deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index 20b9431..aefad55 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
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We can now expand the integral over the hidden states:  $\idotsint \prod_j^N \prod_u^{T_j} $$\idotsint \prod_{j=1}^N \prod_{u=1}^{T_j}  dx_u^j \sum_i [ \sum_t (d_t^i - m(x_t^i))^2 ]P(X|D,\Theta_{k-1})$ ]P(X|D,\Theta_{k-1})$$  $ $$  = \sum_{i,t} \idotsint \prod_j^N \prod_u^{T_j} \prod_j \prod_u  dx_u^j (d_t^i - m(x_t^i))^2 P(X|D,\Theta_{k-1})$ P(X|D,\Theta_{k-1})$$  $ $$  = \sum_{i,t} \iint dx_t^i (d_t^i - m(x_t^i))^2 \idotsint \prod_{j\neq i}^N i}  \prod_{u\neq t}^{T_j} t}  dx_u^j P(X|D,\Theta_{k-1})$ P(X|D,\Theta_{k-1})$$  $ $$  = \sum_{i,t} \iint dx_t^i (d_t^i - m(x_t^i))^2 P(x_t^i|D,\Theta)$ P(x_t^i|D,\Theta)$$  Where $P(x_t^i|D,\Theta)$ is the posterior distribution of our hidden states at time $t$ for a trace $i$, and is computed as a discrete table indexed by integer $l \in [1,..N_\lambda]$ and $m \in [1,..N_\phi]$: $P_t^i[l,m]$ . Since we work in discretized space we rewrite the above integral as: