According to the paper One-class document classification via Neural Networks of Manevitz and Yousef it seems to be possible to construct a one-class Naive Bayes classifier, even without a standard deviation.
I cite the relevant passage where the authors mention how to implement the core of the classifier:
We calculate $p(d|E)$ as the product of $p(w|E)$ for all words in the
dictionary that appear in the document $d$. Each of the $p(w|E)$ is
estimated independently using the formula:
$p(w|E) = \dfrac{n_w + 1}{n + |dictionary|}$,
where $n_w$ is the number of times word $w$ occurs in $E$, and $n$ is the total number of words in $E$.
We calculate a threshold $\delta$ by the minimum over all examples in
$E$, of the value $p(d|E)$ for each document in the set of examples.
Then we experiment with values $\lambda\cdot\delta$ for $0 < \lambda \leq 1$ as in the previous algorithms using $F_1$ to find the optimal
threshold for acceptance. That is, given a new document $d$, we accept
it if the calculated value $p(d|E)$ is larger than the determined
$\lambda\cdot\delta$. For this classifier algorithm we store $\delta$
and $\lambda$.
A more detailed picture of the algorithm is explained in the doctoral dissertation Characteristic Concept Representations of Piew Datta.
