Naive Bayes Classifier - Discriminant Function
In Naive Bayes, for the case of two classes, a discriminant function could be $$D(\boldsymbol{x}) = \frac{P(\boldsymbol{x}, c=1)}{P(\boldsymbol{x}, c=0)}$$ which can be anywhere in $[0, +\infty)$, and decides $c=1$ if $D(\boldsymbol{x})&gt;1$, $c=0$ otherwise, or it could be the logarithm of that value

$$d(\boldsymbol{x}) = \text{log}\frac{P(\boldsymbol{x}, c=1)}{P(\boldsymbol{x}, c=0)}=\text{log}P(\boldsymbol{x}, c=1)-\text{log}P(\boldsymbol{x}, c=0)$$
which can be anywhere in $(-\infty, +\infty)$ (handling zero probability as a special case), and decides $c=1$ if $d(\boldsymbol{x})&gt;0$, $c=0$ otherwise.

As a side note, $P(\boldsymbol{x}, c=k)$ in Naive Bayes is calculated as
$$P(\boldsymbol{x}, c=k)=P(c=k)\prod_{i=1}^{d}P(x_i|c=k)$$
or equivalently for log probabilities as
$$\text{log}P(\boldsymbol{x}, c=k)=\text{log}P(c=k) + \sum_{i=1}^{d}\text{log}P(x_i|c=k)$$