# What does the notation $\mathcal{N}(z; \mu, \sigma)$ stand for in statistics?

I know that the notation $$\mathcal{N}(\mu, \sigma)$$ stands for a normal distribution. But I'm reading the book "An Introduction to Variational Autoencoders" and in it, there is this notation: $$\mathcal{N}(z; 0, I)$$ What does it mean?

picture of the book:

• To understand the notation you have to be familiar with multivariate probability distributions. Are you familiar with it? – DuttaA Aug 23 at 18:27
• random normal variable $z$ with mean 0 and unit variance. – Brale Aug 23 at 18:47
• @Brale isn't it the probability of $N(0,I)$ at the point $z$? because the notation of what you said should be something like $Z\sim N(0,I)$ shouldn't it? – Peyman Aug 23 at 20:36
• @peyman The probability of a normal distribution at any specific point is 0. You are referring to the density function evaluated at that point (which is not a probability). – David Ireland Aug 23 at 21:15

It means that $$z$$ has a (multivariate) normal distribution with 0 mean and identity covariance matrix. This essentially means each individual element of the vector $$z$$ has a standard normal distribution.
• $P(z)$ means the density function of $z$, so when they say $P(z)=...$ they are saying $z$ has a certain distribution which follows on from the equals sign. Note my other comment that for a continuous distribution such as Gaussian, the probability of getting any exact point is 0, that is $\mathbb{P}(Z=z) = 0$ for all $z$. – David Ireland Aug 23 at 21:30
• @DuttaA I find that a lot of ML authors abuse notation. The problem here is that you're not defining a distribution over a random variable. I guess you would read it loosely as 'function $P(z)$ is equal to the normal distribution' which makes no sense. Maybe I am just being pedantic as obviously I knew what it should mean, but the correct notation would be that $Z \sim \text{N}(0, \textbf{I})$, i.e. $Z$ is a random variable that has a normal distribution with the given parameters. – David Ireland Aug 24 at 9:19
• @DuttaA No it isn't just when you are sampling data, it is how you should write it in general -- if you take any probability course from a maths or stats department you would see this notation used universally. If you want to make clear that some data has a distribution you write $X \sim \text{'the distribution'}$. If you are doing Bayesian statistics you don't assign a prior over $z$ by saying $p(z) = ...$, you write $z \sim ...$. What they are doing here in VAEs is assigning a Gaussian prior to $z$, thus all you need to write is $z \sim \text{N}(\mu, \sigma)$. – David Ireland Aug 24 at 12:29