# Clarification in Anomaly Detection Algorithm

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

• Should be $p(X>x) < \epsilon$ or $p(X<x)<\epsilon$. Jul 21, 2019 at 13:02

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.

• Thanks. This explanation makes sense. Jul 23, 2019 at 9:57