Short question: Why is this true??
Long question:
Very simply, I am trying to figure out what justifies this first equation. The author of the book I am reading, (context here if you want it, but not necessary), claims the following:
Due to the assumption of near-gaussianity, we can write:
$$ p_0(\xi) = A \; \phi(\xi) \; exp( a_{n+1}\xi + (a_{n+2} + \frac{1}{2})\xi^2 + \sum_{i=1}^{n} a_i G_i(\xi)) $$
Where $p_0(\xi)$ is the PDF of your observed data that has maximum entropy, given that you had only observed a series of expectations, (simple numbers) $c_i, i = 1 ... n$, where $c_i = \mathbb{E}\{G_i(\xi)\}$, and $\phi(\xi)$ is the PDF of a standardized gaussian variable, that is, 0 mean, and unit variance.
Where all this is going is that he uses the above equation as a starting point for making the PDF, $p_0(\xi)$ simpler, and I get how he does it, but I do not get how he justifies the above equation, ie, the starting point.
I have tried to keep it brief to as not to obfuscate anyone, but if you want additional details please let me know in the comments. Thanks!