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Chapter6_Priorities/Chapter6.ipynb

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"A very simple example follows: suppose we wish to estimate the parameter $\\mu$ of a Normal distribution, with $\\sigma = 5$. Since $\\mu$ could range over the whole real line, we can use a Normal distribution as a prior for $\\mu$. How to select the prior's hyperparameters, denoted ($\\mu_p, \\sigma_p^2$)? The $\\sigma_p^2$ parameter can be chosen to reflect the uncertainty we have. For $\\mu_p$, we have two options:\n",
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"1. Empirical Bayes suggests using the empirical sample mean, which will center the prior around the observed empirical mean:\n",
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"$$ \\mu_p = \\frac{1}{N} \\sum_{i=0}^N X_i $$\n",
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"$$\n",
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"\\mu_p = \\frac{1}{N} \\sum_{i=0}^N X_i \n",
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"$$\n",
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"2. Traditional Bayesian inference suggests using prior knowledge, or a more objective prior (zero mean and fat standard deviation).\n",
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