Table 2:

Glossary of variables and symbols used in this paper.

Symbol	Used for
X	Data point, X ∈ $\mathcal{X}$
n	Number of data points
Ys	Label for one of the t classification tasks, Ys ∈ {1,...,ks}
t	Number of tasks
Y	Vector of task labels Y = [Y1, Y2,...,Yt]T
r	Cardinality of the output space, r = |$\mathcal{Y}$|
G task	Task structure graph
$\mathcal{Y}$	Output space of allowable task labels defined by Gtask, Y ∈ $\mathcal{Y}$
$\mathcal{D}$	Distribution from which we assume (X, Y) data points are sampled i.i.d.
si	Weak supervision source, a function mapping X to a label vector
${\mathbf{\lambda }}_i$	Label vector ${\mathbf{\lambda }}_i$ ∈ $\mathcal{Y}$ output by the ith source for X
m	Number of sources
λ	m × t matrix of labels output by the m sources for X
$\mathcal{Y}$ 0	Source output space, which is $\mathcal{Y}$ augmented to include elements set to zero
τi	Coverage set of ${\mathbf{\lambda }}_i$- the tasks si gives non-zero labels to; for convenience, τ0 = {1,…,t}
$\mathcal{Y}$ τi	The output space for ${\mathbf{\lambda }}_i$ given coverage set τi
${\mathcal{Y}}_{{\tau }_i}^{\text{min}}$	The output space ${\mathcal{Y}}_{{\tau }_i}$ with all but the first value, for defining a minimal set of statistics
G source	Source dependency graph, Gsource = (V, E), V = {Y, ${\mathbf{\lambda }}_1$,...,${\mathbf{\lambda }}_m$}
$\mathcal{C}$	Cliqueset (maximal and non-maximal) of Gsource
$\tilde{\mathcal{C}},\mathcal{S}$	The maximal cliques (nodes) and separator sets of the junction tree over Gsource
Ψ(C, yC)	The indicator variable for the variables in clique C ∈ $\mathcal{C}$ taking on values yC, (yC)i ∈ $\mathcal{Y}$τi
μ	The parameters of our label model we aim to estimate; $\mu =\mathbb{E}[\psi ]$
O	The set of observable cliques, i.e. those corresponding to cliques without Y
Σ	Generalized covariance matrix of O ⊆ $\mathcal{S}$, Σ ≡ Cov [Ψ(O ⊆ $\mathcal{S}$)]
K	The inverse generalized covariance matrix K = Σ−1
dO, d $\mathcal{S}$	The dimensions of O and $\mathcal{S}$ respectively
G aug	The augmented source dependencies graph Gaug = (Ψ, Eaug)
Ω	The edge set of the inverse graph of Gaug
P	Diagonal matrix of class prior probabilities, P(Y)
Pμ (Y, λ)	The label model parameterized by μ
$\tilde{\mathbf{Y}}$	The probabilistic training label, i.e. Pμ(Y|λ)
fw (X)	The end model trained using (X, $\tilde{\mathbf{Y}}$)