#latent-variables

Question 1: Single choice

mediumAI-checkedProbabilistic & Graphical Models

$X$ is the input, $Y$ the output and $Z$ the index of the linear regression model (or expert) that produces $Y$ . In a mixture of linear regressions, $P(y \mid x) = \sum_z p_z\, \mathcal{N}(y; w_{z0} + w_{z1}x, \sigma_z^2)$ with fixed mixing weights $p_z$ ; in a mixture of experts, a gating network makes the choice of $z$ depend on $x$ . Which directed graphs represent the two models?

Mixture of regressions: $X \to Z$ , $X \to Y$ , $Z \to Y$ ; mixture of experts: $X \to Y \leftarrow Z$ 1
Mixture of regressions: $X \to Y \leftarrow Z$ ; mixture of experts: $X \to Z$ , $X \to Y$ , $Z \to Y$ 2
Both: $X \to Z \to Y$ , without an edge from $X$ to $Y$ 3
Both: $X \to Y \leftarrow Z$ ; the gating network is a parameter, not an edge4

Timed test