Skip to content

Assignment 3 #89

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
wants to merge 1 commit into
base: main
Choose a base branch
from
Open

Assignment 3 #89

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
238 changes: 238 additions & 0 deletions Assignment_3/200074_Akash.ipynb
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -0,0 +1,238 @@
{
"cells": [
{
"cell_type": "markdown",
"id": "dfb7cd7a",
"metadata": {},
"source": [
"# 1."
]
},
{
"cell_type": "markdown",
"id": "20db50bb",
"metadata": {},
"source": [
"For standard Cauchy distribution, we have the pdf as $\\pi(x) = \\frac{1}{\\pi(1+x^2)}$ \n",
"As, we use standard normal distibution as the importance distribution, so we have its pdf as $ g(x) = \\frac{1}{\\sqrt{ 2 \\pi}}e^{-\\frac{x^2}{2}}$ \n",
"A, mean is first moment so, $h(x) = x$ \n",
"We can estimate the mean as $\\hat{\\theta_g} = \\frac{1}{N} \\sum_{t=1}^{N}\\frac{h(Z_t)\\pi(Z_t)}{g(Z_t)}$ , where $Z_{i}$s are obtained from importance sampling estimator of the proposal distribution. \n",
"Putting respective values, we get mean as $\\hat{\\theta_g} = \\frac{1}{N} \\sum_{t=1}^{N} \\frac{1}{N} \\sum_{t=1}^{N} \\frac{\\sqrt{2} Z_t e^{-\\frac{Z_t^2}{2}}}{\\sqrt{\\pi}(Z_t^2 + 1)}$ \n",
"$\\sigma_g(\\hat{\\theta_g}) = \\sigma_g^2 + \\theta^2 = \\int_{-\\infty}^{\\infty} \\frac{h^2(x)\\pi^2(x)}{g^2(x)} dx$ \n",
"$ \\sigma_g^2 = \\int_{-\\infty}^{\\infty} \\frac{h^2(x)\\pi^2(x)}{g^2(x)} dx = \\int_{-\\infty}^{\\infty} \\frac{2x^2 e^{x^2}}{\\pi (1+x^2)^2} dx \\rightarrow \\infty \\implies$ Estimator does not has finite variance."
]
},
{
"cell_type": "markdown",
"id": "3d04a737",
"metadata": {},
"source": [
"# 2."
]
},
{
"cell_type": "markdown",
"id": "6ec1f356",
"metadata": {},
"source": [
"(a). We are given that sup $\\frac{f(x)}{g(x)} < \\infty$ \n",
"As, $\\tilde{f(x)} \\propto f(x)$ and $\\tilde{g(x)} \\propto g(x)$, so $\\frac{\\tilde{f(x)}}{\\tilde{g(x)}} \\propto \\frac{f(x)}{g(x)} < \\infty$ \n",
"Hence, the weighted importance estimator should have finite variance."
]
},
{
"cell_type": "markdown",
"id": "1c42440d",
"metadata": {},
"source": [
"(b). In accept-reject, a proposal might get rejected even after several iterations but in importance sampling, we get output after every iteration."
]
},
{
"cell_type": "markdown",
"id": "82e98bdb",
"metadata": {},
"source": [
"# 3."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "59555451",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"MersenneTwister(1)"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Distributions\n",
"using Random\n",
"\n",
"Random.seed!(1)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "a40a35d1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"genrte (generic function with 2 methods)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function genrte(v, n=50)\n",
" tvec = TDist(v)\n",
" y = rand(tvec, n)\n",
" product = 1\n",
" \n",
" numr = 0\n",
" denr = 0\n",
" reslt = zeros(10000)\n",
" for i = 1:10000\n",
" x = rand(Normal())\n",
" for i = 1:n\n",
" product *= (1 + ((y[i] - x)^2)/v)^(-(v + 1)/2)\n",
" end\n",
" r = product * (sqrt(2 * π))\n",
" numr += x * r\n",
" denr += r\n",
" reslt[i] = numr / denr\n",
" end\n",
" return reslt\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "6937564f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean is -1.5477036982569745\n",
"Variance is 2.416128135052854e-30\n"
]
}
],
"source": [
"generate5 = genrte(5)\n",
"print(\"Mean is \")\n",
"println(mean(generate5))\n",
"print(\"Variance is \")\n",
"println(var(generate5))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "4c9fa654",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean is -0.8377053458293047\n",
"Variance is 4.437786370505242e-31\n"
]
}
],
"source": [
"generate1 = genrte(1)\n",
"print(\"Mean is \")\n",
"println(mean(generate1))\n",
"print(\"Variance is \")\n",
"println(var(generate1))"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "36828f7e",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mean is -0.801275682761385\n",
"Variance is 1.232718436251456e-30\n"
]
}
],
"source": [
"generate2 = genrte(2)\n",
"print(\"Mean is \")\n",
"println(mean(generate2))\n",
"print(\"Variance is \")\n",
"println(var(generate2))"
]
},
{
"cell_type": "markdown",
"id": "a14810da",
"metadata": {},
"source": [
"# 4."
]
},
{
"cell_type": "markdown",
"id": "4a220a57",
"metadata": {},
"source": [
"$$\n",
"p (\\lambda | Y) \\propto \\lambda ^{\\alpha -1}e^{-\\beta\\lambda}\\prod_{i=1}^{n}\\frac{\\lambda^{Y_{i}}e^{-\\lambda}}{Y_{i}!}\\\\\n",
"p (\\lambda | Y) \\propto \\lambda ^{\\alpha +\\sum_{i=1}^{n}Y_{i}-1}e^{-\\beta\\lambda}\\prod_{i=1}^{n}\\frac{e^{-\\lambda}}{Y_{i}!}\\\\\n",
"p (\\lambda | Y) \\propto \\lambda ^{\\alpha +\\sum_{i=1}^{n}Y_{i}-1}e^{-\\beta\\lambda-n\\lambda}\\prod_{i=1}^{n}\\frac{1}{Y_{i}!}\\\\\n",
"p (\\lambda | Y) \\propto \\lambda ^{\\alpha +\\sum_{i=1}^{n}Y_{i}-1}e^{-\\beta\\lambda-n\\lambda}\\\\\n",
"$$\n",
"$p (\\lambda | Y)$ represents a Gamma Distribution ~ $Gamma(\\alpha +\\sum_{i=1}^{n}Y_{i},\\beta+n)$ "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "dc2ccf51",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.6.2",
"language": "julia",
"name": "julia-1.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.6.2"
}
},
"nbformat": 4,
"nbformat_minor": 5
}