Consider the following conditional probability density function: $$\mathbb{P}_{X_1,\ldots,X_n|Y_1,\ldots,Y_{n},Q}(x_1,\ldots,x_n|y_0,\ldots,y_{n},q)=\prod_{i=1}^n \lambda e^{-\lambda (x_i-r_i)}$$ where $r_i=\max{\{0,\sum_{j=1}^{i}{y_j}}-\sum_{j=1}^{i-1}{x_j}-q\}$, and $$\mathbb{P'}_{X_1,\ldots,X_n|Y_1,\ldots,Y_{n},Q}(x_1,\ldots,x_n|y_0,\ldots,y_{n},q)=\prod_{i=1}^n (\lambda/\alpha_i) e^{-\lambda (x_i/\alpha_i-r'_i)}$$ where $r'_i=\max{\{0,\sum_{j=1}^{i}{y_j}}-\sum_{j=1}^{i-1}{x_j/\alpha_j}-q\}$. Find a (preferably or relatively) tight upperbound for $$\mathcal{D}_{KL}(\mathbb{P' ||P})$$where $\mathcal{D}_{KL}(\cdot)$ is the Kullback_Liebler divergene.

My Solution: \begin{align} \mathcal{D}_{KL}(\mathbb{P' ||P}) &= \int \mathbb{P'} \log{\frac{\mathbb{P'}}{\mathbb{P}}}dx_1 \ldots dx_n\\ &= \int \mathbb{P'}\sum_{i=1}^{n} \left(-\log{(\alpha_i)}-\lambda (x_i/\alpha_i-r'_i)+\lambda (x_i-r_i)\right)dx_1 \ldots dx_n\\ &= -\sum_{i=1}^{n}\log(\alpha_i)+\int \mathbb{P'}\sum_{i=1}^{n} \left({\left({\mathbb{{-\lambda (x_i/\alpha_i-r'_i)}}+{\lambda (x_i-r_i)}}\right)}\right)dx_1 \ldots dx_n\\ \end{align}

Because of the $r_i$s, I have difficulty in calculating this, any idea how to do this?

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