Space Emerges from What We Know—Spatial Categorisations Induced by Information Constraints

. 2020 Oct 19;22(10):1179. doi: 10.3390/e22101179

Algorithm 1

[p ({\tilde{S}}_{2}^{*}), {\tilde{D}}_{2}]

= SuccessiveRefinement(

p (S), R_{1}, Δ R

)

1:
$[p (S_{1}^{*}), p (S_{1}^{*} | S)] = G R D (p (S), R_{1})$
2:
$p (S | S_{1}^{*}) = \frac{p (S_{1}^{*} | S) p (S)}{p (S_{1}^{*})}$
3:
for $s_{1}^{*} \in S$ do
4:
$[p ({\tilde{S}}_{2}^{*} | S_{1}^{*} = s_{1}^{*}), p ({\tilde{S}}_{2}^{*} | S, S_{1}^{*} = s_{1}^{*})] = G R D (p (S | S_{1}^{*} = s_{1}^{*}), Δ R)$
5:
end for
6:
$p ({\tilde{S}}_{2}^{*} | S) = \sum_{s_{1}^{*} \in S} p ({\tilde{S}}_{2}^{*} | S, S_{1}^{*} = s_{1}^{*}) p (s_{1}^{*})$
7:
$p ({\tilde{S}}_{2}^{*}) = \sum_{s \in S} p ({\tilde{S}}_{2}^{*} | S = s) p (s)$
8:
${\tilde{D}}_{2} = E [d (S, {\tilde{S}}_{2}^{*})] = \sum_{s, s_{2}^{*} \in S} p (s) p (s_{2}^{*} | s) d (s, s_{2}^{*})$
9:
return $p ({\tilde{S}}_{2}^{*}), {\tilde{D}}_{2}$