Probability Theory for State Estimation

• $X$ is the covariance matrix
• $m_x,m_y$ are the means

The conditional probability can be expressed in terms of the joint probability distribution $p(x,y)$ and the marginal probability $p(y)$:

• $p(x\mid y)=\frac{p(x,y)}{p(y)}$:To find the probability of $x$ given $y$, look at how often $(x,y)$ happens compared to how often $y$ happens overall
• $p(x,y)$ = joint probability that $X=x$ and $Y=y$ happen together.
• $p(y)$ = probability that $Y=y$, regardless of $X$.

The law of total probability says that the marginal probability of X, denoted $p(x)$, can be recovered from a joint distribution by summing or integrating out the other variable $Y$.