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I am reading a section in a book about translation. The formula below is given as a mathematical way to calculate word translation entropy. So for a source token s and all available translations t_i...n one could calculate the word translation entropy of s like so.

formula

But I am confused about the raised star. The accompanied text reads:

Word translation entropy H(s) is the sum over all observed word translation probabilities (i.e. expectations) of a given ST word s into TT words t_i...n multiplied with their information content.

So I am assuming that it is multiplication. But why is it a raised star in superscript? Why isn't it written it normally, inline? (I am not at all a mathematician, so apologies if this is a basic notation question.)

The formula can be found on p. 31 in the following book.

Carl, M., Schaeffer, M. J., & Bangalore, S. (2016). The CRITT Translation Process Research Database. In M. Carl, S. Bangalore, & M. J. Schaeffer (Eds.), New Directions in Empirical Translation Process Research (pp. 13–54). Cham, Switzerland: Springer International Publishing.

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  • $\begingroup$ What's the book? $\endgroup$ – Wojowu Jan 11 at 11:11
  • $\begingroup$ @Wojowu Please see my edit. $\endgroup$ – Bram Vanroy Jan 11 at 11:14
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The asterisk is indeed multiplication: checking a little further in the book (on page 33) there is another equation: $$\frac{1}{n}^*\sum_j^n abs(cross^* Htra) $$ where that first raised asterisk is unmistakeably multiplication. Also checking other definitions of entropy in this context confirms that it is the sum of proabilities multiplied by their logarithms.

It appears be be poor typesetting and no other reason. I vaguely remember seeing this some years ago ($10$-$15$?) when Word was used for typesetting either without using the Equation Editor package or before it was improved.

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  • $\begingroup$ Thank you for the reply. Just to be sure: the way it is written (as a superscript) is not the correct mathematical approach? It should just be a normal asterisk? $\endgroup$ – Bram Vanroy Jan 11 at 11:47
  • $\begingroup$ Mathematicians would probably not put the asterisk in at all, and would move the $-$ sign associated with the logarithm in front of the summation sign. But no, superscript asterisk is normally (mathematically) used for dual-spaces and sometimes complex conjugation $\endgroup$ – postmortes Jan 11 at 11:49
  • $\begingroup$ Okay, that's what I found when looking for the concept in information theory, indeed. Thanks! $\endgroup$ – Bram Vanroy Jan 11 at 11:51

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