 ### shorter text in math

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 ... ... @@ -81,17 +81,17 @@ In this way we can make sure to approximately focus on the physically relevant c Doing this requires us to rewrite the integral as \$\$\begin{eqnarray} \int p_\text{real}(R) A(R) dR = & \int p_\text{sampling}(R) \underbrace{\frac{p_\text{real}(R)}{p_\text{sampling}(R)}}_{=w(R)} A(R) dR\\ = & \int p_\text{sampling}(R) w(R) A(R) dR\\ \int p_\text{sample}(R) \underbrace{\frac{p_\text{real}(R)}{p_\text{sample}(R)}}_{=w(R)} A(R) dR\\ = & \int p_\text{sample}(R) w(R) A(R) dR\\ \approx & \frac{1}{N} \sum_{i=1}^N w(R_i) A(R_i) \tag{3} \end{eqnarray}\$\$ where the configurations \$R_i\$ are now sampled from \$p_\text{sampling}(R)\$. When using this approximate probability distribution where the configurations \$R_i\$ are now sampled from \$p_\text{sample}(R)\$. When using this approximate probability distribution we thus have to introduce *weights* \$w(R)\$ into the average. ### Why approximate importance sampling eventually fails Approximate importance sampling is an attractive way of sampling: if we have a convenient and computationally efficient \$p_\text{sampling}\$ we can apply the Monte Carlo integration and seemingly focus on the relevant part of the configuration space. \$p_\text{sample}\$ we can apply the Monte Carlo integration and seemingly focus on the relevant part of the configuration space. Unfortunatley, this approach becomes increasingly worse as the dimension of the configuration space increases. This is related to the the very conter-intuitive fact thatin high-dimensional space "all the volume is near the surface". This defies our intuition that ... ... @@ -115,10 +115,10 @@ This effect directly shows in the weights. Let us demonstrate this using a simpl where \$\$p_\text{real}(x_1, \dots, x_d) = (2 \pi \sigma_\text{real})^{d/2} e^{-\frac{\sum_{k=1}^d x_k^2}{2\sigma_\text{real}}}\$\$ is a normal distribution with standard deviation \$\sigma_\text{real}\$. For the sampling distribution we use also a normal distribution, but with a slightly differen standard deviation \$\sigma_\text{sampling}\$: \$\$p_\text{sampling}(x_1, \dots, x_d) = (2 \pi \sigma_\text{sampling})^{d/2} e^{-\frac{\sum_{k=1}^d x_k^2}{2\sigma_\text{sampling}}}\,.\$\$ We will now compute how the weights \$p_\text{real}/p_\text{sampling}\$ are distributed for different dimensionality \$d\$. In the example below we have chosen \$\sigma_\text{real} = 1\$ and \$\sigma_\text{sampling} = 0.9\$ and sampling over \$N=10000\$ also a normal distribution, but with a slightly differen standard deviation \$\sigma_\text{sample}\$: \$\$p_\text{sample}(x_1, \dots, x_d) = (2 \pi \sigma_\text{sample})^{d/2} e^{-\frac{\sum_{k=1}^d x_k^2}{2\sigma_\text{sample}}}\,.\$\$ We will now compute how the weights \$p_\text{real}/p_\text{sample}\$ are distributed for different dimensionality \$d\$. In the example below we have chosen \$\sigma_\text{real} = 1\$ and \$\sigma_\text{sample} = 0.9\$ and sampling over \$N=10000\$ configurations: ![Vanishing weights in high-dimensional space](figures/weights.svg) We see that as dimensionality increses, the distribution of weights gets more and more skewed towards 0. For a large dimensionality, ... ...
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