Clarification in Anomaly Detection Algorithm

Question

I am referring to Prof Andrew Ng Coursera ML notes (Week 9). He says that to identify outliers we first model the training data and then fit a Gaussian distribution with probability density $ p(X; \mu $, $\sigma^2 $ ). He then suggests that we can classify a new example $x$ as an outlier if "probability": $ p(x) <\epsilon $, where $\epsilon $ is a hyperparameter.

However, it doesn't make sense to me that how is he using term density and probability interchangeably without even explicitly mentioning it. I find this puzzling because if we assume $X$ to be a continuous random variable then $p(X=x)=0$ $\forall x$. Please help me in understanding what is actually going on in this algorithm at a finer level.

https://math.stackexchange.com/questions/521575/difference-between-probability-and-probability-density

Answer 1

Putting a threshold on the density of probability equals to putting a limit on the probability in the sense that you say

$$ \int_{-\infty}^{-\epsilon}p(y)dy + \int_{\epsilon}^{\infty}p(y)dy \equiv \alpha $$

and then define the rule (for a univariate Gaussian): if

$$ \int_{-\infty}^{-x}p(y)dy + \int_{x}^{\infty}p(y)dy < \alpha $$

then $x$ is an outlier. And this is the idea behind more sophisticated approaches like one-class SVMs for outlier detection. You say any sample from sample space corresponding to regions with very (according to some threshold) low probability are considered outliers.