Binary-feature Bernoulli model

Let $d\ge 1$ be an integer, let $\theta \in \{-1,+1{\}}^d$, and let $0<\tau \le 1\mathrm{/}2$. The $(\theta ,\tau )$ binary-feature Bernoulli model is the probability law $P_{\theta ,\tau }$ on $\{-1,+1{\}}^d\times \{-1,+1\}$ defined by $P_{\theta ,\tau }(\{(x,y)\})=\frac{1}{2}{\prod }_{j=1}^d[(\frac{1}{2}+\tau ){\mathbf{1}}_{\{x_j=y{\theta }_j\}}+(\frac{1}{2}-\tau ){\mathbf{1}}_{\{x_j=-y{\theta }_j\}}]\mathrm{.}$ Thus $Y$ is uniform on $\{-1,+1\}$ and, conditional on $Y=y$, the feature coordinates are independent, with $X_j=y{\theta }_j$ with probability $1\mathrm{/}2+\tau$ and $X_j=-y{\theta }_j$ with probability $1\mathrm{/}2-\tau$. The displayed masses sum to one because each coordinate factor sums to one over its two possible signs and the label weights sum to one. The uniform-prior experiment used below draws $\mathrm{\Theta }$ uniformly from $\{-1,+1{\}}^d$ and then draws observations conditionally independently from $P_{\mathrm{\Theta },\tau }$.

Probability measures and probability spaces: a probability law has total mass one.

Independent random elements: for each fixed $y$, coordinate independence is taken under the conditional probability law given $Y=y$; in the prior experiment, observation independence is taken under each conditional law given $\mathrm{\Theta }=\theta$.

Schmidt et al., Definition 7 and the Bernoulli model setup.