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examples/mixture_models/dp_mix.ipynb

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"\n",
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" $$P = \\sum_{i = 1}^\\infty w_i \\delta_{\\omega_i}.$$\n",
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"\n",
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"3. The [stick-breaking process](https://en.wikipedia.org/wiki/Dirichlet_process#The_stick-breaking_process) gives an explicit construction of the weights $w_i$ and samples $\\omega_i$ above that is straightforward to sample from. If $\\beta_1, \\beta_2, \\ldots \\sim \\textrm{Beta}(1, \\alpha)$, then $w_i = \\beta_i \\prod_{j = 1}^{n - 1} (1 - \\beta_j)$. The relationship between this representation and stick breaking may be illustrated as follows:\n",
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"3. The [stick-breaking process](https://en.wikipedia.org/wiki/Dirichlet_process#The_stick-breaking_process) gives an explicit construction of the weights $w_i$ and samples $\\omega_i$ above that is straightforward to sample from. If $\\beta_1, \\beta_2, \\ldots \\sim \\textrm{Beta}(1, \\alpha)$, then $w_i = \\beta_i \\prod_{j = 1}^{i - 1} (1 - \\beta_j)$. The relationship between this representation and stick breaking may be illustrated as follows:\n",
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" 1. Start with a stick of length one.\n",
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" 2. Break the stick into two portions, the first of proportion $w_1 = \\beta_1$ and the second of proportion $1 - w_1$.\n",
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" 3. Further break the second portion into two portions, the first of proportion $\\beta_2$ and the second of proportion $1 - \\beta_2$. The length of the first portion of this stick is $\\beta_2 (1 - \\beta_1)$; the length of the second portion is $(1 - \\beta_1) (1 - \\beta_2)$.\n",

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