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You have looked at the Wikipedia article on perplexity. It gives the perplexity of a discrete distribution as $$2^{-\sum_x p(x)\log_2 p(x)}$$ which could also be written as $$\exp\left({\sum_x p(x)\log_e \frac{1}{p(x)}}\right)$$ i.e. as a weighted geometric average of the inverses of the probabilities. For a continuous distribution, the sum would turn into a integral. The article also gives a way of estimating perplexity for a model using $N$ pieces of test data $$2^{-\sum_{i=1}^N \frac{1}{N} \log_2 q(x_i)}$$ which could also be written $$\exp\left(\frac{{\sum_{i=1}^N \log_e \left(\dfrac{1}{q(x_i)}\right)}}{N}\right) \text{ or } \sqrt[N]{\prod_{i=1}^N \frac{1}{q(x_i)}}$$ or in a variety of other ways, and this should make it even clearer where "log-average inverse probability" comes from. |
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