Hey, from this equation, it looks like the probability that xn is in cluster k ( (\pi)k) doesn't depend on n, which means that all the data points have the same probability distribution to be in every cluster, shouldn't it be ( (\pi)kn) ? which means that every point has it's own vector ( (\pi)n) which expresses the probability that this specific point to be in each cluster? otherwise it is the same for all of them,
Thanks,
You should understand \(\pi_k\) as the relative size of the k-th cluster. The probability of a data point \(x_n\) being in cluster k is \(P(x_n\mid z_n=k)\) which is modeled as a gaussian distribution. One very important point is that \(z_n\) is not observed, only \(x_n\) is, so to decide to which cluster a data point \(x_n\) belongs, you should compute the posterior \(P(z_n\mid x_n)\).
soft-clustering
Hey, from this equation, it looks like the probability that xn is in cluster k ( (\pi)k) doesn't depend on n, which means that all the data points have the same probability distribution to be in every cluster, shouldn't it be ( (\pi)kn) ? which means that every point has it's own vector ( (\pi)n) which expresses the probability that this specific point to be in each cluster? otherwise it is the same for all of them,
Thanks,
You should understand \(\pi_k\) as the relative size of the k-th cluster. The probability of a data point \(x_n\) being in cluster k is \(P(x_n\mid z_n=k)\) which is modeled as a gaussian distribution. One very important point is that \(z_n\) is not observed, only \(x_n\) is, so to decide to which cluster a data point \(x_n\) belongs, you should compute the posterior \(P(z_n\mid x_n)\).
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