Fig. 4

Graphical interpretation of the third rule in Eq. (7) for the update control test. Given a point z, this test can be implemented as following: (1) draw a sample \(v' \sim \mathcal {U}([0,\max \{\pi (z),q_{t}(z|\mathcal {S}_{t})\}])\), (2) then if v^′≤d _t (z), add z to the set of support points, i.e., \(\mathcal {S}_{t+1}=\mathcal {S}_{t} \cup \{z\}\). a The interval \([0,\max \{\pi (z),q_{t}(z|\mathcal {S}_{t})\}]\) and the distance d _t (z). b The case when v^′≤d _t (z) so that the point is incorporated in the set of support points whereas c illustrates the case when v^′>d _t (z); hence, \(\mathcal {S}_{t+1}=\mathcal {S}_{t}\). Note that as the proposal function q _t becomes closer and closer to π (i.e., d _t (z) decreases for any z), the probability of adding a new node to \(\mathcal {S}_{t}\) decreases