deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index 9d4f18a..0859603 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 $\prod_i^N P(d_t^i,x_i^t|\Theta) = \prod_i^N P(d_t^i|x_i^t,\Theta)P(x_i^t|\Theta)$. Here $P(d_t^i|x_i^t,\Theta)$ is simply the emission probability and is proportional to $\exp[ $\prod_t^{T_i} \exp[  -(d_t^i - m(x_t^i))^2]$. The joint distribution of hidden states $P(X|D,\Theta)$ is given by $\prod_i^N P(x_t^i|d_t^i,\Theta)$.