{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Alpha的后验分解\n",
    "\n",
    "最近在学习量化分析, 其中有一个作业是这样:\n",
    "\n",
    "> Alpha的后验分解：给定一个组合P，估算该组合相对于业绩基准组合B的各参数：$β_p$ 、$α_p$、 以及组合风险 $σ_p$ 和残差风险 $ω_p$\n",
    "\n",
    "总觉得自己哪里算错了, 比如[华夏上证50ETF(代码: 510050)](http://cn.morningstar.com/quicktake/F0000003X3)的alpha算出来总是-0.00091, \n",
    "\n",
    "一时检查不出来, 暂时先摆出来供讨论\n",
    "<!-- TEASER_END -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import tushare as ts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 设定目标"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "target=\"510050.XSHG\" # 华夏上证50\n",
    "benchmark=\"000300.XSHG\" # 沪深300指数\n",
    "start=\"2015-1-1\"\n",
    "end=\"2018-9-1\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 获得历史价格"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 如果有基金净值数据, 不能使用get_price取得, 因此分别提取数据\n",
    "p_target   =get_price(target   , start_date=start, end_date=end, fields=\"close\")\n",
    "p_benchmark=get_price(benchmark, start_date=start, end_date=end, fields=\"close\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>510050.XSHG</th>\n",
       "      <th>000300.XSHG</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2018-08-31</th>\n",
       "      <td>2.521</td>\n",
       "      <td>3334.5036</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-30</th>\n",
       "      <td>2.533</td>\n",
       "      <td>3351.0942</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-29</th>\n",
       "      <td>2.553</td>\n",
       "      <td>3386.5736</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-28</th>\n",
       "      <td>2.560</td>\n",
       "      <td>3400.1705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-27</th>\n",
       "      <td>2.571</td>\n",
       "      <td>3406.5735</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            510050.XSHG  000300.XSHG\n",
       "date                                \n",
       "2018-08-31        2.521    3334.5036\n",
       "2018-08-30        2.533    3351.0942\n",
       "2018-08-29        2.553    3386.5736\n",
       "2018-08-28        2.560    3400.1705\n",
       "2018-08-27        2.571    3406.5735"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 为避免投资组合的净值(价格)采样时间与基准的采样时间不同, 取两者的交集\n",
    "price= pd.concat([p_target, p_benchmark], axis=1, join='inner') \n",
    "price.columns=[target,benchmark]\n",
    "# 通过ts获取基金净值的时间顺序, 与通过ts获取股票价格的时间顺序, 这两者可能不同, 因此统一到时间降序上\n",
    "price.sort_index(ascending=False, inplace=True)\n",
    "price.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>510050.XSHG</th>\n",
       "      <th>000300.XSHG</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2018-08-31</th>\n",
       "      <td>2.533</td>\n",
       "      <td>3351.0942</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-30</th>\n",
       "      <td>2.553</td>\n",
       "      <td>3386.5736</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-29</th>\n",
       "      <td>2.560</td>\n",
       "      <td>3400.1705</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-28</th>\n",
       "      <td>2.571</td>\n",
       "      <td>3406.5735</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-27</th>\n",
       "      <td>2.523</td>\n",
       "      <td>3325.3347</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            510050.XSHG  000300.XSHG\n",
       "date                                \n",
       "2018-08-31        2.533    3351.0942\n",
       "2018-08-30        2.553    3386.5736\n",
       "2018-08-29        2.560    3400.1705\n",
       "2018-08-28        2.571    3406.5735\n",
       "2018-08-27        2.523    3325.3347"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 前一个交易日价格\n",
    "price.shift(-1).head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 计算收益率\n",
    "\n",
    "每日收益率: 通过投资组合权益计算出的日收益率。\n",
    "\n",
    "$$\n",
    "每日收益率=\\frac{当前交易日总权益-前一交易日总权益}{前一交易日总权益}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>510050.XSHG</th>\n",
       "      <th>000300.XSHG</th>\n",
       "      <th>riskfree</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2018-08-31</th>\n",
       "      <td>-0.004737</td>\n",
       "      <td>-0.004951</td>\n",
       "      <td>0.018291</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-30</th>\n",
       "      <td>-0.007834</td>\n",
       "      <td>-0.010476</td>\n",
       "      <td>0.017971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-29</th>\n",
       "      <td>-0.002734</td>\n",
       "      <td>-0.003999</td>\n",
       "      <td>0.019096</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-28</th>\n",
       "      <td>-0.004278</td>\n",
       "      <td>-0.001880</td>\n",
       "      <td>0.018622</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-08-27</th>\n",
       "      <td>0.019025</td>\n",
       "      <td>0.024430</td>\n",
       "      <td>0.018767</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            510050.XSHG  000300.XSHG  riskfree\n",
       "2018-08-31    -0.004737    -0.004951  0.018291\n",
       "2018-08-30    -0.007834    -0.010476  0.017971\n",
       "2018-08-29    -0.002734    -0.003999  0.019096\n",
       "2018-08-28    -0.004278    -0.001880  0.018622\n",
       "2018-08-27     0.019025     0.024430  0.018767"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r= (price-price.shift(-1))/price.shift(-1)\n",
    "r.drop(r.index[-1], inplace=True) # last row is NaN, I think it should be droped\n",
    "# 取得中债国债收益率曲线作为无风险收益率\n",
    "r_riskfree=get_yield_curve(start_date=start, end_date=end, tenor='0S')\n",
    "r = pd.merge(r, r_riskfree, left_index=True, right_index=True,how=\"inner\")\n",
    "r.columns=[target, benchmark, \"riskfree\"]\n",
    "r.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 计算$\\beta$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "计算方法参考[RiceQuant API文档](https://www.ricequant.com/api/python/chn#backtest-results-factors)\n",
    "\n",
    "> 贝塔(beta, $\\beta$ ): [CAPM](https://en.wikipedia.org/wiki/Capital_asset_pricing_model)模型中市场基准组合项的系数，表示资产收益对市场整体收益波动的敏感程度。\n",
    " $$\n",
    " \\beta=\\frac{Cov(r_{p,e}, r_{b,e})}{Var(r_{b,e})}\n",
    " $$\n",
    "\n",
    "> 其中$r_{p,e}$为策略超额收益率（策略收益率 - 无风险组合收益率）；$r_{b,e}$为市场基准组合超额收益率（市场基准组合收益率 - 无风险组合收益率）；Cov 表示协方差；Var 表示方差\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "注意: numpy中, np.cov(a,b)的计算结果是$2\\times2$矩阵\n",
    "$$\n",
    "np.cov(a,b)=\\left[ \\begin{array}{cc} cov\\left( a,\\; a \\right) & cov\\left( a,\\; b \\right) \\\\ cov\\left( a,\\; b \\right) & cov\\left( b,\\; b \\right) \\end{array} \\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9511718021320907"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_pe=r[target]-r[\"riskfree\"]\n",
    "r_be=r[benchmark]-r[\"riskfree\"]\n",
    "beta=np.cov(r_pe, r_be)[0,1]/np.var(r_be)\n",
    "beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 求解$\\alpha$\n",
    "\n",
    "## 方法1\n",
    "计算方法参考[RiceQuant API文档](https://www.ricequant.com/api/python/chn#backtest-results-factors)\n",
    "\n",
    "> 阿尔法(alpha, $\\alpha$): [CAPM](https://en.wikipedia.org/wiki/Capital_asset_pricing_model)模型表达式中的残余项。表示策略所持有投资组合的收益中和市场整体收益无关的部分，是策略选股能力的度量。当策略所选股票的总体表现优于市场基准组合成分股时，阿尔法取正值；反之取负值。\n",
    "$$\n",
    "\\alpha=E[r_p-[r_f+\\beta \\times (r_b-r_f)]]\n",
    "$$\n",
    "\n",
    "> 其中$r_p$为策略所持有投资组合收益；$r_f$为无风险组合收益；$\\beta$为CAPM模型中的贝塔系数；E表示随机变量的期望。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.00091193563738157"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha=(r[target]-(r[\"riskfree\"]+beta*(r[benchmark]-r[\"riskfree\"]))).mean()\n",
    "alpha"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 方法2\n",
    "\n",
    "$$\n",
    "\\theta_p(t)= \\alpha_p+ \\epsilon(t)\n",
    "\\\\\n",
    "E[r_p(t)] = \\beta_p E[r_B(t)] + E[\\theta_p(t)]\n",
    "\\\\\n",
    "\\alpha_p=E[\\theta_p(t)]\n",
    "\\\\\n",
    "\\alpha_p=E[r_p(t)]-\\beta_p E[r_B(t)]\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0009119356373815703"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(r_pe.mean()-beta*r_be.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 求解投资组合风险 $\\sigma_p$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01640696817905404"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma_p=r[target].std()\n",
    "sigma_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 求解残差风险 $\\omega_p$\n",
    "\n",
    "$$\n",
    "\\omega _{p}=\\; \\sqrt{\\sigma _{p}^{2}-\\beta _{p}^{2}\\cdot \\sigma _{B}^{2}\\; }\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.005885928156058898"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "omega_p=np.sqrt(sigma_p**2-beta**2*(r[benchmark].std()**2))\n",
    "omega_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 年化\n",
    "这里假设一年有244个交易日。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.008566663958171634, 0.005885928156058898)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "day_count=len(get_trading_dates(start_date=start,end_date=end))\n",
    "annual_correct=np.sqrt(244/(day_count-1))\n",
    "annual_sigma_p=annual_correct*sigma_p\n",
    "annual_omega_p=annual_correct*omega_p\n",
    "annual_sigma_p, omega_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# RiceQuant上其他一些指标"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 年化波动率(volatility, $\\sigma_t$): \n",
    ">策略收益率的标准差，最常用的风险度量。波动率越大，策略承担的风险越高。这里假设一年有244个交易日。\n",
    "\n",
    "$$\n",
    "\\sigma =\\sqrt{\\frac{244}{n-1}\\sum_{i=1}^{n}{\\left[ r_{p}\\left( i \\right)\\; -\\; \\bar{r_{p}} \\right]^{2}}}\n",
    "$$\n",
    "\n",
    ">其中，n为回测期内交易日数目；$r_p(i)$ 表示第 i 个交易日策略所持有投资组合的日收益率；$ \\bar{r_p} $ 为回测期内策略日收益率的均值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.008566663958171633"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "volatility=np.sqrt(244/(day_count-1)*(r[target].std()**2)) # 定义与年化投资组合风险sigma_p一致\n",
    "volatility"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 年化跟踪误差(tracking error,$\\sigma_t$ ): \n",
    "\n",
    ">纯多头主动交易策略（阿尔法策略和基准择时策略）收益和市场基准组合收益之间差异的度量。跟踪误差越大，意味着策略所持有投资组合偏离基准组合的程度越大。需要注意，跟踪误差不适用于多-空结合的对冲策略的风险评估\n",
    "\n",
    "\n",
    "$$\n",
    "\\sigma =\\sqrt{\\frac{244}{n-1}\\sum_{i=1}^{n}{\\left[ r_{pa}\\left( i \\right)\\; -\\; \\bar{r_{pa}} \\right]^{2}}}\n",
    "\\\\\n",
    "r_{pa}(i)=r_p(i)-r_b(i)\n",
    "$$\n",
    "其中，n为回测期内交易日数量；$r_{pa}(i), r_p(i), r_b(i)$分别表示第i个交易日策略所持有投资组合的日主动收益、日收益率和基准组合的日收益率。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5218443423891064"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tracking_error=np.sqrt(244/(day_count-1 * (r_pe.std()**2)))\n",
    "tracking_error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 年化下行波动率(downside risk, $\\sigma_d$):\n",
    "\n",
    "> 相比波动率，下行波动率对收益向下波动和向上波动两种情况做出了区分，并认为只有收益向下波动才意味着风险。在实际计算中，我们统一使用基准组合收益为目标收益，作为向上波动和向下波动的判断标准。\n",
    "\n",
    "$$\n",
    "\\sigma =\\sqrt{\\frac{244}{n-1}\\sum_{i=1}^{n}{\\left[ r_{p}\\left( i \\right)\\; -\\; r_{b}(i) \\right]^{2} I(i)}}\n",
    "$$\n",
    "$$\n",
    "I(i)=\\begin{cases}1 & r_p(i)<r_b(i) \\\\ 0 & r_p(i) \\geq r_b(i)\\end{cases}\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    ">其中，n为回测期内交易日数量；$r_p,r_b$ 分别表示第i个交易日策略所持有投资组合的日收益率、基准组合的日收益率；I(i)为指示函数(indicator function)，如果第i个交易日策略所持有投资组合收益低于基准组合收益，则标记为1（向下波动），否则标记为0（向上波动）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.06079720493473829"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "downside_r=r.where(r[target]<r[benchmark]).dropna()\n",
    "downside_risk=np.sqrt(244/(day_count-1)*((downside_r[target]-downside_r[benchmark])**2).sum())\n",
    "downside_risk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 最大回撤(max drawdown): \n",
    "\n",
    "> 在回测期内，在任一交易日往后推，策略总权益走到最低点时收益率回撤幅度的最大值。最大回撤是评估策略极端风险管理能力的重要指标。其计算方式如下：\n",
    "$$\n",
    "Drawdown_t=\\begin{cases}0 & NET_t = min_{j \\geq t} NET_j\\\\\\frac{NET_t-min_{j \\geq t}NET_j}{NET_t} & ELSE\\end{cases}\n",
    "\\\\\n",
    "NET=某期净值\n",
    "\\\\\n",
    "MaxDrawdown=Max(Drawdown_t)\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Timestamp('2015-06-08 00:00:00'), 0.44965219768536796)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NET=p_target\n",
    "NET.sort_index(ascending=False, inplace=True)\n",
    "NET_min=NET.copy()\n",
    "for date in NET.index:\n",
    "    NET_min.loc[date]=(NET.loc[:date].min())\n",
    "Drawdown_t=(NET-NET_min)/NET\n",
    "max_Drawdown=Drawdown_t.max()\n",
    "max_Drawdown_date=Drawdown_t.idxmax()\n",
    "(max_Drawdown_date, max_Drawdown)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.5"
  },
  "nikola": {
   "category": "",
   "date": "2018-9-23 1:00 UTC+08:00",
   "description": "",
   "link": "",
   "slug": "Alpha_tear",
   "tags": "python",
   "title": "Alpha的后验分解",
   "type": "text"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}