= Multiplicative entropy power inequality
For independent positive <random variables> $Y_1,Y_2$ with densities, finite <differential entropies> and finite $\mu_i=\mathbb E\log_2Y_i$, applying the <entropy power inequality> to $Z_i=\ln Y_i$ gives
$$
2^{2h(Y_1Y_2)}\geq2^{2\mu_2}2^{2h(Y_1)}+2^{2\mu_1}2^{2h(Y_2)}.
$$
Indeed $h(Z_i)=h(Y_i)-\mu_i$, while $h(Y_1Y_2)=h(Z_1+Z_2)+\mu_1+\mu_2$, provided the sum's <differential entropy> is defined. Positivity alone does not ensure these expressions exist: $Y=e^T$ with $T$ having <Cauchy distribution> has undefined $\mathbb E\log_2Y$.
// Target: algebra.bigb
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