Information Field Theory II

Last time we calculated a “map” of our signal, which was the expectation value of s with respect to the a posteriori probability P(s|d), and I stated that there where several ways of doing it. One important method is to introduce the notion of a generating function into the definition of the partition function Z.

Z_d[J] \equiv \int \!\! \mathcal{D}s~ e^{-H[s] + J^{\dagger}s}

where Z_d[0]=Z. In this way we can calculate any higher moment of the signal field via differentiation;

\langle s(x_1) \cdots s(x_n) \rangle = \frac{1}{Z}\frac{\delta^n Z_d[J]}{\delta J(x_1) \cdots \delta J(x_n)} \vert_{J=0} .

We should also introduce the connected correlation functions, which includes corrections for lower moments to the correlation function.

\langle s(x_1) \cdots s(x_n) \rangle_c = \frac{\delta^n ln(Z_d[J])}{\delta J(x_1) \cdots \delta J(x_n)} \vert_{J=0} .

In this way, the map is now the first derivative

m_d = \langle s \rangle_c =\frac{\delta ln(Z_d[J])}{\delta J(x)} \vert_{J=0} = Dj .

I’m still leaving the evaluation of the integral as an exercise, mostly because I don’t want to type out the latex! The important thing is that the generating function is what we’ll use once the Gaussian “free theory” has extra terms added on, and we will use Wick’s theorem to calculate the perturbation expansion in terms of the two point correlation function \langle s(x) s(y) \rangle_c = D(x,y), which is just the propagator of the free theory.

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