Mixture Distributions

147. Mixture Distributions

A mixture is what you get when you randomly pick one of several distributions and then draw from it. You’re not combining the random variables — you’re choosing between them.

𝑓 (𝑥) = \sum_{𝑖} 𝑤_{𝑖} 𝑓_{𝑖} (𝑥) 𝑤_{𝑖} \geq 0, \sum_{𝑖} 𝑤_{𝑖} = 1

Flip a biased coin: with probability $𝑤_{1}$ , pick component 1, with probability $𝑤_{2} = 1 - 𝑤_{1}$ pick component 2.
Draw your value from whichever component the coin selected.

Example

𝑓 (𝑥) = 0.7 \cdot 𝒩︀ (𝑥; 0, 1) + 0.3 \cdot 𝒩︀ (𝑥; 5, 1)

𝑓 (𝑥) = 0.6 \cdot 𝒩︀ (0, 1) + 0.4 \cdot 𝒩︀ (3, 1)

𝑓 (𝑥) = 0.5 \cdot 𝒩︀ (2, 0.6) + 0.5 \cdot 𝒩︀ (2, 3)