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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!!!

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