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\end{equation}$$
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However, in practice we're using _importance sampling_ which needs to take the $pdf$ of the distribution
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- into account and adds a term $\frac{\left< \VoH\right> }{D(h_i, \alpha)\left< \NoH\right> }$.
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+ into account and adds a term $\frac{4 \left< \VoH\right> }{D(h_i, \alpha)\left< \NoH\right> }$.
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See Importance Sampling For The IBL section:
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$$\begin{equation}\label{iblImportanceSampling}
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**do not** have the same PDF as $h_i$. The PDF of a transformed distribution is given by:
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$$\begin{equation}
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- p(T_r(x)) = p(x) |J(T_r)|
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+ p(T_r(x)) = p(x) |J(T_r)|^{-1}
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\end{equation}$$
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Where $|J(T_r)|$ is the determinant of the Jacobian of the transform. In our case we're considering
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the transform from $h_i$ to $l_i$ and the determinant of its Jacobian is given in \ref{iblPDF}.
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$$\begin{equation}\label{iblPDF}
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- p(l,v,\Theta) = D(h,\alpha) \left< \NoH \right> |J_{h \rightarrow l}| \\
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- |J_{h \rightarrow l}| = \frac{1}{ 4 \left< \VoH \right> }
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+ p(l,v,\Theta) = D(h,\alpha) \left< \NoH \right> |J_{h \rightarrow l}|^{-1} \\
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+ |J_{h \rightarrow l}| = 4 \left< \VoH \right>
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\end{equation}$$
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### Choosing important directions
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