{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# MCMC: Slice sampling"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Florent Leclercq,
\n",
"Imperial Centre for Inference and Cosmology, Imperial College London,
\n",
"florent.leclercq@polytechnique.org"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from scipy.integrate import quad\n",
"from scipy.stats import norm, cauchy\n",
"from math import pi\n",
"from matplotlib import pyplot as plt\n",
"from matplotlib.ticker import NullFormatter\n",
"np.random.seed(123456)\n",
"%matplotlib inline\n",
"plt.rcParams.update({'lines.linewidth': 2})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Slice sampling tutorial"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"gaussian=norm(0,1)\n",
"a=-5.\n",
"b=5.\n",
"x_arr=np.linspace(a,b,100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1- Drawing $y$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Starting from an initial $x$, we draw $y$ uniformly in $[0,f(x)]$."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"x=1.3\n",
"f_of_x=gaussian.pdf(x)\n",
"y=np.random.uniform(0.,f_of_x)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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