deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index 7e0d57b..f2a7b45 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
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Our data are composed of a set of $N$ times traces $D=\{d_t^i\}$, with $i \in [1,..N]$ and $t \in [1,.. T_i]$.  Given the data, the hidden states $X=\{x_t^i\}=\{(\phi_t^i,\lambda_t^i)\}$ and the parameters $\Theta$, we want to maximimze maximize  the likelihood of the data $P(D|\Theta)$ over $\Theta$. In the standard expectation-maximization framework \cite{bilmes1998gentle}, we aim to maximize the expected value of the log of the joint distribution of hidden states and data, $P(D,X|\Theta)$, with respect to the unknown states, conditioned on the data and the current parameters: 
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\argmax_{\Theta_k} \int_X \log( P(D,X|\Theta_k) ) P(X|D,\Theta_{k-1})  \end{equation}  In the context of our HMM the joint distribution $P(D,X|\Theta)=P(D|X,\Theta)P(X|\Theta)$ is given by $\prod_i^N P(d_1^i,..,d_{T_i}^i ,x_1^i,..,x_{T_i}^i|\Theta) = \prod_i^N P(d_1^i,..,d_{T_i}^i|x_1^i,..,x_{T_i}^i,\Theta)P(x_1^i,..,x_{T_i}^i|\Theta)$.   Here $P(d_1^i,..,d_{T_i}^i|x_1^i,..,x_{T_i}^i,\Theta)$ is the emission probability for a given trace and is proportional to $\prod_t^{T_i} \exp[ -(d_t^i - m(x_t^i))^2]$.