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| For a given probability distribution $\mathbb{P}$ with parameter $\theta$, we can extract feature(s) $h^\theta = g(\mathbb{P}^\theta)$. We can also calculate the features for the empirical distribution $\hat{h} = g(\hat{\mathbb{P}})$. Then solve for $\theta$ by setting $h^\theta = \hat{h}$. | For a given probability distribution $\mathbb{P}$ with parameter $\theta$, we can extract feature(s) $h^\theta = g(\mathbb{P}^\theta)$. We can also calculate the features for the empirical distribution $\hat{h} = g(\hat{\mathbb{P}})$. Then solve for $\theta$ by setting $h^\theta = \hat{h}$. | ||
| + | ==== Method of moments ==== | ||
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| + | Moments of distributions are commonly used as features for feature matching. The $k$-th moment of a random variable $X$ is $\mathbb{E}[X^k]$. | ||
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| + | To estimate the moment from empirical data $X_1, ... X_n$, replace expectation with the average: | ||
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| + | $$ \hat{\mathbb{E}}[X^k] = \frac{1}{n} \sum_{i=1}^n X_i^k $$ | ||
| ===== Maximum likelihood estimator ===== | ===== Maximum likelihood estimator ===== | ||