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Probabilistic k-nearest neighbours

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Probabilistic k-nearest nerighbours

Let $\mathcal D = \big\{(y_n, {\bf x}_n) \vert y_n \in \{c_q\}_{q=1}^Q, {\bf x}_n\in\mathbb{R}^M\big\}$ be a dataset.

A pknn model attempts to find the posterior distribution of neighbours $p(\beta, k \vert \mathcal{D})$ given the likelihood function

$p(y_i \vert {\bf x}, \beta, k) = \frac{\exp\left(\frac{\beta}{k}\sum_{n\vert {\bf x}_n \in \mathcal{N}({\bf x}_i)}\mathbb{1}(y_n = y_i)\right)}{\sum_{q=1}^Q\exp\left(\frac{\beta}{k}\sum_{n\vert {\bf x}_n \in \mathcal{N}({\bf x}_i)}\mathbb{1}(y_n = q)\right)}$

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