Law of Total Probability

In robotics literature, we observe endless number of equations written in probabilities. I’d like to point out few ones which I found typically useful to remember. You will find all these in any book with probability theory.

\begin{aligned}p(x) &= \sum_y p(x, y) \\ &= p(x|y)\cdot p(y)\end{aligned} or \begin{aligned}p(x) &= \int p(x, y) dy \\ &= \int p(x|y) \cdot p(y) dy \end{aligned} depending on whether the random variable x is a continuous or discrete variable.


One can see that the law of total probability is a variant of marginalization, which states following:
p(x) = \sum_y p(x, y) or p(x) = \int p(x, y) dy for discrete or continuous random variable x, respectively.

Product Rule

One will also get to use/see the product rule a lot.
\begin{aligned}p(x,y) &= p(x|y) \cdot p(y) \\ &= p(y|x) \cdot p(x) \end{aligned}

Note that the joint distribution of p(x,y) can be expressed in two different equations as shown above. And whenever you see a conditional distribution, remember that it can also be expressed differently using Bayes rule.


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