{
"cells": [
{
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"R version 4.2.2 (2022-10-31)\n",
"Platform: x86_64-apple-darwin17.0 (64-bit)\n",
"Running under: macOS Catalina 10.15.7\n",
"\n",
"Matrix products: default\n",
"BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib\n",
"LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib\n",
"\n",
"locale:\n",
"[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8\n",
"\n",
"attached base packages:\n",
"[1] stats graphics grDevices utils datasets methods base \n",
"\n",
"loaded via a namespace (and not attached):\n",
" [1] fansi_1.0.4 crayon_1.5.2 digest_0.6.31 utf8_1.2.2 \n",
" [5] IRdisplay_1.1 repr_1.1.5 lifecycle_1.0.3 jsonlite_1.8.4 \n",
" [9] evaluate_0.20 pillar_1.8.1 rlang_1.0.6 cli_3.6.0 \n",
"[13] uuid_1.1-0 vctrs_0.5.2 IRkernel_1.3.2 tools_4.2.2 \n",
"[17] glue_1.6.2 fastmap_1.1.0 compiler_4.2.2 base64enc_0.1-3\n",
"[21] pbdZMQ_0.3-9 htmltools_0.5.4"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sessionInfo()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"library(ggplot2)\n",
"library(statmod)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Numerical integration\n",
"\n",
"### Goal\n",
"\n",
"To evaluate $\\int_a^b f(x) dx$, $f:[a, b] \\to \\mathbb{R}$.\n",
"\n",
"* Only a few functions can be integrated analytically\n",
"* For the rest, we resort to numerical approximation\n",
"\n",
"Why do we need integrals?\n",
"\n",
"1. Expectation\n",
" - $X$ has some density on $[0, 1]$; $Y = \\sin(X)$\n",
" - $\\mathbf{E}[Y] = \\int_0^1 \\sin(x)f_X(x)dx$\n",
" \n",
"2. Bayesian inference\n",
" - Parameter $\\theta$ has a prior density $f_{\\Theta}(\\theta)$ on $\\mathbb{R}$\n",
" - Likelihood of data $x$ is $f_{X|\\Theta}(x|\\theta)$\n",
" - Want to evaluate the posterior density\n",
"$$\n",
" f_{\\Theta|X}(\\theta|x) = \\frac{f_{X|\\Theta}(x|\\theta)f_{\\Theta}(\\theta)}{\\int_{-\\infty}^{\\infty} f_{X|\\Theta}(x|\\theta')f_{\\Theta}(\\theta')d\\theta'}\n",
"$$\n",
" - The integral in the denominator may not be analytically solved.\n",
"\n",
"3. MLE of structured data\n",
" - Data: $y_{ij}$ = # recalled words by $i$th patient of Alzheimer's disease in $j$th month, $i=1,\\dotsc, n$, $j=1, \\dotsc, T$\n",
" - Model:\n",
"\\begin{align*}\n",
" Y_{ij} | \\lambda_{ij} &\\stackrel{indep}{\\sim} \\text{Poi}(\\lambda_{ij}), \\quad j=1,\\dotsc, T\\\\\n",
" \\lambda_{ij} &= \\exp(\\gamma_i + \\beta_0 + j\\beta_1) \\\\\n",
" \\gamma_i &\\stackrel{i.i.d.}{\\sim} N(0, \\sigma^2), \\quad i=1, \\dots, n\n",
"\\end{align*}\n",
" - Called a generaized linear mixed model (here, Poisson GLM with a *random intercept*)\n",
" - Likelihood:\n",
"\\begin{align*}\n",
" L(\\beta_0, \\beta_1, \\sigma^2) &= \\prod_{i=1}^n p(y_{i1}, \\dotsc, y_{iT}) \\\\\n",
" &= \\prod_{i=1}^n \\int_{-\\infty}^{\\infty} \\prod_{j=1}^T p(y_{ij} | \\lambda_{ij}(\\gamma_i) ) f(\\gamma_i)d\\gamma_i \\\\\n",
" &= \\prod_{i=1}^n \\int_{-\\infty}^{\\infty} \\prod_{j=1}^T e^{-\\lambda_{ij}(\\gamma_i)}\\frac{\\lambda_{ij}^{y_{ij}}(\\gamma_i)}{y_{ij}!}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{\\gamma_i^2}{2\\sigma^2}}d\\gamma_i, \\\\\n",
" \\text{where}~~ & \\lambda_{ij}(\\gamma_i) = \\exp(\\gamma_i + \\beta_0 + j\\beta_1). \n",
"\\end{align*}\n",
" - Likelihood function (hence score function) consists of integrals, which in most case do not have closed form.\n",
" \n",
"What should we do?\n",
"\n",
"* Quadrature methods (now) - goes back to Archimedes\n",
"* Monte Carlo methods (next) - 20th century invention"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Newton-Côtes Quadrature\n",
"\n",
"* Key idea: \n",
" 1. Subdivide the integration domain $[a, b]$ into $n$ *equally spaced* subintervals $[x_i, x_{i+1}]$ of length $h=\\frac{b - a}{n}$, i.e., $x_i = a + i h$, $i=0, 1, \\dotsc, n-1$.\n",
" 2. Approximate the integrand $f$ on $[x_i, x_{i+1}]$ by an *interpolating polynomial* $p_i$ of order $m$.\n",
" 3. Approximate $I_i = \\int_{x_i}^{x_{i+1}}f(x)dx$ by $\\int_{x_i}^{x_{i+1}}p_i(x)dx$. We know how to compute the latter exactly.\n",
" 4. Approximate the integral $I=\\int_a^b f(x)dx$ by $\\sum_{i=0}^{n-1} I_i$.\n",
" \n",
"* If $n$ is sufficiently large, the approximation will be accurate enough. \n",
"\n",
"* Choice of order $m$ makes difference:\n",
" - $m = 0$: Riemann (rectangular) rule\n",
" - $m = 1$: Trapezoidal rule\n",
" - $m = 2$: Simpson's rule\n",
"\n",
"![](\"images/quadrature.png\")
\n",
"(Taken from Givens and Hoeting)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Riemann rule\n",
"\n",
"* Approximate $f$ on $[x_i, x_{i+1}]$ by a constant function $f(x_i)$:\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx \\approx \\int_{x_i}^{x_{i+1}}f(x_i)dx = (x_{i+1} - x_i)f(x_i) = h f(x_i).\n",
"$$\n",
"\n",
"* How accurate is the Riemann rule? Assume that $f$ is four-times continuously differentiable. Then,\n",
"$$\n",
" f(x) = f(x_i) + (x - x_i)f'(x_i) + \\frac{1}{2}(x - x_i)^2 f''(x_i) + \\frac{1}{6}(x - x_i)^3f^{(3)}(x_i) + O(|x - x_i|^4)\n",
"$$\n",
"and\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx = h f(x_i) + \\frac{h^2}{2}f'(x_i) + \\frac{h^3}{6}f''(x_i) + O(h^4)\n",
" .\n",
"$$\n",
"Therefore, \n",
"\\begin{align*}\n",
" \\text{(error)} = \\sum_{i=0}^{n-1}(I_i - h f(x_i)) &= \\frac{h^2}{2}\\sum_{i=0}^{n-1}f'(x_i) + n O(h^3) \\\\\n",
" &\\le \\frac{(b-a)^2}{2n^2} n \\sup_{x\\in [a, b]}|f'(x)| + n O\\left(\\frac{(b-a)^3}{n^3}\\right) \\\\\n",
" &= O(n^{-1})\n",
" .\n",
"\\end{align*}"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true,
"jupyter": {
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}
},
"outputs": [],
"source": [
"riemann <- function(f, a, b, n) {\n",
" h <- (b - a) / n \n",
" \n",
" xi <- seq.int(a, b, length.out = n + 1)\n",
" xi <- xi[-1]\n",
" xi <- xi[-length(xi)]\n",
"\n",
" intgrl <- h * (f(a) + sum(f(xi)))\n",
" \n",
" return(intgrl)\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
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"[1] 0.4596977"
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],
"source": [
"f <- function(x) sin(x) # integrand: example above with uniform X\n",
"(truth <- 1 - cos(1)) # true integral"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true,
"jupyter": {
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},
"outputs": [
{
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"[1] 0.4554865"
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"source": [
"riemann(f, 0, 1, 100)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true,
"jupyter": {
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}
},
"outputs": [],
"source": [
"numsub <- c(10, 100, 1000, 10000, 100000)\n",
"err_R <- numeric(length(numsub))\n",
"for (i in seq_along(numsub)) {\n",
" n <- numsub[i]\n",
" R <- riemann(f, 0, 1, n)\n",
" err_R[i] <- R - truth\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
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"outputs": [
{
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AIIIICAbwJRvh3GUaESOHPmjJw8eVL2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"text/plain": [
"plot without title"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"ggplot() + geom_point(aes(x=numsub, y=abs(err_R))) + geom_line(aes(x=numsub, y=abs(err_R))) + \n",
"scale_x_log10() + scale_y_log10() + theme_bw(base_size = 15)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Trapezoidal rule\n",
"\n",
"* Approximate $f$ on $[x_i, x_{i+1}]$ by a linear function interpolating $(x_i, f(x_i))$ and $(x_{i+1}, f(x_{i+1}))$, i.e.,\n",
"$$\n",
" p_i(x) = f(x_i) + \\frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i}(x - x_i)\n",
" .\n",
"$$\n",
"So\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx \\approx \\int_{x_i}^{x_{i+1}}p_i(x)dx = \\frac{x_{i+1} - x_i}{2}[f(x_i) + f(x_{i+1})] = \\frac{h}{2} [f(x_i) + f(x_{i+1})].\n",
"$$\n",
"and\n",
"$$\n",
" I = \\int_a^b f(x)dx \\approx h\\left(\\frac{1}{2}f(a) + f(x_1) + \\dotsb + f(x_{n-1}) + \\frac{1}{2}f(b)\\right) =: T(n)\n",
" .\n",
"$$\n",
"* How accurate is the Trapezoidal rule? Assume again that $f$ is four-times continuously differentiable. Then,\n",
"$$\n",
" f(x) = f(x_i) + (x - x_i)f'(x_i) + \\frac{1}{2}(x - x_i)^2 f''(x_i) + \\frac{1}{6}(x - x_i)^3f^{(3)}(x_i) + O(|x - x_i|^4)\n",
"$$\n",
"and\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx = h f(x_i) + \\frac{h^2}{2}f'(x_i) + \\frac{h^3}{6}f''(x_i) + \\frac{h^4}{24}f^{(3)}(x_i) + O(h^5)\n",
" .\n",
"$$\n",
"Also,\n",
"\\begin{align*}\n",
" \\int_{x_i}^{x_{i+1}}p_i(x)dx &= \\frac{h}{2} [f(x_i) + f(x_i+h)] \\\\\n",
" &= \\frac{h}{2}\\left( f(x_i) + f(x_i) + h f'(x_i) + \\frac{h^2}{2}f''(x_i) + \\frac{h^3}{6}f^{(3)}(x_i) + O(h^4) \\right) \\\\\n",
" &= h f(x_i) + \\frac{h^2}{2}f'(x_i) + \\frac{h^3}{4}f''(x_i) + \\frac{h^4}{12}f^{(3)}(x_i) + O(h^5)\n",
"\\end{align*}\n",
"Therefore, \n",
"\\begin{align*}\n",
" T(n) - I = \\sum_{i=0}^{n-1}\\left(I_i - \\int_{x_i}^{x_{i+1}}p_i(x)dx\\right) &= \\frac{h^3}{12}\\sum_{i=0}^{n-1}f''(x_i) + n O(h^4) \\\\\n",
" &\\le \\frac{(b-a)^3}{12n^3} n \\sup_{x\\in [a, b]}|f''(x)| + n O\\left(\\frac{(b-a)^4}{n^4}\\right) \\\\\n",
" &= O(n^{-2})\n",
" .\n",
"\\end{align*}"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"trapezoidal <- function(f, a, b, n) {\n",
" h <- (b - a) / n \n",
" \n",
" xi <- seq.int(a, b, length.out = n + 1)\n",
" xi <- xi[-1]\n",
" xi <- xi[-length(xi)]\n",
"\n",
" intgrl <- h * (0.5 * f(a) + sum(f(xi)) + 0.5 * f(b))\n",
" \n",
" return(intgrl)\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.459693863311358"
],
"text/latex": [
"0.459693863311358"
],
"text/markdown": [
"0.459693863311358"
],
"text/plain": [
"[1] 0.4596939"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# f <- function(x) sqrt(1-x^2) # integrand\n",
"trapezoidal(f, 0, 1, 100) "
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.45969769413186"
],
"text/latex": [
"0.45969769413186"
],
"text/markdown": [
"0.45969769413186"
],
"text/plain": [
"[1] 0.4596977"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"truth"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"numsub <- c(10, 100, 1000, 10000, 100000)\n",
"err_T <- numeric(length(numsub))\n",
"for (i in seq_along(numsub)) {\n",
" n <- numsub[i]\n",
" T <- trapezoidal(f, 0, 1, n)\n",
" err_T[i] <- T - truth\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
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aFqBBBAAAEEEEAAAQTSFCBASpOFG/0h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\nAggg4F8BA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IIIAAAggggICVBAiQrDRa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0bdpUNm3alOFz3iCAAAJ2EyBAstsWoTwI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\nQCbwABuaahyVorqAAAAAAElFTkSuQmCC",
"text/plain": [
"plot without title"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"ggplot() + geom_point(aes(x=numsub, y=abs(err_T))) + geom_line(aes(x=numsub, y=abs(err_T))) + \n",
"scale_x_log10() + scale_y_log10() + theme_bw(base_size = 15)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Simpson's rule\n",
"\n",
"* Approximate $f$ on $[x_i, x_{i+1}]$ by a quadratic function interpolating $(x_i, f(x_i))$, $\\left(\\frac{x_i + x_{i+1}}{2}, f\\left(\\frac{x_i + x_{i+1}}{2}\\right)\\right)$, and $(x_{i+1}, f(x_{i+1}))$, i.e., *equal-spacing* of interpolation points within the equally spaced intervals $[x_i, x_{i+1}]$. The interpolating quadratic is\n",
"\\begin{align*}\n",
" p_i(x) &= f(x_i)\\frac{x - \\frac{x_i + x_{i+1}}{2}}{x_i - \\frac{x_i + x_{i+1}}{2}}\\frac{x - x_{i+1}}{x_i - x_{i+1}} \\\\\n",
" & + f\\left(\\frac{x_i + x_{i+1}}{2}\\right)\\frac{x - x_i}{\\frac{x_i + x_{i+1}}{2} - x_i}\\frac{x - x_{i+1}}{\\frac{x_i + x_{i+1}}{2} - x_{i+1}} \\\\\n",
" & + f(x_{i+1})\\frac{x - x_i}{x_{i+1} - x_i}\\frac{x - \\frac{x_i + x_{i+1}}{2}}{x_{i+1} - \\frac{x_i + x_{i+1}}{2}}\n",
" .\n",
"\\end{align*}\n",
"So\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx \\approx \\int_{x_i}^{x_{i+1}}p_i(x)dx = \\frac{h}{6} \\left[f(x_i) + 4f\\left(\\frac{x_i + x_{i+1}}{2}\\right) + f(x_{i+1})\\right].\n",
"$$\n",
"and\n",
"\\begin{align*}\n",
" I &= \\int_a^b f(x)dx \\approx h\\left(\\frac{1}{6}f(a) + \\frac{2}{3}f(\\frac{x_0 + x_1}{2}) + \\frac{1}{3}f(x_1) + \\frac{2}{3}f(\\frac{x_1 + x_2}{2}) + \\dotsb + \\frac{1}{3}f(x_{n-1}) + \\frac{2}{3}f(\\frac{x_{n-1} + x_n}{2}) + \\frac{1}{6}f(b)\\right) \\\\\n",
" &=: S(n)\n",
" .\n",
"\\end{align*}\n",
"* How accurate is the Simpson rule? Assume that $f$ is five-times continuously differentiable. Then,\n",
"$$\n",
" f(x) = f(x_i) + (x - x_i)f'(x_i) + \\frac{1}{2}(x - x_i)^2 f''(x_i) + \\frac{1}{6}(x - x_i)^3f^{(3)}(x_i) + \\frac{1}{24}(x - x_i)^4f^{(4)}(x_i) + O(|x - x_i|^5)\n",
"$$\n",
"and\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx = h f(x_i) + \\frac{h^2}{2}f'(x_i) + \\frac{h^3}{6}f''(x_i) + \\frac{h^4}{24}f^{(3)}(x_i) + \\frac{h^5}{120}f^{(4)}(x_i) + O(h^6)\n",
" .\n",
"$$\n",
"Also,\n",
"\\begin{align*}\n",
" \\int_{x_i}^{x_{i+1}}p_i(x)dx &= \\frac{h}{6} \\left[f(x_i) + 4f\\left(x_i + \\frac{h}{2}\\right) + f(x_i + h)\\right] \\\\\n",
" &= \\frac{h}{6}\\left( f(x_i) + 4f(x_i) + 2h f'(x_i) + \\frac{h^2}{2}f''(x_i) + \\frac{h^3}{12}f^{(3)}(x_i) + \\frac{h^4}{96}f^{(4)}(x_i) + O(h^5) \\right.\\\\\n",
" & \\left. \\quad + f(x_i) + h f'(x_i) + \\frac{h^2}{2}f''(x_i) + \\frac{h^3}{6}f^{(3)}(x_i) + \\frac{h^4}{24}f^{(4)}(x_i)\n",
" + O(h^5) \\right) \\\\\n",
" &= h f(x_i) + \\frac{h^2}{2}f'(x_i) + \\frac{h^3}{6}f''(x_i) + \\frac{h^4}{24}f^{(3)}(x_i) + \\frac{5h^5}{576}f^{(4)}(x_i) + O(h^6)\n",
"\\end{align*}\n",
"Therefore, \n",
"\\begin{align*}\n",
" S(n) - I = \\sum_{i=0}^{n-1}\\left(I_i - \\int_{x_i}^{x_{i+1}}p_i(x)dx\\right) &= -\\frac{h^5}{2880}\\sum_{i=0}^{n-1}f^{(4)}(x_i) + n O(h^6) \\\\\n",
" &\\le \\frac{(b-a)^5}{2880n^5} n \\sup_{x\\in [a, b]}|f^{(4)}(x)| + n O\\left(\\frac{(b-a)^6}{n^6}\\right) \\\\\n",
" &= O(n^{-4})\n",
" .\n",
"\\end{align*}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note the jump of accuracy order from $O(n^{-2})$ (Trapezoidal) to $O(n^{-4})$ (Simpson)!\n",
"\n",
"Also note that Simpson's method use **twice** more points than the Trapezoidal method. Hence for fair comparison, compare them with the same number of evaluation points."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"simpson <- function(f, a, b, n) {\n",
" h <- (b - a) / n \n",
" i <- seq_len(n - 1)\n",
" xi <- a + i * h \n",
" xmid <- c(xi - h / 2, b - h / 2)\n",
" intgrl <- h * (f(a) + 2 * sum(f(xi)) + 4 * sum(f(xmid)) + f(b)) / 6\n",
" \n",
" return(intgrl)\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.459697694157399"
],
"text/latex": [
"0.459697694157399"
],
"text/markdown": [
"0.459697694157399"
],
"text/plain": [
"[1] 0.4596977"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# f <- function(x) sqrt(1-x^2) # integrand\n",
"simpson(f, 0, 1, 50) # in fact about 100 pts"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.45969769413186"
],
"text/latex": [
"0.45969769413186"
],
"text/markdown": [
"0.45969769413186"
],
"text/plain": [
"[1] 0.4596977"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"truth"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's equate the total number of points."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"simpson <- function(f, a, b, n) {\n",
" h <- (b - a) / n\n",
" \n",
" xi <- seq.int(a, b, length.out = n + 1)\n",
" xi <- xi[-1]\n",
" xi <- xi[-length(xi)]\n",
" \n",
" intgrl <- (h / 3) * (f(a) + 2 * sum(f(xi[seq.int(2, length(xi), 2)])) + \n",
" 4 * sum(f(xi[seq.int(1, length(xi), 2)])) + f(b))\n",
" \n",
" return(intgrl)\n",
" \n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.459697694157399"
],
"text/latex": [
"0.459697694157399"
],
"text/markdown": [
"0.459697694157399"
],
"text/plain": [
"[1] 0.4596977"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"simpson(f, 0, 1, 100)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"0.45969769413186"
],
"text/latex": [
"0.45969769413186"
],
"text/markdown": [
"0.45969769413186"
],
"text/plain": [
"[1] 0.4596977"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"truth"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"numsub <- c(10, 100, 1000, 10000, 100000)\n",
"err_S <- numeric(length(numsub))\n",
"for (i in seq_along(numsub)) {\n",
" n <- numsub[i]\n",
" S <- simpson(f, 0, 1, n)\n",
" err_S[i] <- S - truth\n",
"}"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
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hJ39uxZqVSpkklQMjIy5P7775dHH31UwsLC8iyvAyzo0qBBg/O286z7/vvvzel3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\nIIAAAggEJvB/H9vZpsBwsDwAAAAASUVORK5CYII=",
"text/plain": [
"plot without title"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"ggplot() + geom_point(aes(x=numsub, y=abs(err_S))) + geom_line(aes(x=numsub, y=abs(err_S))) + \n",
"scale_x_log10() + scale_y_log10() + theme_bw(base_size = 15)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [],
"source": [
"comp <- data.frame(method = factor(c(rep(\"riemann\", length(numsub)), rep(\"trapezoidal\", length(numsub)), \n",
" rep(\"simpson\", length(numsub)) )),\n",
" numsub = c(numsub = rep(numsub, 3)), \n",
" error = c(err_R, err_T, err_S) )"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"A data.frame: 15 × 3\n",
"\n",
"\t | method | numsub | error |
|---|
\n",
"\t | <fct> | <dbl> | <dbl> |
|---|
\n",
"\n",
"\n",
"\t| numsub1 | riemann | 1e+01 | -4.245669e-02 |
|---|
\n",
"\t| numsub2 | riemann | 1e+02 | -4.211186e-03 |
|---|
\n",
"\t| numsub3 | riemann | 1e+03 | -4.207738e-04 |
|---|
\n",
"\t| numsub4 | riemann | 1e+04 | -4.207393e-05 |
|---|
\n",
"\t| numsub5 | riemann | 1e+05 | -4.207359e-06 |
|---|
\n",
"\t| numsub6 | trapezoidal | 1e+01 | -3.831453e-04 |
|---|
\n",
"\t| numsub7 | trapezoidal | 1e+02 | -3.830821e-06 |
|---|
\n",
"\t| numsub8 | trapezoidal | 1e+03 | -3.830814e-08 |
|---|
\n",
"\t| numsub9 | trapezoidal | 1e+04 | -3.830813e-10 |
|---|
\n",
"\t| numsub10 | trapezoidal | 1e+05 | -3.830658e-12 |
|---|
\n",
"\t| numsub11 | simpson | 1e+01 | 2.556920e-07 |
|---|
\n",
"\t| numsub12 | simpson | 1e+02 | 2.553913e-11 |
|---|
\n",
"\t| numsub13 | simpson | 1e+03 | 2.664535e-15 |
|---|
\n",
"\t| numsub14 | simpson | 1e+04 | 5.551115e-17 |
|---|
\n",
"\t| numsub15 | simpson | 1e+05 | 5.551115e-17 |
|---|
\n",
"\n",
"
\n"
],
"text/latex": [
"A data.frame: 15 × 3\n",
"\\begin{tabular}{r|lll}\n",
" & method & numsub & error\\\\\n",
" & & & \\\\\n",
"\\hline\n",
"\tnumsub1 & riemann & 1e+01 & -4.245669e-02\\\\\n",
"\tnumsub2 & riemann & 1e+02 & -4.211186e-03\\\\\n",
"\tnumsub3 & riemann & 1e+03 & -4.207738e-04\\\\\n",
"\tnumsub4 & riemann & 1e+04 & -4.207393e-05\\\\\n",
"\tnumsub5 & riemann & 1e+05 & -4.207359e-06\\\\\n",
"\tnumsub6 & trapezoidal & 1e+01 & -3.831453e-04\\\\\n",
"\tnumsub7 & trapezoidal & 1e+02 & -3.830821e-06\\\\\n",
"\tnumsub8 & trapezoidal & 1e+03 & -3.830814e-08\\\\\n",
"\tnumsub9 & trapezoidal & 1e+04 & -3.830813e-10\\\\\n",
"\tnumsub10 & trapezoidal & 1e+05 & -3.830658e-12\\\\\n",
"\tnumsub11 & simpson & 1e+01 & 2.556920e-07\\\\\n",
"\tnumsub12 & simpson & 1e+02 & 2.553913e-11\\\\\n",
"\tnumsub13 & simpson & 1e+03 & 2.664535e-15\\\\\n",
"\tnumsub14 & simpson & 1e+04 & 5.551115e-17\\\\\n",
"\tnumsub15 & simpson & 1e+05 & 5.551115e-17\\\\\n",
"\\end{tabular}\n"
],
"text/markdown": [
"\n",
"A data.frame: 15 × 3\n",
"\n",
"| | method <fct> | numsub <dbl> | error <dbl> |\n",
"|---|---|---|---|\n",
"| numsub1 | riemann | 1e+01 | -4.245669e-02 |\n",
"| numsub2 | riemann | 1e+02 | -4.211186e-03 |\n",
"| numsub3 | riemann | 1e+03 | -4.207738e-04 |\n",
"| numsub4 | riemann | 1e+04 | -4.207393e-05 |\n",
"| numsub5 | riemann | 1e+05 | -4.207359e-06 |\n",
"| numsub6 | trapezoidal | 1e+01 | -3.831453e-04 |\n",
"| numsub7 | trapezoidal | 1e+02 | -3.830821e-06 |\n",
"| numsub8 | trapezoidal | 1e+03 | -3.830814e-08 |\n",
"| numsub9 | trapezoidal | 1e+04 | -3.830813e-10 |\n",
"| numsub10 | trapezoidal | 1e+05 | -3.830658e-12 |\n",
"| numsub11 | simpson | 1e+01 | 2.556920e-07 |\n",
"| numsub12 | simpson | 1e+02 | 2.553913e-11 |\n",
"| numsub13 | simpson | 1e+03 | 2.664535e-15 |\n",
"| numsub14 | simpson | 1e+04 | 5.551115e-17 |\n",
"| numsub15 | simpson | 1e+05 | 5.551115e-17 |\n",
"\n"
],
"text/plain": [
" method numsub error \n",
"numsub1 riemann 1e+01 -4.245669e-02\n",
"numsub2 riemann 1e+02 -4.211186e-03\n",
"numsub3 riemann 1e+03 -4.207738e-04\n",
"numsub4 riemann 1e+04 -4.207393e-05\n",
"numsub5 riemann 1e+05 -4.207359e-06\n",
"numsub6 trapezoidal 1e+01 -3.831453e-04\n",
"numsub7 trapezoidal 1e+02 -3.830821e-06\n",
"numsub8 trapezoidal 1e+03 -3.830814e-08\n",
"numsub9 trapezoidal 1e+04 -3.830813e-10\n",
"numsub10 trapezoidal 1e+05 -3.830658e-12\n",
"numsub11 simpson 1e+01 2.556920e-07\n",
"numsub12 simpson 1e+02 2.553913e-11\n",
"numsub13 simpson 1e+03 2.664535e-15\n",
"numsub14 simpson 1e+04 5.551115e-17\n",
"numsub15 simpson 1e+05 5.551115e-17"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"comp"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": true,
"jupyter": {
"outputs_hidden": true
}
},
"outputs": [
{
"data": {
"image/png": 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Y3N8FTW9KgnnXSSbKD0/fffO9l0mjRp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\nAElFTkSuQmCC",
"text/plain": [
"plot without title"
]
},
"metadata": {
"image/png": {
"height": 420,
"width": 420
}
},
"output_type": "display_data"
}
],
"source": [
"ggplot(data = comp, aes(numsub, abs(error))) + geom_point(aes(colour=method)) + geom_line(aes(colour=method)) + \n",
"scale_x_log10() + scale_y_log10() + theme_bw(base_size = 15)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### General $m$\n",
"\n",
"Use the Lagrange polynomial \n",
"$$\n",
"L_{ij}^m(x) = \\prod_{k\\neq j}^{m+1} \\frac{x - x_{ik}^*}{x_{ij}^* - x_{ik}^*}\n",
"$$\n",
"for equally spaced $x_{ij}^*$ in $[x_i, x_{i+1}]$ such that\n",
"$$\n",
" p_i(x) = \\sum_{j=0}^m f(x_{ij}^*)L_{ij}^m(x)\n",
" .\n",
"$$\n",
"\n",
"It is easy to check\n",
"$$\n",
"L_{ij}^m(x_{ik}^*) = \\begin{cases} 1, & k = j , \\\\\n",
" 0, & k \\neq j.\n",
" \\end{cases}\n",
"$$\n",
"Thus $p_i(x)$ is a polynomial of degree at most $m$ that interpolates $(x_{ij}^*, f(x_{ij}^*))$.\n",
"\n",
"The Lagrange polynomial proves the *interpolation theorem*:\n",
"\n",
"**Theorem 1**. Given $m+1$ distinct points $x_0, \\dotsc, x_m \\in \\mathbb{R}$ and the corresponding values $y_0, \\dotsc, y_m \\in \\mathbb{R}$, there exists a unique polynomial of degree at most $m$ that interpolates $\\{(x_0, y_0), \\dotsc, (x_m, y_m)\\} \\subset \\mathbb{R}^2$.\n",
"\n",
"*Proof* The existence of such a polynomial is shown above. To show uniqueness, let \n",
"$$\n",
" p(x) = \\sum_{j=0}^m y_j L_j^m(x)\n",
"$$\n",
"be the interpolating polynomial constructed using the Lagrange polynomial, and $q(x)$ be another interpolating polynomial of degree at most $m$. Then we have\n",
"\\begin{align*}\n",
" p(x_0) &= q(x_0) \\\\\n",
" &\\vdots \\\\\n",
" p(x_m) &= q(x_m),\n",
"\\end{align*}\n",
"so that $f(x) = p(x) - q(x)$ has $m+1$ distinct roots. Since $f(x)$ is a polynomial of degree at most $m$, by the fundamental theorem of algebra, it can have at most $m$ roots. Therefore, $f(x) = 0$ for all $x$ or \n",
"$$\n",
" p(x) = q(x)\n",
"$$\n",
"for all $x$. \n",
"\n",
"**Corollary 1**. Let $f(x)$ is a polynomial of degree at most $m$. Suppose $x_0, \\dotsc, x_m$ subdivide $[a, b]$ so that $x_0 = a$, $x_m = b$, and $x_j$'s are distinct.\n",
"Then\n",
"$$\n",
" \\int_a^b f(x)dx = \\sum_{j=0}^m w_j f(x_j),\n",
"$$\n",
"where $w_j = \\int_a^b L_j^m(x)dx$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$f(x) = \\sum_{j=0}^m f(x_j)L_j^m(x)$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Gaussian quadrature\n",
"\n",
"In Newton-Côtes quadrature, evaluation points $x_{ij}^*$ are **equally spaced** in $[x_i, x_{i+1}]$ and the integration $I_i$ is approximated as\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx \\approx \\sum_{j=0}^m w_{ij}f(x_{ij}^*)\n",
"$$\n",
"by choosing $w_{ij}$ optimally, in the sense that the approximation is *exact* if $f$ is up to $m$th degree polynomial (Corollary 1).\n",
"\n",
"We may remove the *constraint* of equal spacing and optimally choose both $w_{ij}$ and $x_{ij}^*$.\n",
"\n",
"**Theorem 2**. If $f$ is a polynomial of degree at most $2m+1$, then there exist $\\{x_{i0}^*, \\dotsc, x_{im}^*\\} \\subset [x_i, x_{i+1}]$ and $\\{w_{i0}, \\dotsc, w_{im}\\} \\subset \\mathbb{R}$ such that\n",
"$$\n",
" I_i = \\int_{x_i}^{x_{i+1}}f(x)dx = \\sum_{j=0}^m w_{ij}f(x_{ij}^*)\n",
" .\n",
"$$\n",
"Furthermore, $w_{ij} > 0$ for all $j$.\n",
"\n",
"Thus the quadrature rule using $m+1$ points can be exact up to $2m+1$ degree polynomial, a big improvement from Newton-Côtes."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Review: Linear Algebra\n",
"\n",
"Let $\\mathcal{P}^m$ be the set of all polynomials of degree at most $m$. We all know that $\\mathcal{P}^m$ is a vector space (over the real numbers) of dimension $m+1$, and $\\{1, x, x^2, \\dotsc, x^m\\}$ forms a basis.\n",
"\n",
"It is easy to check for any given distinct points $\\{x_0, \\dotsc, x_m\\}$, $\\{L_0^m(x), \\dotsc, L_m^m(x)\\}$ also forms a basis of $\\mathcal{P}^m$.\n",
"\n",
"Now consider an interval $(a, b) \\subset \\mathbb{R}$, and $\\mathcal{P}^m(a,b)$ be the $\\mathcal{P}^m$ restricted to $(a, b)$. If we define\n",
"$$\n",
" \\langle \\mathbf{p}, \\mathbf{q} \\rangle = \\int_a^b p(x)q(x)dx: \\quad \\mathbb{P}^m(a,b) \\times \\mathbb{P}^m(a,b) \\to \\mathbb{R}\n",
"$$\n",
"for $\\mathbf{p}=p(x)$ and $\\mathbf{q}=q(x)$ in $\\mathbb{P}^m(a,b)$, then $\\langle \\mathbf{p}, \\mathbf{q} \\rangle$ is an inner product of $\\mathbf{p}$ and $\\mathbf{q}$, and $\\mathbb{P}^m(a,b)$ is an inner product space:\n",
"1. $\\langle \\mathbf{p}, \\mathbf{p} \\rangle \\ge 0$;\n",
"2. $\\langle \\mathbf{p}, \\mathbf{p} \\rangle = 0 \\iff \\mathbf{p} = 0$, i.e., $p(x)=0$ for all $x\\in[a, b]$;\n",
"3. $\\langle \\alpha \\mathbf{p}, \\mathbf{q} \\rangle = \\alpha \\langle \\mathbf{p}, \\mathbf{q} \\rangle$, $\\alpha \\in \\mathbb{R}$;\n",
"4. $\\langle \\mathbf{p}, \\mathbf{q} \\rangle = \\langle \\mathbf{q}, \\mathbf{p} \\rangle$;\n",
"5. $\\langle \\mathbf{p} + \\mathbf{q}, \\mathbf{r} \\rangle = \\langle \\mathbf{p}, \\mathbf{r} \\rangle + \\langle \\mathbf{q}, \\mathbf{r} \\rangle$.\n",
"\n",
"This inner product allows us to define a norm $\\|\\mathbf{p}\\|$ over $\\mathcal{P}^m(a,b)$:\n",
"$$\n",
" \\|\\mathbf{p}\\| = \\langle \\mathbf{p}, \\mathbf{p} \\rangle^{1/2} =\n",
" \\left(\\int_a^b p^2(x)dx\\right)^{1/2}.\n",
"$$\n",
"\n",
"An orthonormal basis of $\\mathcal{P}^m(a, b)$ can be constructed by using the Gram-Schmidt procedure: let $\\mathbf{z}_j = z_j(x) = x^{j}$, $j=0, \\dotsc, m$. Then set $\\mathbf{q}_0 = \\mathbf{z}_0 / \\|\\mathbf{z}_0\\|$ and\n",
"\\begin{align*}\n",
" \\tilde{\\mathbf{q}}_j &= \\mathbf{z}_j - \\sum_{k=0}^{j-1}\\langle \\mathbf{z}_j, \\mathbf{q}_k \\rangle \\mathbf{q}_k \\\\\n",
" \\mathbf{q}_j &= \\tilde{\\mathbf{q}}_j / \\|\\tilde{\\mathbf{q}}_j\\|\n",
"\\end{align*}\n",
"for $j=0, \\dotsc, m$. This amounts to\n",
"\\begin{align*}\n",
" \\tilde{q}_j(x) &= x^j - \\sum_{k=0}^{j-1}\\left(\\int_a^b t^j q_k(t) dt\\right) q_k(x) \\\\\n",
" q_j(x) &= \\frac{\\tilde{q}_j(x)}{\\left(\\int_a^b \\tilde{q}_j^2(t)dt\\right)^{1/2}}\n",
"\\end{align*}\n",
"\n",
"Orthogonal polynomial $q_j(x)$ are called the Legendre polynomial. Note that Legendre polynomials are nested:\n",
"$$\n",
"\\text{span}\\{\\mathbf{q}_0, \\dotsc, \\mathbf{q}_m\\}\n",
"= \\mathcal{P}^{m}(a,b) \\subset \\mathcal{P}^{m+1}(a, b)\n",
"= \\text{span}\\{\\mathbf{q}_0, \\dotsc, \\mathbf{q}_m, \\mathbf{q}_{m+1} \\}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Proof of Theorem 2\n",
"\n",
"*Proof of Theorem 2*. WLOG change the notation to $\\{x_0=a, x_1, \\dotsc, x_{m-1}, x_m = b\\}$ and $\\{w_0, \\dotsc, w_m\\}$. \n",
"\n",
"Since $\\mathbf{f} = f(x) \\in \\mathcal{P}^{2m+1}(a, b)$, polynomial division yields\n",
"$$\n",
" f(x) = g(x)q_{m+1}(x) + r(x),\n",
"$$\n",
"where $\\mathbf{q}_{m+1} = q_{m+1}(x)$ is the $m+1$st Legendre polynomial in $\\{\\mathbf{q}_0, \\dotsc, \\mathbf{q}_{2m+1}\\}$, which forms an orthonomal basis of $\\mathcal{P}^{2m+1}(a, b)$; $\\mathbf{g} = g(x)$ and $\\mathbf{r} = r(x)$ are polynomials of degree at most $m$. By the construction of the Legendre polynomials, we see that \n",
"$$\n",
"\\langle \\mathbf{g}, \\mathbf{q}_{m+1} \\rangle = \\int_a^b g(x)q_{m+1}(x)dx = 0\n",
".\n",
"$$\n",
"Thus $\\int_a^b f(x)dx = \\int_a^b r(x)dx$.\n",
"\n",
"Now choose $x_0, \\dotsc, x_m$ as the distinct real roots of the $m+1$st degree polynomial $q_{m+1}(x)$. \n",
"(Such roots exist indeed; see Lemma 1 below.)\n",
"Then\n",
"$$\n",
" f(x_j) = g(x_j)q_{m+1}(x_j) + r(x_j) = r(x_j), \\quad j=0, \\dotsc, m.\n",
"$$\n",
"From Corollary 1, \n",
"$$\n",
" \\int_a^b r(x) dx = \\sum_{j=0}^m w_j r(x_j), \\quad w_j = \\int_a^b L_j^m(x) dx,\n",
"$$\n",
"where $L_j^m(x)$ is the $j$th Lagrange polynomial given $\\{x_0, \\dotsc, x_m\\}$ and $\\{r(x_0), \\dotsc, r(x_m)\\}$.\n",
"\n",
"Thus\n",
"$$\n",
" \\int_a^b f(x)dx = \\int_a^b r(x) dx = \\sum_{j=0}^m w_j r(x_j) = \\sum_{j=0}^m w_j f(x_j),\n",
"$$\n",
"as desired.\n",
"\n",
"It remains to show that $w_k > 0$ for any $k\\in \\{0, 1, \\dotsc, m\\}$. Set $f(x) = [L_k^m(x)]^2$, a $2m$-degree polynomial. By the definition of the Lagrange polynomial, \n",
"$$\n",
" f(x_j) = \\begin{cases} 1 & j = k \\\\\n",
" 0 & j \\neq k\n",
" \\end{cases}\n",
"$$\n",
"So,\n",
"$$\n",
" 0 < \\int_a^b f(x) dx = \\sum_{j=0}^m w_j f(x_j) = \\sum_{j=0}^m w_j\\delta_{jk} = w_k\n",
" .\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Lemma 1**. The roots of $k$th Legendre polynomial $q_k(x)$ are all real and distinct. \n",
"\n",
"*Proof*. Suppose the contrary is true. Then $q_k(x)$ changes its sign fewer than $k$ times. \n",
"Let the roots of sign changes be $r_1 < \\dotsb < r_l$ for $l < k$.\n",
"Then $q_k(x)\\prod_{i=1}^l (x - r_i)$ will not suffer sign changes, only being zero at $r_1, \\dotsc, r_l$. Thus\n",
"$$\n",
" \\int_a^b q_k(x)\\prod_{i=1}^l (x - r_i) dx \\neq 0,\n",
"$$\n",
"which is a contradiction since the degree of polynomial $\\prod_{i=1}^l (x - r_i)$ is less than $k$ and is orthogonal to $q_k(x)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Degree $2m+1$ is maximal\n",
"\n",
"Consider $f(x) = \\prod_{j=0}^m (x-x_j)^2$, where $x_j$ are the roots of $q_{m+1}$. Obviously $f$ is a $(2m+2)$-degree polynomial. However,\n",
"$$\n",
" 0 < \\int_a^b f(x) dx \\neq \\sum_{j=0}^m w_j f(x_j)\n",
"$$\n",
"since $f(x_j) = 0$ for all $j = 0, \\dotsc, m$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Orthogonal polynomials\n",
"\n",
"If we change the inner product of $\\mathcal{P}^m(a, b)$ to \n",
"$$\n",
" \\langle \\mathbf{p}, \\mathbf{q} \\rangle = \\int_a^b p(x)q(x)w(x) dx\n",
"$$\n",
"for a function $w(x)$ proportional to a density function on $(a, b)$, then the Gram-Schmidt procedure on $\\{1, x, \\dotsc, x^m\\}$ generates a different sequence of orthogonal polynomials, where the orthogonality is with respect to $w$ (called $w$-orthogonal). The Legendre polynomial corresponds to the uniform distribution on $(a, b)$.\n",
"\n",
"Typical choice of $w$ and associated $w$-orthogonal polynomials are tabulated:\n",
"\n",
"| othogonal polynomial | weight function ($w(x)$) | domain | distribution|\n",
"|:---------------------|:---------------------------------:|:-------------------:|:------------|\n",
"| Legendre | 1 | $(-1, 1)$ | uniform |\n",
"| Laguerre | $x^{\\alpha}e^{-x}$ | $(0, \\infty)$ | gamma |\n",
"| Hermite | $e^{-x^2}$ | $(-\\infty, \\infty)$ | normal |\n",
"| Jacobi | $(1-x)^{\\alpha}(1+x)^{\\beta}$ | $(-1, 1)$ | beta* |\n",
"\n",
"(*) After change of variable."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Legendre polynomial\n",
"\n",
"The $m$th degree Legendre polynomial (on $(-1, 1)$) is a solution to Legendre's ordinary differential equation \n",
"$$\n",
" (1 - x^2)u'' - 2x u' + m(m + 1)m = 0\n",
"$$\n",
"given by\n",
"$$\n",
" q_m(x) = \\frac{(-1)^m}{2^m m!}\\frac{d^m}{dx^m}[(1 - x^2)^m].\n",
"$$\n",
"\n",
"#### Laguerre polynomial\n",
"\n",
"The $m$th degree Laguerre polynomial is a solution to Laguerre's ordinary differential equation \n",
"$$\n",
" xu'' + (\\alpha + 1 - x)u' + m u = 0\n",
"$$\n",
"given by\n",
"$$\n",
" q_m(x) = \\Gamma(m + \\alpha)\\sum_{k=0}^m\\binom{m}{k}\\frac{(-1)^k x^k}{\\Gamma(k + \\alpha)}.\n",
"$$\n",
"\n",
"\n",
"#### Hermite polynomial\n",
"\n",
"The $m$th degree Hermite polynomial is a solution to Hermites's ordinary differential equation \n",
"$$\n",
" u'' - 2xu' + 2m u = 0\n",
"$$\n",
"given by\n",
"$$\n",
" q_m(x) = (-1)^m e^{x^2}\\frac{d^m}{dx^m}e^{-x^2}.\n",
"$$\n",
"\n",
"#### Jacobi polynomial\n",
"\n",
"The $m$th degree Jacobi polynomial is a solution to Jacobi's ordinary differential equation \n",
"$$\n",
" (1 - x^2)u'' + (\\beta - \\alpha + (\\alpha + \\beta + 2)x)u' + m(m + \\alpha + \\beta + 1) u = 0\n",
"$$\n",
"given by\n",
"$$\n",
" q_m(x) = \\frac{\\Gamma(\\alpha + m + 1)}{m!\\Gamma(\\alpha + \\beta + m + 1)}\\sum_{k=0}^m\n",
" \\frac{\\Gamma(\\alpha + \\beta + n + m + 1)}{m!\\Gamma(\\alpha + m + 1)}\n",
" \\left(\\frac{x - 1}{2}\\right)^k\n",
" .\n",
"$$\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Evaluation of orthogonal polynomials\n",
"\n",
"What we need in Gaussian quadrature is the roots of the $m$th degree orthogonal polynomial (with respect to $w$). These roots are usually found by using Newton's method. The following properties of orthogonal polynomials are useful:\n",
"\n",
"#### 3-term recursion\n",
"\n",
"$$\n",
" q_m(x) = (\\alpha_m + x\\beta_m)q_{m-1}(x) - \\gamma_m q_{m-2}(x)\n",
"$$\n",
"\n",
"for some sequence $\\alpha_m$, $\\beta_m$, and $\\gamma_m$.\n",
"\n",
"\n",
"#### Interlacing property\n",
"\n",
"If $\\mu_1 < \\dotsb < \\mu_m$ are the roots of $q_m(x)$, then the roots of $q_{m+1}(x)$ lie in each of the intervals\n",
"$$\n",
" (-\\infty, \\mu_1), ~ (\\mu_1, \\mu_2), \\dotsc, (\\mu_m, \\infty).\n",
"$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Gauss-Hermite quadrature\n",
"\n",
"If we want to integrate $f(x)$ from $-\\infty$ to $\\infty$, then Hermite polynomial is the choice:\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x)dx\n",
" = \\int_{-\\infty}^{\\infty} \\left(\\frac{f(x)}{e^{-x^2}}\\right) e^{-x^2}dx\n",
" \\approx\n",
" \\sum_{j=0}^m w_j f(x_j)e^{x_j^2}\n",
" ,\n",
"$$\n",
"where $\\{w_j\\}$ and $\\{x_j\\}$ are determined by the (scaled) Hermite polynomial.\n",
"\n",
"* The approximation is good if $\\hat{f}(x) = f(x)e^{x^2}$ is closed to a polynomial.\n",
"\n",
"* If $\\hat{f}$ is far from a polynomial, e.g., $f$ is concentrated on a point far from the origin, then $x_j$ will not be placed where most of the mass of $f$ is located, and GHQ may result in a poor approximation.\n",
" - Example: $f(x) = \\exp(-(x - \\mu)^2/(2\\sigma^2))$.\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x) dx \n",
" = \\int_{-\\infty}^{\\infty} \\exp\\left(-\\frac{(x - \\mu)^2}{2\\sigma^2} + x^2\\right)\\exp\\left(-x^2\\right)dx\n",
"$$\n",
" - Then, \n",
"$$\n",
" \\hat{f}(x) = \\exp\\left(-\\frac{(x - \\mu)^2}{2\\sigma^2} + x^2\\right)\n",
" \\propto \\exp\\left(\\left(1 - \\frac{1}{2\\sigma^2}\\right)\\left(x - \\frac{\\mu}{2\\sigma^2 - 1}\\right)^2 \\right).\n",
"$$\n",
" - If $\\mu \\gg 0$ or $\\sigma^2 \\gg 1$, then $\\hat{f}$ will be difficult to approximate with a polynomial.\n",
" \n",
"* Remedy: change of variable. Set $z = \\frac{x - \\mu}{\\sqrt{2}\\sigma}$. Then,\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x) dx = \\sqrt{2}\\sigma\\int_{-\\infty}^{\\infty} e^{-z^2}dz\n",
" = \\sigma w_0, \n",
"$$\n",
"a 1-point GHQ suffices!"
]
},
{
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"
\n"
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"\\begin{description}\n",
"\\item[\\$nodes] 0\n",
"\\item[\\$weights] 1.77245385090552\n",
"\\end{description}\n"
],
"text/markdown": [
"$nodes\n",
": 0\n",
"$weights\n",
": 1.77245385090552\n",
"\n",
"\n"
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"$nodes\n",
"[1] 0\n",
"\n",
"$weights\n",
"[1] 1.772454\n"
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],
"source": [
"# statmod::gauss.quad() calculates nodes and weights for Gaussian quadrature\n",
"statmod::gauss.quad(1, \"hermite\") # this computes w_0"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
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"source": [
"sqrt(pi) # int_{-\\infty}^{\\infty}exp(-x^2)dx"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Example: Binomial-logit mixture\n",
"\n",
"Suppose $x$ is a random sample from a Binomial distribution with $n$ trials and success probability $p=\\frac{e^{\\theta}}{1 + e^{\\theta}}$.\n",
"Suppose the prior on $\\theta$ is $N(0, \\sigma^2)$. \n",
"\n",
"* Prior density: $f_{\\Theta}(\\theta) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-x^2/(2\\sigma^2)}$.\n",
"* Likelihood: $f_{X|\\Theta}(x|\\theta) = \\binom{n}{x}p^x(1 - p)^{n-x}$, with\n",
"$p=\\frac{e^{\\theta}}{1 + e^{\\theta}}$.\n",
"* Posterior density:\n",
"\\begin{align*}\n",
" f_{\\Theta|X}(\\theta|x) &= \\frac{f_{X|\\Theta}(x | \\theta)f_{\\Theta}(\\theta)}{\\int_{-\\infty}^{\\infty}f_{X|\\Theta}(x | \\theta')f_{\\Theta}(\\theta')d\\theta'}\n",
" = \\frac{p(\\theta)^x[1-p(\\theta)]^{n-x}e^{-\\theta^2/(2\\sigma^2)}}{\\int_{-\\infty}^{\\infty}p(\\theta)^x[1-p(\\theta)]^{n-x}e^{-\\theta^2/(2\\sigma^2)}d\\theta}\n",
" \\\\\n",
" &= \\frac{e^{\\theta x}[1+e^{\\theta}]^{-n}e^{-\\theta^2/(2\\sigma^2)}}{\\int_{-\\infty}^{\\infty}e^{\\theta x}[1+e^{\\theta}]^{-n}e^{-\\theta^2/(2\\sigma^2)}d\\theta}\n",
"\\end{align*}\n",
"\n",
"Thus if we define $g(\\theta) = \\frac{e^{\\theta x}}{[1 + e^{\\theta}]^n}$,\n",
"then the normalizing constant is\n",
"$$\n",
"\\sigma\\sqrt{2}\\int_{-\\infty}^{\\infty} g(\\sigma\\sqrt{2}y)w(y)dy\n",
"$$ \n",
"where $w(y) = e^{-y^2}$,\n",
"the weight function for the Hermite polynomials. However, this turns out to be a bad idea:"
]
},
{
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"text/plain": [
"0.002829073 with absolute error < 1.5e-09"
]
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"source": [
"g <- function(theta, x, n) exp(theta * x - n * log(1 + exp(theta)))\n",
"\n",
"x <- 18 # data\n",
"n <- 20 # sample size\n",
"sigma <- 10 # standard deviation of the prior\n",
"\n",
"m <- 40 # degree of Hermite polynomial\n",
"ghq <- statmod::gauss.quad(m + 1, \"hermite\")\n",
"denom <- sigma * sqrt(2) * sum(g(sigma * sqrt(2) * ghq$nodes, x, n) * ghq$weights)\n",
"denom\n",
"# R built-in adaptive quadrature\n",
"integrate(function(y) sigma * sqrt(2) * g(sigma * sqrt(2) * y, x, n) * exp(-y^2) , -Inf, Inf) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The reason is that the integrand $g(\\theta)$ has maximum at $p(\\theta) = x/n = 0.9$ or $\\theta = \\log\\frac{p}{1-p} = 2.197 \\gg 0$.\n",
"\n",
"A better apporach is the set $\\bar{g}(\\theta) = \\frac{e^{\\theta x}}{[1 + e^{\\theta}]^n}e^{-\\theta^2/(2\\sigma^2)}=g(\\theta)e^{-\\theta^2/(2\\sigma^2)}$ and find the maximizer $\\theta^*$ of $\\bar{g}$, and change the variable to $z = \\theta - \\theta^*$. Then, the normalizing constant is\n",
"$$\n",
" \\int_{-\\infty}^{\\infty}\\bar{g}(\\theta)d\\theta \n",
" = \\int_{-\\infty}^{\\infty} \\bar{g}(z + \\theta^*)dz\n",
" = \\int_{-\\infty}^{\\infty} \\bar{g}(z + \\theta^*)e^{z^2}e^{-z^2}dz\n",
" = \\int_{-\\infty}^{\\infty}\\tilde{g}(z)w(z)dz,\n",
"$$\n",
"where \n",
"$$\n",
" \\tilde{g}(z) = \\bar{g}(z + \\theta^*)e^{z^2}\n",
" .\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 27,
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"source": [
"theta_hat <- optimize(function(y) -g(y, x, n) * exp(-y^2 / sigma^2), c(-5, 5))$minimum # univariate minimizer\n",
"theta_hat"
]
},
{
"cell_type": "code",
"execution_count": 28,
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"source": [
"g2 <- function(z, x, n, sigma) g(z + theta_hat, x, n) * exp(-(z + theta_hat)^2 / 2 / sigma^2 + z^2) # g_tilde(z+theta_hat)\n",
"denom2 <- sum(g2(ghq$nodes, x, n, sigma) * ghq$weights)\n",
"denom2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Laplace approximation\n",
"\n",
"Laplace approximation concerns replacing the integral\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x)e^{-a g(x)} dx\n",
"$$\n",
"by optimization. WLOG assume that $g$ has its minimum at 0. If $g$ is three times continuously differentiable, then Taylor expansion around 0 yields\n",
"$$\n",
" g(x) = g(0) + g'(0)x + \\frac{1}{2}g''(0)x^2 + O(x^3) = g(0) + \\frac{1}{2}g''(0)x^2 + o(x^2).\n",
"$$\n",
"Thus\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x)e^{-a g(x)} dx\n",
" \\approx\n",
" f(0)e^{-a g(0)}\\int_{-\\infty}^{\\infty}\\exp\\left(-\\frac{a g''(0)x^2}{2}\\right)dx\n",
" = f(0)e^{-ag(0)}\\sqrt{\\frac{2\\pi}{ag''(0)}}\n",
"$$\n",
"as $a \\to \\infty$, as in this case contributions to the integral is dominated by those around 0.\n",
"This amounts to reduce the integral to integration against $N(0, 1/[ag''(0)])$. A formal statement of the Laplace approximation is given below.\n",
"\n",
"**Theorem 3**. If $g$ satisfies the following conditions\n",
"1. for every $\\delta > 0$ there exists a $\\rho > 0$ such that $g(x) - g(0) \\ge \\rho$ for all $x$ with $|x| \\ge \\delta$;\n",
"2. $g$ is twice continuously differentiable in a neighborhood of 0 and $g''(0) > 0$;\n",
"3. $f$ is continuous in a neighborhood of 0 and $f(0) > 0$;\n",
"4. the integral $\\int_{-\\infty}^{\\infty}f(x)e^{-ag(x)}dx$ is absolutely convergent for $a \\ge a_0$;\n",
"\n",
"then, \n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x)e^{-a g(x)} dx\n",
" \\asymp\n",
" f(0)e^{-ag(0)}\\sqrt{\\frac{2\\pi}{ag''(0)}}\n",
"$$\n",
"as $a\\to\\infty$. That is, \n",
"$$\n",
" \\lim_{a\\to\\infty} \\left(\\int_{-\\infty}^{\\infty} f(x)e^{-a g(x)} dx\\right)/\\left(f(0)e^{-ag(0)}\\sqrt{\\frac{2\\pi}{ag''(0)}}\\right) = 1.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Applications\n",
"\n",
"#### Stirling's formula\n",
"\n",
"The gamma function\n",
"$$\n",
" \\Gamma(t) = \\int_0^{\\infty} x^{t - 1}e^{-x} dx\n",
"$$\n",
"generalizes the factorial operation: $\\Gamma(n+1) = n!$. If we define $z = x / t$, then\n",
"$$\n",
" \\Gamma(t+1) = t^{t+1}\\int_0^{\\infty}e^{-t g(z)}dz\n",
"$$\n",
"for $g(z) = z - \\log z$. Since $g$ has its minimum at $z=1$, applying the Laplace approximation yields\n",
"$$\n",
" \\Gamma(t+1) \\asymp \\sqrt{2\\pi}t^{t+1/2}e^{-t}\n",
"$$\n",
"as $t\\to \\infty$, which is the famous Stirling's formula for $t!$."
]
},
{
"cell_type": "code",
"execution_count": 29,
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"source": [
"n <- 70\n",
"factorial(n)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true,
"jupyter": {
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"source": [
"sqrt(2*pi) * n^(n + 0.5) * exp(-n)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Posterior expectation\n",
"\n",
"Recall Bayesian inference:\n",
"\n",
"- Parameter $\\theta$ has a prior density $f_{\\Theta}(\\theta)$ on $\\mathbb{R}$\n",
"- Likelihood of data $x$ is $f_{X|\\Theta}(x|\\theta)$\n",
"- Posterior density\n",
"$$\n",
" f_{\\Theta|X}(\\theta|x) = \\frac{f_{X|\\Theta}(x|\\theta)f_{\\Theta}(\\theta)}{\\int_{-\\infty}^{\\infty} f_{X|\\Theta}(x|\\theta')f_{\\Theta}(\\theta')d\\theta'}\n",
"$$\n",
"\n",
"We may want to evaluate posterior expectation $\\mathbf{E}[h(\\Theta) | X_1, \\dotsc, X_n]$ given $n$ independent observations. The required expectation takes the form\n",
"$$\n",
" \\frac{\\int e^{\\tilde{h}(\\theta)}e^{\\ell_n(\\theta) + \\pi(\\theta)}d\\theta}{\\int e^{l_n(\\theta) + \\pi(\\theta)}d\\theta}\n",
"$$\n",
"where $\\tilde{h}(\\theta) = \\log h(\\theta)$, $\\ell_n(\\theta) = \\sum_{i=1}^n \\log f_{X|\\Theta}(x_i|\\theta)$ is the the log likelihood of the data, and $\\pi(\\theta) = \\log f_{\\Theta}(\\theta)$.\n",
"\n",
"If $n$ is large, usually the log posterior $\\ell_n(\\theta) + \\pi(\\theta)$ is sharply peaked around $\\hat{\\theta}$, the posterior mode.\n",
"\n",
"Using the Laplace approximation, the denominator is approximated by\n",
"$$\n",
" \\int e^{\\ell_n(\\theta) + \\pi(\\theta)}d\\theta \n",
" \\approx e^{\\ell_n(\\hat{\\theta})+\\pi(\\hat{\\theta})} \n",
" \\sqrt{\\frac{2\\pi}{-[\\ell_n''(\\hat{\\theta}) + \\pi''(\\hat{\\theta})]}}\n",
" .\n",
"$$\n",
"\n",
"If $\\tilde{h}(\\theta) + \\ell_n(\\theta) + \\pi(\\theta)$ takes its maximum at $\\tilde{\\theta}$, then the numerator is approximated by\n",
"$$\n",
" \\int e^{\\tilde{h}(\\theta) + \\ell_n(\\theta) + \\pi(\\theta)}d\\theta \n",
" \\approx e^{\\tilde{h}(\\tilde{\\theta}) + \\ell_n(\\tilde{\\theta})+\\pi(\\tilde{\\theta})} \n",
" \\sqrt{\\frac{2\\pi}{-[\\tilde{h}(\\tilde{\\theta}) + \\ell_n''(\\tilde{\\theta}) + \\pi''(\\tilde{\\theta})]}}\n",
" .\n",
"$$\n",
"\n",
"Thus the expectation is approximated by\n",
"$$\n",
" \\exp\\left(\\tilde{h}(\\tilde{\\theta}) + \\ell_n(\\tilde{\\theta}) + \\pi(\\tilde{\\theta}) \n",
" - \\ell_n(\\hat{\\theta}) - \\pi(\\hat{\\theta})\\right)\n",
" \\sqrt{\\frac{\\ell_n''(\\hat{\\theta}) + \\pi''(\\hat{\\theta})}{\\tilde{h}''(\\tilde{\\theta}) + \\ell_n''(\\tilde{\\theta}) + \\pi''(\\tilde{\\theta})}} \n",
" .\n",
"$$"
]
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"#### Example: binomial-logit mixture\n",
"\n",
"Recall from the binomial-logit mixture example above, set \n",
"$$\n",
"f(\\theta) = e^{-\\theta^2/(2\\sigma^2)},\n",
"\\quad\n",
"g(\\theta) = -\\theta x + n\\log(1 + e^{\\theta})\n",
".\n",
"$$\n",
"\n",
"Then the Laplace approximation of the normalization constant of the posterior density is\n",
"$$\n",
"f(\\theta^*)e^{-g(\\theta^*)}\\sqrt{\\frac{2\\pi}{g''(\\theta^*)}}\n",
"$$\n",
"where $\\theta^* = \\log\\frac{x/n}{1 - x/n}$. It is easy to see that\n",
"$$\n",
" g''(\\theta^*) = n \\frac{e^{\\theta^*}}{(1 + e^{\\theta^*})^2} = n p (1 - p),\n",
" \\quad\n",
" p = x / n.\n",
"$$\n",
"For the numerical example of $x=18$ and $n=20$, we have the following:"
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"x <- 18; n <- 20\n",
"p <- x / n\n",
"theta_star <-log(p / (1-p))\n",
"lap <- exp(-theta_star^2 / 2/ sigma^2) * exp(theta_star * x) / \n",
" (1 + exp(theta_star))^n * sqrt(2 * pi / n / p / (1 - p))\n",
"lap # Laplace approximation\n",
"denom2 # Gauss-Hermite quadrature"
]
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"source": [
"#### Example: moments of beta distribution\n",
"\n",
"Recall that the beta distribution has density \n",
"$$\n",
"f_{\\Theta}(\\theta) \n",
"= \\frac{\\Gamma(\\alpha+\\beta)\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}}{\\Gamma(\\alpha)\\Gamma(\\beta)} \n",
"\\propto \\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}\n",
"$$ \n",
"on $(0, 1)$. If the likelihood of the data $X$ is $\\text{Binom}(n, \\theta)$, then the posterior density of the success probability $\\theta$ is also beta: $f_{\\Theta|X}(\\theta|x) \\propto \\theta^{\\alpha + x -1}(1-\\theta)^{\\beta + n - x -1}$. Thus, finding the moment of a beta distribution amounts to computing the posterior moment of the success probability.\n",
"\n",
"The exact value of $\\mathbf{E}[\\Theta^k]$ is\n",
"$$\n",
" \\mathbf{E}[\\Theta^k] = \\frac{\\int_0^1 \\theta^{k+\\alpha-1}(1-\\theta)^{\\beta-1}d\\theta}{\\int_0^1\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}d\\theta}\n",
" = \\frac{\\Gamma(k + \\alpha)\\Gamma(\\beta)}{\\Gamma(k + \\alpha + \\beta)}\n",
" \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}\n",
" = \\frac{\\Gamma(k + \\alpha)/\\Gamma(\\alpha)}{\\Gamma(k + \\alpha + \\beta)/\\Gamma(\\alpha + \\beta)}\n",
" = \\frac{(k+\\alpha -1)\\dotsb \\alpha}{(k+\\alpha+\\beta-1)\\dotsb(\\alpha+\\beta)}\n",
"$$\n",
"\n",
"In order to apply the Laplace approximation, observe that $-\\log(\\theta^{\\alpha-1}(1-\\theta)^{\\beta-1}) = -[(\\alpha-1)\\log\\theta + (\\beta-1)\\log(1-\\theta)] =: g(\\theta)$, which has the unique minimum at $\\hat{\\theta} = \\frac{\\alpha-1}{\\alpha+\\beta-2}\\in(0, 1)$. Therefore,\n",
"$$\n",
" \\int_0^1 \\theta^{\\alpha-1}(1-\\theta)^{\\beta-1} d\\theta\n",
" \\approx e^{-g(\\hat{\\theta})}\\sqrt{\\frac{2\\pi}{g''(\\hat{\\theta})}}\n",
" = \\left(\\frac{\\alpha-1}{\\alpha+\\beta-2}\\right)^{\\alpha-1}\\left(\\frac{\\beta-1}{\\alpha+\\beta-2}\\right)^{\\beta-1}\n",
" \\sqrt{2\\pi}\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{(\\alpha+\\beta-2)^3}}\n",
"$$\n",
"since $g(\\hat{\\theta}) = -\\log\\left[\\left(\\frac{\\alpha-1}{\\alpha+\\beta-2}\\right)^{\\alpha-1}\\left(\\frac{\\beta-1}{\\alpha+\\beta-2}\\right)^{\\beta-1}\\right]$ and $g''(\\hat{\\theta}) = (\\alpha-1)\\left(\\frac{\\alpha+\\beta-2}{\\alpha-1}\\right)^2 + (\\beta-1)\\left(\\frac{\\alpha+\\beta-2}{\\beta-1}\\right)^2$.\n",
"\n",
"Likewise,\n",
"$$\n",
" \\int_0^1 \\theta^{k+\\alpha-1}(1-\\theta)^{\\beta-1} d\\theta\n",
" \\approx \n",
" \\left(\\frac{k+\\alpha-1}{k+\\alpha+\\beta-2}\\right)^{k+\\alpha-1}\\left(\\frac{\\beta-1}{k+\\alpha+\\beta-2}\\right)^{\\beta-1}\n",
" \\sqrt{2\\pi}\\sqrt{\\frac{(k+\\alpha-1)(\\beta-1)}{(k+\\alpha+\\beta-2)^3}}\n",
"$$\n",
"\n",
"Therefore,\n",
"$$\n",
" \\mathbf{E}[\\Theta^k] \\approx\n",
" \\left(\\frac{(k+\\alpha-2)(\\alpha+\\beta-2)^3}{(\\alpha-1)(k+\\alpha+\\beta-2)^3}\\right)^{1/2}\n",
" \\cdot\n",
" \\frac{\\left(\\frac{k+\\alpha-1}{k+\\alpha+\\beta-2}\\right)^{k+\\alpha-1}\\left(\\frac{\\beta-1}{k+\\alpha+\\beta-2}\\right)^{\\beta-1}}{\\left(\\frac{\\alpha-1}{\\alpha+\\beta-2}\\right)^{\\alpha-1}\\left(\\frac{\\beta-1}{\\alpha+\\beta-2}\\right)^{\\beta-1}}\n",
"$$\n",
"\n",
"It can be shown that $\\alpha, \\beta \\to \\infty$ in such a way that $\\frac{\\alpha}{\\alpha+\\beta} \\to p \\in (0,1)$, then\n",
"the approximation of $\\mathbf{E}[\\Theta^k]$ tends to $p^k$. Also note that\n",
"$$\n",
" \\frac{(k+\\alpha -1)\\dotsb \\alpha}{(k+\\alpha+\\beta-1)\\dotsb(\\alpha+\\beta)} \\to p^k\n",
"$$\n",
"in this setting. Thus the Laplace approximation of the beta moment is correct in the limit."
]
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"### Multidimensional integration\n",
"\n",
"Laplace approximation naturally extends to multidimensional integration. On the contrary, quadrature methods needs number of points that grows exponentially with dimension.\n",
"\n",
"We will see later that Monte Carlo methods can also alleviate this difficulties.\n"
]
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"### Proof of Theorem 3\n",
"\n",
"Assume WLOG $g(0) = 0$ (multiply $e^{ag(0)}$ on both sides and set $g(x) := g(x) - g(0)$). \n",
"From Condition 2,\n",
"$$\n",
" g(x) = g(0) + g'(0)x + \\frac{1}{2}g''(0)x^2 + o(x^2) = \\frac{1}{2}g''(0)x^2 + o(x^2)\n",
"$$\n",
"as $x \\to 0$.\n",
"Combining with condition 3, for a given small $\\epsilon \\in (0, \\frac{1}{2}g''(0))$ there exists $\\delta > 0$ such that\n",
"\\begin{align*}\n",
" (1-\\epsilon) f(0) &\\le f(x) \\le (1+\\epsilon) f(0) \\\\\n",
" -\\epsilon x^2 &\\le g(x) - \\frac{1}{2}g''(0)x^2 \\le \\epsilon x^2\n",
"\\end{align*}\n",
"for all $x$ with $|x| < \\delta$.\n",
"\n",
"Now examine the contributions to the integral from the region $|x| \\ge \\delta$. From condition 1, $g(x) \\ge \\rho$. Then for $a \\ge a_0$,\n",
"\\begin{align*}\n",
" \\left|\\int_{\\delta}^{\\infty}f(x)e^{-ag(x)}dx\\right|\n",
" &\\le \\int_{\\delta}^{\\infty}|f(x)|e^{-(a - a_0)g(x)}e^{-a_0 g(x)}dx \\\\\n",
" &\\le e^{-(a - a_0)\\rho}\\int_{\\delta}^{\\infty}|f(x)|e^{-a_0 g(x)}dx \\\\\n",
" &\\le e^{-(a - a_0)\\rho}\\int_{-\\infty}^{\\infty}|f(x)|e^{-a_0 g(x)}dx \n",
" = O(e^{-\\rho a})\n",
" .\n",
"\\end{align*}\n",
"The last equality is due to condition 4. By the same argument, we also have \n",
"$$\n",
"\\int_{-\\infty}^{-\\delta}f(x)e^{-ag(x)}dx = O(e^{-\\rho a})\n",
".\n",
"$$\n",
"\n",
"The central portion of the integral can be bounded above:\n",
"\\begin{align*}\n",
" \\int_{-\\delta}^{\\delta} f(x)e^{-ag(x)}dx \n",
" & \\le (1+\\epsilon)f(0)\\int_{-\\delta}^{\\delta} e^{-\\frac{a}{2}[g''(0)-2\\epsilon]x^2} dx \\\\\n",
" & \\le (1+\\epsilon)f(0)\\int_{-\\infty}^{\\infty} e^{-\\frac{a}{2}[g''(0) - 2\\epsilon]x^2} dx\\\\\n",
" &= (1+\\epsilon)f(0)\\sqrt{\\frac{2\\pi}{a[g''(0)- 2\\epsilon]}}\n",
" .\n",
"\\end{align*}\n",
"Therefore,\n",
"$$\n",
" \\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx \\le (1+\\epsilon)f(0)\\sqrt{\\frac{2\\pi}{a[g''(0)- 2\\epsilon]}} + O(e^{-\\rho a}),\n",
"$$\n",
"yielding\n",
"$$\n",
" \\limsup_{a\\to\\infty} \\sqrt{a}\\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx\n",
" \\le (1+\\epsilon) f(0) \\sqrt{\\frac{2\\pi}{g''(0)- 2\\epsilon}}\n",
" .\n",
"$$\n",
"Sending $\\epsilon$ to zero, we have\n",
"$$\n",
" \\limsup_{a\\to\\infty} \\sqrt{a}\\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx\n",
" \\le f(0) \\sqrt{\\frac{2\\pi}{g''(0)}}\n",
" .\n",
"$$\n",
"\n",
"\n",
"We can also find a lower bound of the central portion of the integral:\n",
"\\begin{align*}\n",
" \\int_{-\\delta}^{\\delta} f(x)e^{-ag(x)}dx \n",
" & \\ge (1-\\epsilon)f(0)\\int_{-\\delta}^{\\delta} e^{-\\frac{a}{2}[g''(0)+2\\epsilon]x^2} dx \n",
" .\n",
"\\end{align*}\n",
"\n",
"\n",
"\n",
"For $|x| \\ge \\delta$, $(\\frac{1}{2}g''(0) + \\epsilon)x^2 \\ge (\\frac{1}{2}g''(0) + \\epsilon)\\delta^2 =: \\lambda$. \n",
"Furthermore, $\\int_{-\\infty}^{\\infty} e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx < \\infty$ for any $a > 0$.\n",
"Hence by repeating the same argument leading to $\\int_{\\delta}^{\\infty}f(x)e^{-ag(x)}dx = O(e^{-\\rho a})$, we have\n",
"$$\n",
" \\int_{-\\infty}^{-\\delta}e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2}dx\n",
" + \\int_{\\delta}^{\\infty}e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2}dx\n",
" = O(e^{-\\lambda a}).\n",
"$$\n",
"Therefore,\n",
"\\begin{align*}\n",
" (1-\\epsilon)f(0)\\int_{-\\delta}^{\\delta} e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx \n",
" &= (1-\\epsilon)f(0)\\int_{-\\infty}^{\\infty} e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx \\\\\n",
" & \\quad - (1-\\epsilon)f(0)\\left[\\int_{-\\infty}^{-\\delta}e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx \n",
" + \\int_{\\delta}^{\\infty}e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx \\right]\n",
" \\\\\n",
" &\\ge (1-\\epsilon)f(0)\\int_{-\\infty}^{\\infty} e^{-\\frac{a}{2}[g''(0) + 2\\epsilon]x^2} dx - O(e^{-\\lambda a}) \\\\\n",
" &= (1-\\epsilon)f(0)\\sqrt{\\frac{2\\pi}{a[g''(0)+ 2\\epsilon]}} + O(e^{-\\lambda a})\n",
" ,\n",
"\\end{align*}\n",
"yielding\n",
"$$\n",
" \\liminf_{a\\to\\infty} \\sqrt{a}\\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx\n",
" \\ge (1-\\epsilon) f(0) \\sqrt{\\frac{2\\pi}{g''(0)+ 2\\epsilon}}\n",
" .\n",
"$$\n",
"Sending $\\epsilon$ to zero, we have\n",
"$$\n",
" \\liminf_{a\\to\\infty} \\sqrt{a}\\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx\n",
" \\ge f(0) \\sqrt{\\frac{2\\pi}{g''(0)}}\n",
" .\n",
"$$\n",
"\n",
"\n",
"Thus\n",
"$$\n",
" \\lim_{a\\to\\infty} \\sqrt{a}\\int_{-\\infty}^{\\infty} f(x)e^{-ag(x)}dx\n",
" = f(0) \\sqrt{\\frac{2\\pi}{g''(0)}}\n",
" .\n",
"$$\n",
"as desired."
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