Task Set-Up

In this Chapter, we will work through a very simple task and then see how this relates to normalizing flows.

Suppose that the random variable $x$ follows a $\mathcal{N}(\mu, 1)$

$$ \begin{equation} p(x) = \frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{(x - \mu)^2}{2}\right\} \end{equation} $$

and that we want to generate samples from it by first sampling from a standard normal distribution and then shifting such samples accordingly

$$ \begin{align} z^{(i)} &\sim \mathcal{N}(0, 1) \newline \newline x^{(i)} &= z^{(i)} + \mu. \end{align} $$

In order to apply this transformation $g(z) = z + \mu$ we need to know the parameter $\mu$. We can find such parameter value from i.i.d. data samples $x^{(1)}, \ldots, x^{(n)} \sim \mathcal{N}(\mu, 1)$ using Normalizing Flows. Here is how we will proceed:

1. Initialize our mean randomly, for instance to $\mu=-1.0$ (keeping $\sigma=1.0$).
2. Get $n$ i.i.d. samples from the standard normal distribution $z^{(1)}, \ldots, z^{(n)}\sim \mathcal{N}(0, 1)$.
3. Use the samples to compute an estimates of the gradients of the log-pdf for $x$ with respect to $\mu$.
4. Use the gradient estimates to improve your mean estimate.