= Second moment of a mixture likelihood ratio
{title2=$\chi^2(\sum_jw_jP_j\Vert Q)=\sum_{j,k}w_jw_k\mathbb E_Q[L_jL_k]-1$}
If $L_j=dP_j/dQ$ and the nonnegative <mixture weights> sum to one, the <likelihood ratio> of the <mixture model> is $L=\sum_jw_jL_j$. Expanding $\mathbb E_QL^2-1$ proves the displayed <chi-squared divergence> identity. For <independent random variables> sampled from a fixed component, one-observation cross moments multiply across observations. The component is drawn once for the whole sample; redrawing it per observation gives a different <mixture model>.
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