haji_thesis/Thesis/app2.tex

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\chapter{اثبات‌ها}\label{Chap:App2}

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\section*{اثبات گزاره \ref{thm:4c-lglk}}\label{app:4c-lglk}
با استفاده از قانون زنجیر در احتمالات می‌توان نوشت
\begin{align}
&f(\mathcal{D} \vert \theta) = \prod_{i=1}^K f\left((t_i,u_i,p_i)| \mathcal{D}(t_i)\right) \prod_{u=1}^{N} S(T,u) \nonumber 
\end{align}
که $t_0=0$ و $S_u(T)$ احتمال بقای فرآیند $\lambda_u(t)$ بعد از آخرین رویدادش است.
\begin{align}
S_u(T) = \exp\left(-\int_{t_{\vert\mathcal{D}_u\vert}}^T  \lambda_{u}(s) ds\right) \nonumber
\end{align}
اکنون با استفاده از رابطه فوق می‌توان درستنمایی را محاسبه کرد.
 \begin{align}
&f(\mathcal{D} \vert \theta) 
= \prod_{u=1}^{N}	\prod_{i=1}^{\vert\mathcal{D}_u\vert} f\left((t_i,u_i,p_i)| \mathcal{D}(t_i)\right) \prod_{u=1}^{N} S(T,u) \nonumber \\
&= \prod_{u=1}^{N}	\prod_{i=1}^{\vert\mathcal{D}_u\vert} \lambda_{u}(t_i) \exp\left(-\int_{t_{i-1}}^{t_i} \lambda_{u}(s) ds\right) f_{u}(p_i | t_i)
 \prod_{u=1}^{N} S(T,u) \nonumber 
\end{align}
\begin{align}
&= \prod_{u=1}^{N} \exp\left(-\int_0^{t_{\vert\mathcal{D}_u\vert}} \lambda_{u}(s) ds\right)  \prod_{i=1}^{\vert\mathcal{D}_u\vert} f_{u}(p_i | t_i) \lambda_{u}(t_i)  \prod_{u=1}^{N} S(T,u) \nonumber \\
&= \prod_{u=1}^{N} \exp\left(-\int_0^{t_{\vert\mathcal{D}_u\vert}}  \lambda_{u}(s) ds\right) S(T,u)
\prod_{i=1}^{\vert\mathcal{D}_u\vert} f_{u}(p_i | t_i) \lambda_{u}(t_i) \nonumber\\
&= \prod_{u=1}^{N} \exp\left(-\int_0^T  \lambda_{u}(s) ds\right) 
\prod_{i=1}^{\vert\mathcal{D}_u\vert} f_{u}(p_i | t_i) \lambda_{u}(t_i) \nonumber
\\
&= \prod_{u=1}^{N} \exp\left(-\int_0^T  \lambda_{u}(s) ds\right) 
\prod_{u=1}^{N} \prod_{i=1}^{\vert\mathcal{D}_u\vert} f_{u}(p_i | t_i) \lambda_{u}(t_i) \nonumber \\
&=\exp\left(-\int_0^T \sum_{u=1}^N \lambda_u(s) ds \right) \prod_{i=1}^K \lambda_{u_i}(t_i) f_{u_i}(p_i|t_i) \nonumber
\end{align}