Probability Tips

• The Probability Density Functions (PDF) describes the relative likelihood for a random variable to take on a given value
• joint distribution representing the probability that the random variable X takes on the value x and that Y takes on the value y i.e. P(x,y) = P(x).P(y) (if independent)
• conditional probability that describes the probability that the random variable X takes on the value x conditioned on the knowledge that Y for sure takes y. i.e. P(X|Y)
• Theorem of Total probability builds on the above
• Bayes Theorem builds on the above as well

Bayes Theorem simplest case

$\mathbf { P(A|B) = \frac{P(A \cap B)}{P(B)} }$

Read as - Probability of A given B is equal to Probability of A & B divided by probability of B.

You can combine the above with this formula derivation

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

$P(A \cap B) = P(A) * P(B|A)$

And turn it into

$\mathbf {P(A|B) = \frac{P(A) * P(B|A)}{P(B)} }$

which can be used to calculate posterior probability (probability based on prior condition)

Theory of Total Probability

$p(a) = \int_{b} p(a|b).p(b)db \text{ for continous probabilities}$

$p(a) = \sum_{b} p(a|b).p(b) \text{ for discrete probabilities}$

You can also re-write the latter as

$\mathbf {p(a) = p(a|b).p(b) + p(a| \lnot b).p(\lnot b) }$

The above was in case of only two values of b (b and ) but you get the gist.

Combining Bayes Theorem and Total Probability

$\mathbf {P(A|B) = \frac{P(A) * P(B|A)}{p(B|A).p(A) + p(B| \lnot A).p(\lnot A) } }$

Useful links

• Columbia Slides - summarizes localization and probability theory, probability part helpful
• Siraj Rawal does a good job summarizing prob theory
• 3Blue1Brown - Bayes Theorem
• Bayes Theorem Simplest Case