Coarse-graining and equivalence classes

Overview

Coarse-graining a theory or representation of the world is a way of merging states of the world. Proper coarse-graining involves strategically throwing out information about high-resolution data such that a model of the simplified data echoes the real-world process. The key features of coarse-graining are that it reduces the size of the uncertainty problem and cannot be reversed.

concept-question What is the relationship between coarse-graining and equivalence classes?

Related notes:

• Abstraction and idealization
• Renormalization

---

Some examples of coarse-graining include:

• Majority voting and other methods of aggregating the preferences of a region;
• JPEG image compression (Fourier transform), which replicates the coarse-graining process of the retina by replacing features that are undetectable to the human eye.

What are some non-examples of coarse-graining?

---

Formal examples

Shannon entropy

The coarse-graining axiom of Shannon entropy states that a fine-grained uncertainty should be equal to a coarse-grained uncertainty.

Tree-like property of Shannon entropy

Suppose we have a set $X=\{a,b,c\}$ and do not care about distinguishing between elements of the set $G=\{b,c\}$. In order to return an uncertainty value for the reduced set $X’=\{a,G\}$, an equation for uncertainty $H(X)$ should satisfy

$$H(X)=H(X^{\prime })+p_{bc}H(G),$$

where $H(G)$ is the uncertainty of the distribution $\{p_b/p_{bc},p_c/p_{bc}\}$, where $p_{bc}=p_b+p_c$.