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    "# Distributions\n",
    "\n",
    "## Truncated normal\n",
    "\n",
    "`multiple-inference`'s truncated normal distribution allows for concave truncation sets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
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   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import seaborn as sns\n",
    "from scipy.stats import norm, truncnorm as scipy_truncnorm\n",
    "\n",
    "from multiple_inference.stats import truncnorm, quantile_unbiased\n",
    "\n",
    "sns.set()\n",
    "x = np.linspace(-1, 2)\n",
    "sns.lineplot(x=x, y=truncnorm([(-1, 0), (1, 2)]).cdf(x))\n",
    "plt.show()"
   ]
  },
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   "source": [
    "## Quantile unbiased distribution\n",
    "\n",
    "The quantile-unbiased distribution is the distribution of an unknown mean of a normal distribution given\n",
    "\n",
    "1. A realized value of the distribution,\n",
    "2. A truncation set in which the realized value had to fall, and\n",
    "3. A known variance\n",
    "\n",
    "In the example below, the realized value is .5, the truncation set is $[0, \\infty)$, and the variance (scale) is 1 by default. The interpretation of the CDF plot is, \"there is a $CDF(x)$ chance that the mean of the normal distribution from which the realized value (.5) was drawn is less than $x$\".\n",
    "\n",
    "We compare the quantile-unbiased distribution to a normal distribution centered on the realized value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "dist = quantile_unbiased(.5, truncation_set=[(0, np.inf)])\n",
    "x = np.linspace(dist.ppf(.025), dist.ppf(.975))\n",
    "sns.lineplot(x=x, y=norm.cdf(x, .5), label=\"conventional\")\n",
    "sns.lineplot(x=x, y=dist.cdf(x), label=\"quantile-unbiased\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = .5\n",
    "f\"There is a {q} chance that the mean of the normal distribution from which the realized value was drawn is less than {dist.ppf(q)}\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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