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I am seeking an expression of the Jensen-Shannon Distance (JSD) between two normal distributions that only uses the respective means and standard deviations. The continuous version of the JSD (in nats) is given as:

$$\sqrt{\frac{1}{2}\int_{-\infty}^{\infty} \left(f(x)\ln\frac{2f(x)}{f(x)+g(x)}+g(x)\ln\frac{2g(x)}{f(x)+g(x)} \right) dx}$$

As shown in the Wikipedia page, the Kullback-Leibler divergence and the Hellinger Distance can be expressed using only the means and standard deviations - that is, without the variable $x$. Can the JSD be expressed in a similar way?

Mari153
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    https://stats.stackexchange.com/questions/8634/jensen-shannon-divergence-for-bivariate-normal-distributions might suggest a closed form is not possible even in the univariate case – Henry Apr 02 '23 at 10:43
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    What do you mean by "defined only by respective means and standard deviations"? Normal distributions are completely defined by these two parameters. – Xi'an Apr 02 '23 at 11:01
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    @Xi'an - reading way to much into one word. Doesn't change the context of the question. – Mari153 Apr 02 '23 at 11:57
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    The comment by @Xi'an is spot on: the expression you give clearly is determined by the two means and the two SDs. There's nothing left to answer, unless you are implicitly trying to ask something else. A few follow-on questions that would be natural are (1) is there a closed form formula for this integral? and (2) does it really depend on all four parameters or does it depend on fewer parameters, such as the difference of the means and the two SDs? Indeed, any such formula in a location-scale family ought to be a function only of the absolute difference of the means and the ratio of the SDs. – whuber Apr 02 '23 at 14:10
  • @whuber. I'm just asking can the JSD be expressed in terms of the mean and SD? It can be for KL and Hellinger. I am not trying to be tricky. – Mari153 Apr 02 '23 at 19:21
  • The answer is tautological: because the two distributions involved are determined by their means and SDs, those four parameters determine the JSD. How it might be expressed it a matter of what you consider an expression to be. – whuber Apr 03 '23 at 01:13
  • @Xi'an - I have clarified the question. I am seeking an expression for the JSD between two normal distributions that doesn't include the variable, $x$. Such an expression is possible for the Kullback-Leibler divergence and the Hellinger Distance. – Mari153 Apr 04 '23 at 09:33
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    This integral expression doesn't include any variables. The "$x$" in the traditional integral notation is a bound variable and is superfluous in other, equivalent, notations. – whuber Apr 04 '23 at 12:30

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