deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index 6ae096b..bbb25cb 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
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\end{equation}  In the context of our HMM the joint distribution $P(D,X|\Theta)$ is given by $\prod_i^N P(d_1^i,..,d_{T_i}^i ,x_i^1,..,x_{T_i}^1|\Theta) = \prod_i^N P(d_1^i,..,d_{T_i}^i|x_1^i,..,x_{T_i}^i,\Theta)P(x_1^i,..,x_{T_i}^i|\Theta)$. 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 $\prod_t^{T_i} \exp[ -(d_t^i - m(x_t^i))^2]$. The joint distribution of hidden states $P(X|D,\Theta)$ is given by $\prod_i^N P(x_1^i,..,x_{T_i}^i|d_1^i,..,d_{T_i}^i,\Theta)$.