Machine Learning MT23, Covariance and correlation
Flashcards
For a random variable $\pmb x \in \mathbb R^D$, what is the covariance matrix $\text{cov}(\pmb x)$ and what is $(\text{cov}(\pmb x)) _ {ij}$?
\[\mathbb E \left[ (\pmb x - \mathbb E[\pmb x])(\pmb x - \mathbb E[\pmb x])^T \right]\]
and
\[(\text{cov}(\pmb x)) _ {ij} = \text{cov}(X _ i, X _ j)\]Covariance depends on the scale of variables. Can you define the correlation $\text{corr}(X, Y)$, which is normalised between $\pm 1$?
\[\text{corr}(X, Y) = \frac{\text{cov}(X, Y)}{\sqrt{\text{var}(X) \text{var}(Y)}\,}\]
Suppose $\pmb \theta$ represents the parameters of some distribution with density function $p$. Can you define the likelihood of observing $(x _ 1, \ldots, x _ n)$, i.e. the probability of observing the data with parameter $\theta$?
\[\prod^N _ {i=1} p(x _ i \mid \theta)\]