p(x) in an anomaly detection system gives values greater 1

Question

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Probability distribution value exceeding 1 is OK?

I followed the machine learning course by Andrew Ng and decided to implement an anomaly detection system for one of my problems. This is how I have proceeded so far:

1. I have about ($m=$)100k feature vectors which represent the normal behavior
2. The values are scaled to the interval $[-1; 1]$ and the mean $\mu$ is calculated and substracted from all the values.
3. Now the scaled feature vectors (data points) are inserted into a matrix so that each row represents one data point
4. Next the covariance matrix is computed by the following equation $\frac{1}{m}*M^{T}M$
5. In order to classify a new data point, I apply the same scaling as before and substract the mean. Let's call the resulting vector $x$
6. This vector is plugged into the following formular: $p(x) = \frac{1}{(2\pi)^{\frac{n}{2}}|\Sigma|^{\frac{1}{2}}} e^{-0.5 x^{T}\Sigma^{-1}x}$

I assumed that $p(x) \in [0;1]$, but instead I get also values $> 1000$.

Do you have advice where I should go from here? Thanks.

Answer 1

The $p(x)$ you are using is the normal probability density. It is not a probability and can be greater than $1$. Bayes classification is determined by finding the class with the highest probability density when costs for the error types are the same.