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        "author": "William Song",
        "author_email": "songcwzjut@163.com",
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            "Intended Audience :: Science/Research",
            "License :: Public Domain",
            "Natural Language :: English",
            "Operating System :: OS Independent",
            "Programming Language :: Python",
            "Programming Language :: Python :: 3.6",
            "Topic :: Scientific/Engineering :: Mathematics"
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        "description": "# Bayes Classifier\n\n## Principle\n\n### Naive Bayes\n\n\n$$\np(c|x)=\\frac{p(x|c)p(c)}{p(x)}\\sim p(x|c)p(c)\\\\\n\\sim \\prod_ip(x_i|c)p(c) = \\prod_ip(x_i,c)p(c)^{1-n}~~~~~~~~~\\text{(Naive condition)}\\\\\n\\sim\\prod_i\\frac{N(x_i,c)}{N}p(c)^{1-n}\n$$\n\n### Semi Naive Bayes\n\n$$\np(c|x,y)=\\sim p(x|c)p(c|y)\\\\\n\\sim \\prod_ip(x_i|c)p(c|y) ~~~~~~~~~\\text{(Semi-Naive condition)}\n$$\n\nwhere $p(c|y)\u200b$ will be estimated by say of neural networks.\n\n### Hemi Naive Bayes, in more general form\n\nWhen $y$ is empty, it is equiv. to the naive one.\n\n$$\np(c|x,y_1,\\cdots y_m)\n\\sim \\prod_ip(x_i|c)\\prod_ip(c|y_i)p(c)^{1-m}  ~~~~~~~~~~(Hemi-condition)\\\\\n\\sim \\prod_ip(x_i|c)\\prod_if_c(y_i)p(c)^{1-m}\\\\\n\\sim \\prod_ip(x_i,c)\\prod_if_c(y_i)p(c)^{1-m-n}\n$$\n\n## Predict\n\n$$\n\\frac{p(c|x,y)}{p(c'|x,y)}= \\prod_i(\\frac{p(x_i|c)}{p(x_i|c')})\\frac{p(c|y)}{p(c'|y)}\\\\\n= \\prod_i(\\frac{p(x_i,c)}{p(x_i,c')})\\frac{p(c|y)}{p(c'|y)}(\\frac{p(c')}{p(c)})^n\n~~~~~~~~~\\text{(Semi-Naive condition)}\\\\\n\\sim \\prod_i(\\frac{N(x_i,c)}{N(x_i,c')})\\frac{p(c|y)}{p(c'|y)}(\\frac{N(c')}{N(c)})^n  ~~~~~~~~~~~~~~~~~~~~~\\text{(estimate)}\n$$\n\n$$\n\\frac{p(c|x,y_1,\\cdots, y_m)}{p(c'|x,y_1,...,y_m)}\\sim ... (\\frac{N(c')}{N(c)})^{n+m-1}\\prod_i\\frac{p(c|y_i)}{p(c'|y_i)}    ~~~~~~~~~(\\text{Hemi-condition})\n$$\n\n\n\n### 0-1 cases\n\n$$\nr = \\frac{p(1|x,y)}{p(0|x,y)}\\sim \\prod_i(\\frac{N(x_i,1)}{N(x_i,0)})\\frac{p(1|y)}{1-p(1|y)}(\\frac{N(0)}{N(1)})^n  (Semi)\\\\\n\nr \\sim \\prod_i(\\frac{N(x_i,1)}{N(x_i,0)})\\prod_i\\frac{p(1|y_i)}{1-p(1|y_i)}(\\frac{N(0)}{N(1)})^{n+m-1}  (Hemi)\n$$\n\niff $r\\geq 1$, $(x,y)$ is in class 1, else in class 0.\n\n\n\n## Estimate (for continuous rv)\n\n$p(x)\\sim \\frac{N(x)}{N}, N(x):$ the number of samples in a neighborhood of $x$",
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