Suppose we are in the clustering problem set up. Soft clustering will use Expectation Maximisation with the Gaussian distribution. Here we fix a $\sigma^2$ and then we have

$$ \mathbb{P}[X=x_i \vert \mu = \mu_j] = \exp\left[- \frac{1}{2} \sigma^2 (x_i - \mu_j)^2 \right]. $$

We will use a mean to optimise $\mu_j$ at each step.

Correctness

• It will have all the downsides of Expectation Maximisation but we can use restarts to over come them.