# what is “log-average”?

It was mentioned in pLSA paper that perplexity refers to the log-averaged inverse probability on unseen data. Can any one give me the exact formula for calculating perplexity

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Searching Google for "log-average" yields as the first hit the Wikipedia article for geometric mean, with a conveniently highlighted excerpt stating that it is also sometimes called the log-average... – Rahul May 5 '11 at 4:33
@Rahul: I too found the wikipedia article, but im not convinced with that definition in the context of pLSA probabilities. Looking for better information on the same. – Learner May 5 '11 at 4:36
– Henry May 5 '11 at 6:43

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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