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---
id: NMI
aliases:
- Normalized mutual information (NMI)
tags: []
---
## Normalized mutual information (NMI)
- Mutual information:
- Quantifies the amount of shared info between $I(C,T) =
\sum_{i=1}^{r}\sum{j=1}^{k}p_{ij}log\dfrac{p{ij}}{p_{c_i}p_{T_j}}$
- Measures the dependency between the observed joint probability $p_{ij}$ of
$C$ and $T$, and the expected joint probability $p_{Ci} * p_P{Tj}$ under the
independence assumption
- When $C$ and $T$ are independent, $p_{ij} = p_{Ci} * p_{Tj}, I(C, T) = 0$.
However, there is no upper bound on the mutual information
- **Normalized mutual information (NMI)** $$N M I(C, T) =
\sqrt{\dfrac{I(C,T)}{H(C)}*\dfrac{I(C, T)}{H(T)}} = \dfrac{I(C,
T)}{\sqrt{H(C) * H(T)}}$$
- Value range of NMI: [0, 1]. Value close to 1 indicates a good clustering
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