Relationship between $ϕkg,t$⁠, $z$⁠, and $u$ in the case of 5 snippets. Left: Given $ϕkg,t$⁠, if $zd=k$ then $ud$ (black circle) falls below $ϕ˜kg,t$⁠; and if $zd≠k$ then $ud$ (white circle) falls above $ϕ˜kg,t$⁠. Right: With $u$ given, $ϕkg,t$ can be anywhere within the interval defined by the highest black circle and the lowest white circle. Figure is adapted from Mimno et al. (2008).