Add files via upload

Assignment_3/200179_Apurb.ipynb

@@ -0,0 +1,185 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "ecee66c0-86f9-44ac-aa52-79d3b58f840d",
+   "metadata": {},
+   "source": [
+    "# Question 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e0814b1-fe53-40a9-8578-1b4819ff18e8",
+   "metadata": {},
+   "source": [
+    "Here $\\pi$ denotes standard $Cauchy$ and $g$ denotes standard $Normal$ distributions respectively $\\newline$\n",
+    "$\\inf_{z\\in X} \\frac{\\pi(z)}{g(z)}$ where $X$ is the support of the distributions occurs at $z=1$ which can be shown using elementary Calculus $\\newline$\n",
+    "$m = \\inf_{z\\in X} \\frac{\\pi(z)}{g(z)}$ $\\newline$\n",
+    "Expectation of $z^{2}$ wrt $\\pi$ (denoted as $E_1$) is undefined $\\newline$\n",
+    "Expectation of $z$ wrt $\\pi$ is equal to expectation of $\\frac{z.\\pi(z)}{g(z)}$ wrt $g$ (denoted by $E_3$) $\\newline$\n",
+    "Expectation of $\\frac{z^{2} .\\pi(z)^{2}}{g(z)^{2}}$ wrt $g$ (denoted as $E_2$) is equal to $\\newline$\n",
+    "$E_2 = \\int_X \\frac{z^{2} .\\pi(z)^{2}}{g(z)^{2}}.g(z) \\,dx \\newline$\n",
+    "$E_2 >= m.E_1 \\newline$\n",
+    "Hence $E_2$ is undefined $\\newline$\n",
+    "We cannot apply CLT, hence variance is undefined"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "523f542e-638a-4f78-8cbf-9ce92003c031",
+   "metadata": {},
+   "source": [
+    "# Question 2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5cefe374-fc45-4b17-8ebb-45c4de25f3bf",
+   "metadata": {},
+   "source": [
+    "a. The weighted importance estimator will also have a finite variance since there will only be a difference of factor of (b/a) in the inequality and, b and a although unknown are fixed quantitites. \\\n",
+    "b. One possible advantage of importance sampling over accept reject method could be that choosing the constant c in accept reject might be tricky. If not properly chosen, we may tend to reject too many samples due to which speed will be compromised."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b32ed4d5-9123-4f8d-a8e8-23186f8d500d",
+   "metadata": {},
+   "source": [
+    "# Question 3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "1a491d87-2028-4f49-8f05-7468f5fcfaff",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "using Distributions\n",
+    "using Random\n",
+    "\n",
+    "#Random.seed!(1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "875c8371-60dc-4dfd-a781-a4e6ea0e7815",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "func (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function func(t,y,v,n)\n",
+    "    ans = 1\n",
+    "    for i in 1:n\n",
+    "        ans = ans * ((1 + ((t[i]-y)^2)/v)^(-(v+1)/2))\n",
+    "    end\n",
+    "    return ans\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "e72b658b-4cbd-4583-9e98-dcaab8f041b9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Sample (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function Sample(t,v,n) #t is vector, v is another parameter\n",
+    "    y = rand(Normal())\n",
+    "    u = rand(Uniform())\n",
+    "    while u>func(t,y,v,n)\n",
+    "        y = rand(Normal())\n",
+    "        u = rand(Uniform())\n",
+    "    end\n",
+    "    return y        \n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "3bfba61d-9fcf-403d-a7e0-cc2135dff338",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1.7872618100163297e-35"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "n = 50\n",
+    "v = 5\n",
+    "t = zeros(n)\n",
+    "for i in 1:n\n",
+    "    t[i] = rand(TDist(v))\n",
+    "end\n",
+    "#Sample(t,v,n)\n",
+    "func(t,rand(Normal()),v,n)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fed86942-1724-40ab-ba52-519449b21615",
+   "metadata": {},
+   "source": [
+    "# Question 4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "00d1ffa8-8002-4450-af02-5c0e1a9a7dc4",
+   "metadata": {},
+   "source": [
+    "$\\pi(\\lambda | Y_1,Y_2...Y_n) \\propto f(Y_1,Y_2...Y_n | \\lambda).\\pi(\\lambda) \\newline$\n",
+    "$\\pi(\\lambda | Y_1,Y_2...Y_n) \\propto \\frac{\\lambda^{\\alpha-1}.\\beta^{-\\alpha}.e^{-\\frac{\\lambda}{\\beta}}}{\\Gamma(\\alpha)}.\\Pi_{i=1}^{n} \\frac{e^{-\\lambda}. {\\lambda}^{y_i} }{y_i!} \\newline$\n",
+    "$\\pi(\\lambda | Y_1,Y_2...Y_n) \\propto e^{-(n+\\frac{1}{\\beta}).\\lambda}.\\lambda^{\\alpha-1+y_1+y_2....+y_n} \\newline$\n",
+    "By observation, $\\pi(\\lambda | Y_1,Y_2...Y_n) = Gamma(y_1+y_2...+y_n+\\alpha,\\frac{\\beta}{1+n.\\beta})$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.7.2",
+   "language": "julia",
+   "name": "julia-1.7"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.7.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}