ \beta_1 &= \beta_0 + 1 - y. \end{align*}

So this works fine and is mathematically correct, but I would really like to incorporate the probabilistic output, as it provides valuable additional information. When directly assuming \begin{equation*} y = p(C_k|x), \end{equation*} and the update $\alpha$ and $\beta$, I get exactly the kind of behavior I'm trying to achieve. Unfortunately, I guess this violates the assumption of being $Bernoulli$ distributed and hence $Beta$ would not be the right conjugate prior, or is ist?

I could set up a $Normal$ prior and interpret the output of the logistic regression also as a $Normal$ distribution (somehow) but the $Beta$ fits perfectly my needs as it is bounded to $0...1$

• It's not clear what you are asking. What do you mean by "make a overall classification performance more robust"? The procedure that you describe is simply modelling the distribution of outputs from the logistic regression. Why is that useful? – Tom Minka Oct 31 '14 at 18:35