# Why is posterior probability so in BPR paper?

In BPR paper BPR: Bayesian personalized ranking from implicit feedback (Steffen Rendle 2009)
$$\prod_{u \in U} p\left(>_{u} | \Theta\right)=\prod_{(u, i, j) \in U \times I \times I} p\left(i>_{u} j | \Theta\right)^{\delta\left((u, i, j) \in D_{S}\right) } \cdot\left(1-p\left(i>_{u} j | \Theta\right)\right)^{\delta\left((u, j, i) \notin D_{S}\right)}$$
Why the power of second term should be $$\delta\left((u, j, i) \notin D_{S}\right)$$ ?Shouldn't it be $$\delta\left((u, i, j) \notin D_{S}\right)$$? It is obvious that $$(u, i, j) \in D_{S}$$ implies $$(u, j, i) \notin D_{S}$$.
And how can it be reduced to be
$$\prod_{u \in U} p\left(>_{u} | \Theta\right)=\prod_{(u, i, j) \in D_{S}} p\left(i>_{u} j | \Theta\right)$$
And \begin{aligned} \mathrm{BPR}-\mathrm{OPT} & :=\ln p\left(\Theta |>_{u}\right) \\ &=\ln p\left(>_{u} | \Theta\right) p(\Theta) \\ &=\ln \prod_{(u, i, j) \in D_{S}} \sigma\left(\hat{x}_{u i j}\right) p(\Theta) \\ &=\sum_{(u, i, j) \in D_{S}} \ln \sigma\left(\hat{x}_{u i j}\right)+\ln p(\Theta) \\ &=\sum_{(u, i, j) \in D_{S}} \ln \sigma\left(\hat{x}_{u i j}\right)-\lambda_{\Theta}\|\Theta\|^{2} \end{aligned}
Does this formula in the last equation miss $$\ln p(\Theta)$$
Any help will be appreciated!!!