University AI - Estimate Class-Conditional PDFs using MLPs

May we use MLP as an estimate of the class-conditional PDFs?

Since the output of an MLP can be seen as:

$$y_i(\underline{x})\simeq P({\omega }_i|\underline{x})=\frac{p(\underline{x}|{\omega }_i)P({\omega }_i)}{p(\underline{x})}$$

We can write:

$$\frac{y_i(\underline{x})}{P({\omega }_i)}\simeq \frac{p(\underline{x}|{\omega }_i)}{p(\underline{x})}$$

Which is knows as scaled likelihood.

• $p(\underline{x})$ is unknown, but can be estimated.
• Also $p(\underline{x})$ estimates are more robust than $p(\underline{x}|{\omega }_i)$ estimates, because we need to estimate only one estimate instead of $c$ other PDFs ($c$ : number of classes, ${\omega }_1,\dots ,{\omega }_c$ ), also with the same logic if we estimate only $p(\underline{x})$ we will have $c$ times more data.
• Also if $P({\omega }_i)$ changes over time (let’s say it assumes the new value $P^{\prime }({\omega }_i)$) , we can just reuse the same MLP, so no re-training necessary and use the following formula:

$$\begin{array}{rl}{P}^{\prime }({\omega }_i|\underline{x})& =\\ & =\frac{p(\underline{x}|{\omega }_i)}{p(\underline{x})}{P}^{\prime }({\omega }_i)\\ & \simeq \frac{y_i(\underline{x})}{P({\omega }_i)}{P}^{\prime }({\omega }_i)\end{array}$$

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