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Fix more Chap09 tiny typos

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This commit is contained in:
Youcef Rahal
2024-05-11 15:20:11 -04:00
parent 1343b68c60
commit 2ac42e70d3
1 changed files with 4 additions and 4 deletions
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8
Notebooks/Chap09/9_4_Bayesian_Approach.ipynb
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@@ -36,7 +36,7 @@
"# import libraries\n", "# import libraries\n",
"import numpy as np\n", "import numpy as np\n",
"import matplotlib.pyplot as plt\n", "import matplotlib.pyplot as plt\n",
"# Define seed so get same results each time\n", "# Define seed to get same results each time\n",
"np.random.seed(1)" "np.random.seed(1)"
] ]
}, },
@@ -85,7 +85,7 @@
}, },
"outputs": [], "outputs": [],
"source": [ "source": [
"# Draw the fitted function, together win uncertainty used to generate points\n", "# Draw the fitted function, together with uncertainty used to generate points\n",
"def plot_function(x_func, y_func, x_data=None,y_data=None, x_model = None, y_model =None, sigma_func = None, sigma_model=None):\n", "def plot_function(x_func, y_func, x_data=None,y_data=None, x_model = None, y_model =None, sigma_func = None, sigma_model=None):\n",
"\n", "\n",
" fig,ax = plt.subplots()\n", " fig,ax = plt.subplots()\n",
@@ -220,7 +220,7 @@
" &\\propto&\\text{Norm}_{\\boldsymbol\\phi}\\biggl[\\frac{1}{\\sigma^2}\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\mathbf{H}\\mathbf{y},\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\biggr].\n", " &\\propto&\\text{Norm}_{\\boldsymbol\\phi}\\biggl[\\frac{1}{\\sigma^2}\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\mathbf{H}\\mathbf{y},\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\biggr].\n",
"\\end{align}\n", "\\end{align}\n",
"\n", "\n",
"In fact, since this already a normal distribution, the constant of proportionality must be one and we can write\n", "In fact, since this is already a normal distribution, the constant of proportionality must be one and we can write\n",
"\n", "\n",
"\\begin{align}\n", "\\begin{align}\n",
" Pr(\\boldsymbol\\phi|\\{\\mathbf{x}_{i},\\mathbf{y}_{i}\\}) &=& \\text{Norm}_{\\boldsymbol\\phi}\\biggl[\\frac{1}{\\sigma^2}\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\mathbf{H}\\mathbf{y},\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\biggr].\n", " Pr(\\boldsymbol\\phi|\\{\\mathbf{x}_{i},\\mathbf{y}_{i}\\}) &=& \\text{Norm}_{\\boldsymbol\\phi}\\biggl[\\frac{1}{\\sigma^2}\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\mathbf{H}\\mathbf{y},\\left(\\frac{1}{\\sigma^2}\\mathbf{H}\\mathbf{H}^T+\\frac{1}{\\sigma_p^2}\\mathbf{I}\\right)^{-1}\\biggr].\n",
@@ -423,4 +423,4 @@
}, },
"nbformat": 4, "nbformat": 4,
"nbformat_minor": 0 "nbformat_minor": 0
} }