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SECTION 07 / 8

Turning expectations into empirical objectives

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Figure 44. Empirical sample-average objective.
Figure 44. Empirical sample-average objective.
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Illustration: Turning expectations into empirical objectives

For observed real samples x1,…,xnx_1,\ldots,x_n and generated samples x~1,…,x~m\tilde{x}_1,\ldots,\tilde{x}_m, estimate J(T) by

J^(T)=1n∑iT(xi)−1m∑jf∗(T(x~j))\begin{aligned}&\widehat{J}(T)\\&=\frac1n\sum_i T(x_i)\\&\quad-\frac1m\sum_j f^*(T(\tilde{x}_j))\end{aligned}

This is an empirical objective. For a fixed integrable T, sample averages approximate their corresponding expectations. Optimizing the empirical supremum introduces additional statistical considerations; do not assume the optimized estimate is automatically unbiased.

Worked example

Evaluate a fixed empirical objective

  1. Suppose the two real sample values of T are 1 and 3, giving average 1+32=2\frac{1+3}{2}=2.

  2. Suppose the two generated sample values of f∗f^*(T) are 0.8 and 1.2, giving average 0.8+1.22=1\frac{0.8+1.2}{2}=1.

  3. Then the empirical objective at this fixed T is J^(T)=2−1=1\widehat{J}(T)=2-1=1.

  4. This is not automatically the supremum and is not automatically the true divergence.

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