Probability

#machine learning
Probability theory itself is intuitive, but can be confusing when it is overly formalized. For now, this section provides some basic terms and formulas.

The best interactive exploration I know is Seeing Theory.

Term Describe Formalize
Experiment We toss a fair coin twice /
Outcome Head or Tail $\{h, t \}$
Sample Space aka Event Space | State Space $\Omega = \{tt, ht, th, hh \}$
Event of Interest Let's say: Getting head exactly once $E= \{ ht, th\}$
Random Variable Worst naming ever: It is neither random nor a variable! It's a function (lookup table) that maps the sample space Ω to T $X( (h, h) )= 2 \\ X( (t, h) )=1 \\ X( (h,t) )= 1 \\ X( ( t,t ) )=0$
Target Space Number of times we get $h$ when flipping the coin $\mathcal{T} = \{0, 1, 2 \}$
Probability Notation $P(X=1) means the probability for all events where$X((event)) = 1P(X=1) = P((h,t)) \\ \cup P((t,h)) \\ = P(h) * P(t) \\ + P(t) * P(h)= 0.5$Assert [English] Formalize Nothing > 100% probable :$P(\Omega) = 1$and$\forall S \subseteq \mathcal{T} $, we know$P(X(S)) \in [0, 1]\$

Code

Often Code is more tangible than symbols. If you made a snippet for your favorite language, I'd be happy to add it here.

[...more to come...]