To compute the probability distribution \mathbf{P}, we need to consider the distances between each pair of data points.
Given that the data points \mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)} are equidistant (at distance A) from each other in p-dimensional space, we can calculate the probabilities as follows:
p_{12} is the probability that \mathbf{x}^{(1)} and \mathbf{x}^{(2)} are nearest neighbors. Since all three data points are equidistant, there are no other data points closer to either \mathbf{x}^{(1)} or \mathbf{x}^{(2)}. Therefore, we can conclude that p_{12} = 1.
p_{13} is the probability that \mathbf{x}^{(1)} and \mathbf{x}^{(3)} are nearest neighbors. Again, since all three data points are equidistant, there are no other data points closer to either \mathbf{x}^{(1)} or \mathbf{x}^{(3)}. Therefore, p_{13} = 1.
Therefore, we have:
p_{12} = 1
p_{13} = 1
In the problems on this page, we will work through the definitions and minimization described above for a simple data configuration.
Given 3 data points \mathbf{x}^{(1)},\mathbf{x}^{(2)},\mathbf{x}^{(3)} that are equidistant (at distance A) from each other in p-dimensional space, compute the probability distribution \mathbf{P} (given by p_{12}, p_{23}, p_{13}), and enter the value of p_{12} and p_{13} below.
p_{12}=\quad
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p_{13}=\quad
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