= Integrating and rescaling a statistical model
{title2=$e^{-\mathcal H'(S)}=\sum_s B(S\mid s)e^{-\mathcal H(s)}$}
Use a dimensionless <statistical Hamiltonian> $\mathcal H=H/(k_BT)$, so that the <partition function> is $Z=\sum_s e^{-\mathcal H(s)}$. Choose a <normalized blocking kernel> $B(S\mid s)$ with $\sum_S B(S\mid s)=1$. For the majority example it is one when every block has its prescribed majority, and zero otherwise. Define the coarse <statistical Hamiltonian> by the displayed sum. Then an elementary interchange of finite sums proves
$$
Z'=\sum_S e^{-\mathcal H'(S)}=\sum_s e^{-\mathcal H(s)}\sum_S B(S\mid s)=Z.
$$
The probability of a retained pattern is therefore its original marginal probability. For any observable $A(S)$ of the retained variables,
$$
\langle A\rangle'=\frac1Z\sum_s e^{-\mathcal H(s)}\sum_S B(S\mid s)A(S).
$$
This is the sense in which elimination preserves predictions for retained information. It does not reconstruct every microscopic observable from a block sign.
Next express positions in units of the enlarged spacing: $x'=x/b$, where $b>1$. In a continuum version, rescale the retained <scalar field> as well to restore a chosen <gradient energy> coefficient; any change in the integration measure contributes to the field-independent term. The result is a new set of dimensionless couplings. Repeating the procedure defines a <renormalization-group flow>. Additional interactions and the additive constant generally appear, even if the starting <statistical Hamiltonian> was simple. Keeping only a few terms is an approximation; the equality of <partition functions> above applies to the complete coarse weight. See also the <Wilsonian coarse-grained statistical Hamiltonian> for integration over fast <Fourier modes>.
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