Lecture14_PerfectSampling.ipynb

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     "source": [
      "#Lecture 14: Perfect Sampling\n",
      "\n",
      "AM207: Chang-han Rhee, Gurst Lecture\n",
      "\n",
      "March 13, 2014\n",
      "\n",
      "\n",
      "------\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import scipy as sp\n",
      "import matplotlib.pyplot as plt\n",
      "import pandas as pd\n",
      "pd.set_option('display.width', 500)\n",
      "pd.set_option('display.max_columns', 100)\n",
      "import seaborn as sns\n",
      "import scipy.stats\n",
      "from scipy.stats import bernoulli"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Perfect Sampling\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this lecture, we will discuss \"Perfect Sampling\" algorithms. These are relatively recent developments, but it has received an enormous amount of attention when the \"coupling from the past\" algorithm was suggested by Propp and Wilson in 1996, and it still remains as an active research area. The subject has fascinated many researchers, partly because the algorithm itself is really cool, but also because it has great potential to do wonderful things in MCMC and other contexts. \n",
      "\n",
      "Before we start, let's review the big picture first. We have already learned in the previous lectures that Markov chain Monte Carlo algorithms sample from a given target distribution by constructing a Markov chain whose equilibrium distribution is the target distribution. We have seen a number of ways to construct such Markov chains&mdash;Metropolis-Hastings, Gibbs, Slice, Hamiltonina MCMC, etc. Once we have such a Markov chain, we can study the target distribution using the samples obtained by running the Markov chain for a long time, because if we run a Markov chain for a long enough time, the finite time distribution will be sufficiently close to the equilibrium distribution. The question is, of course, how long is long enough? To (incompletely) answer this question, we also have seen a few diagnostics. However, there's no general method to give a definite answer to this question, and determing the convergence of Markov chains remains to be a notoriously hard problem. \n",
      "\n",
      "The class of algorithms we are going to discuss today approaches this problem from a different perspective. Instead of approximating the equilibrium distribution of a given Markov chain and reducing or bounding the approximation error, these algorithms try to sample from the exact equilibrium distribution of the Markov chain with perfect precision. Due to this grand ambition, people like to call them \"perfect simulation\" or \"perfect sampling\" algorithms. Less exciting but more informative terms \"exact simulation\" or \"exact sampling\" are also frequently used.\n",
      "\n",
      "We will focus on the classic perfect sampling algorithm&mdash;Propp and Wilson's coupling from the past (CFTP).\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The Goal"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Given a Markov chain $\\{x^{(i)}\\}$ such that $x^{(i)} \\Rightarrow x^{(\\infty)}$, sample from the equilibrium distribution $\\pi$ of $x^{(\\infty)}$ "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The Classic: Coupling from the Past (CFTP)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As any english speaker can easily guess from the name, coupling from the past algorithm generates exact samples from the equilibrium distribution of a Markov chain by (virtually) running the Markov chain from its infinite past. But, what's mysterious in the name is the word \"coupling.\""
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "What is Coupling?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In probability, coupling refers to the construction of a probability space where two random objects are defined simultaneously. In perfect simulation context, the coupling referes to the coalescence of paths from all possible starting points. Here is an example that we will follow throughout this lecture."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "An Example: Random walk with boundary\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's consider the following Markov chain evolving in state space $S = \\{0, 1, \\cdots, M\\}$:\n",
      "\n",
      "\\begin{align}\n",
      "x^{(n+1)} &= \\min\\big( \\max \\big(x^{(n)} + \\xi^{(n+1)}, 0\\big), M\\big)\n",
      "\\end{align}\n",
      "\n",
      "where $\\xi^{(n)}$ is iid with $P(\\xi^{(n)} = -1) = P(\\xi^{(n)} = 1) = 1/2$. This is a simple random walk that evolves within the boundaries at $0$ and $M$.\n",
      "\n",
      "We can construct the coupling (or force the coalescence) by using common random noise $\\xi^{(n)}$ for all paths.\n",
      "\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Coupling (Coalescence) of paths"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Since we will consider multiple versions (i.e., realized paths) of the Markov chain each of which starts from different points in the state space, it will be convenient to introduce a new notation: let $x^{(n)}(s)$ denote the Markov chain that has started from state $s \\in S$ (at time $0$). Once two or more versions of these paths meet each other, we can see from the above dynamics (recursion) of the Markov chain that those paths will stay together forever afterwards. That is, if $x^{(k)}(s_1) = x^{(k)}(s_2)$ for some $k$, $x^{(k+i)}(s_1) = x^{(k+i)}(s_2)$ for all $i\\geq 0$. The python code below illustrates such coupling/coalescence. "
     ]
    },
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def generate_noise(l):\n",
      "    return 2*bernoulli.rvs(0.5, size=l)-np.ones(l)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
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     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def run(noises,low_bnd,up_bnd,start_pt):\n",
      "    x = np.zeros(len(noises)+1)\n",
      "    x[0] = start_pt\n",
      "    for i,noise in enumerate(noises):\n",
      "            x[i+1] = min( max( x[i]+noise, low_bnd ), up_bnd )\n",
      "    return x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "l=50\n",
      "M=10\n",
      "noises = generate_noise(l)\n",
      "print noises\n",
      "fig = plt.figure()\n",
      "for i in range(M+1):\n",
      "    path = run(noises,0,M,i)\n",
      "    plt.plot(path)"
     ],
     "language": "python",
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        "[ 1.  1. -1. -1.  1. -1. -1. -1.  1. -1. -1.  1.  1. -1.  1. -1.  1.  1.\n",
        "  1. -1. -1.  1.  1. -1. -1.  1. -1. -1.  1. -1. -1.  1. -1. -1. -1.  1.\n",
        " -1.  1. -1.  1.  1.  1. -1. -1. -1. -1.  1.  1.  1. -1.]\n"
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cJG5/yxSDweiMrgrxhfBRsqVrmoGVncxFtvSbLIzJSOUqOFLIR5SicgS7hQPk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\nkwIo4zTJAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109042bd0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Analytic Solution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have intentionally chosen a tractable example to have a reference distribution. The transition matrix of the Markov chain, for example in case $M = 3$, i.e., $\\{x^{(n)}\\}$ has state space $S = \\{0,1,2,3\\}$, is \n",
      "\n",
      "$$ P = \n",
      "\\left(\\begin{array}{cccc}\n",
      "1/2 & 1/2 & 0   & 0\\\\ \n",
      "1/2 & 0   & 1/2 & 0\\\\\n",
      "0   & 1/2 & 0 & 1/2\\\\\n",
      "0   & 0   & 1/2 & 1/2\n",
      "\\end{array}\\right).$$\n",
      "\n",
      "From the form of the transition matrix $P$, we can check that $\\pi = (1/4, 1/4, 1/4, 1/4)$ satisfies the steady-state equation\n",
      "\n",
      "$$\\pi P = \\pi.$$\n",
      "\n",
      "Hence, the equilibrium distribution of $\\{x^{(n)}\\}$ is uniform distribution on state space. This is true for any $M$. This can be also seen by computing the power of the transition matrix $P$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = np.matrix([[0.5, 0.5, 0,   0  ],\\\n",
      "               [0.5, 0,   0.5, 0  ],\\\n",
      "               [0,   0.5, 0,   0.5],\\\n",
      "               [0,   0,   0.5, 0.5]])\n",
      "print P; print\n",
      "print P**2;print\n",
      "print P**5;print\n",
      "print P**10;print\n",
      "print P**20;print\n",
      "print P**50;print\n",
      "print P**200;print\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 0.5  0.5  0.   0. ]\n",
        " [ 0.5  0.   0.5  0. ]\n",
        " [ 0.   0.5  0.   0.5]\n",
        " [ 0.   0.   0.5  0.5]]\n",
        "\n",
        "[[ 0.5   0.25  0.25  0.  ]\n",
        " [ 0.25  0.5   0.    0.25]\n",
        " [ 0.25  0.    0.5   0.25]\n",
        " [ 0.    0.25  0.25  0.5 ]]\n",
        "\n",
        "[[ 0.3125  0.3125  0.1875  0.1875]\n",
        " [ 0.3125  0.1875  0.3125  0.1875]\n",
        " [ 0.1875  0.3125  0.1875  0.3125]\n",
        " [ 0.1875  0.1875  0.3125  0.3125]]\n",
        "\n",
        "[[ 0.265625  0.25      0.25      0.234375]\n",
        " [ 0.25      0.265625  0.234375  0.25    ]\n",
        " [ 0.25      0.234375  0.265625  0.25    ]\n",
        " [ 0.234375  0.25      0.25      0.265625]]\n",
        "\n",
        "[[ 0.25048828  0.25        0.25        0.24951172]\n",
        " [ 0.25        0.25048828  0.24951172  0.25      ]\n",
        " [ 0.25        0.24951172  0.25048828  0.25      ]\n",
        " [ 0.24951172  0.25        0.25        0.25048828]]\n",
        "\n",
        "[[ 0.25000001  0.25        0.25        0.24999999]\n",
        " [ 0.25        0.25000001  0.24999999  0.25      ]\n",
        " [ 0.25        0.24999999  0.25000001  0.25      ]\n",
        " [ 0.24999999  0.25        0.25        0.25000001]]\n",
        "\n",
        "[[ 0.25  0.25  0.25  0.25]\n",
        " [ 0.25  0.25  0.25  0.25]\n",
        " [ 0.25  0.25  0.25  0.25]\n",
        " [ 0.25  0.25  0.25  0.25]]\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "The Bias from the Initial Transience"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this example, the initial transience remains visible to naked eyes until $T \\geq 80$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = 10\n",
      "N = 10000\n",
      "\n",
      "samples = np.zeros(N)\n",
      "\n",
      "fig, axes=plt.subplots(figsize=(10,10), nrows=3, ncols=2)\n",
      "\n",
      "l=20\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[0][0].hist(samples,bins=11)\n",
      "axes[0][0].set_title('At t=20')\n",
      "\n",
      "l=40\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[0][1].hist(samples,bins=11)\n",
      "axes[0][1].set_title('At t=40')\n",
      "\n",
      "l=60\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[1][0].hist(samples,bins=11)\n",
      "axes[1][0].set_title('At t=60')\n",
      "\n",
      "l=80\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[1][1].hist(samples,bins=11)\n",
      "axes[1][1].set_title('At t=80')\n",
      "\n",
      "l=100\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[2][0].hist(samples,bins=11)\n",
      "axes[2][0].set_title('At t=100')\n",
      "\n",
      "l=200\n",
      "for i in range(N):\n",
      "    noises = generate_noise(l)\n",
      "    samples[i] = run(noises,0,M,2)[l]\n",
      "axes[2][1].hist(samples,bins=11)\n",
      "axes[2][1].set_title('At t=200')\n",
      "\n",
      "fig.tight_layout()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dmpoql8slt9stq9V62thAk5gUL5st2bT5UlKSzn1QCOY0q8bBXl+wxMXFSlLErftCRXvd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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10994c7d0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Coupling from the Past"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Yet another notation: we will denote the Markov chain that has started from time $m$ at state $s$ with $x^{(n)}(s;m)$.\n",
      "\n",
      "The classic coupling from the past algorithm proceeds as follows.\n",
      "\n",
      "1. Set starting time $n = -1$.\n",
      "2. Initialize our Markov chain $\\{x^{(i)}\\}$ at time $n$ from each point $s$ of the state space $S$.\n",
      "3. Generate a random noise $\\xi^{(n+1)}$.\n",
      "4. Using the (common) random noise $\\xi^{(n+1)}, \\xi^{(n+2)}, \\cdots, \\xi^{(0)}$ generated in step 3, run each Markov chain upto time 0.\n",
      "5. If all the paths have coupled (coalesce) by time 0, return the common value $x^{(0)}$.\n",
      "    If not, set $n = n-1$ and go to step 2.\n",
      "\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Implementation and Numerics"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's see if CFTP works for our bounded random walk example. Note that in the following implementation, we didn't have to simulate the process from all possible starting points because of the monotonicity of our random walk example. We only had to monitor two versions: one started from $M$ and the other one started from $0$. If those two versions couple, then we know that all the other versions were sandwiched between them, and hence, every path has coalesced. Also, it should be noted that it is not necessary to detect the coalescence from the most recent starting points. In other words, we don't have to scan every starting point -1, -2, -3,$\\ldots$. Instead, we can scan backward using binary search, i.e., doubling the number of time steps we go back (-1, -2, -4, -8,$\\ldots$) instead of increasing by 1. Finally, we also note that we don't have to start from $t = -1$, because there's no way the backward chain coalesce in one step; it requires at least $M$ steps to reach the bottom from the top. Therefore, we started at $t=-M$ instead of $t=-1$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def cycle(innovs,low_bnd,up_bnd):\n",
      "    upper = np.zeros(len(innovs)+1)\n",
      "    lower = np.zeros(len(innovs)+1)\n",
      "    upper[0] = up_bnd\n",
      "    lower[0] = low_bnd\n",
      "    for i,innov in enumerate(innovs):\n",
      "            upper[i+1] = min(max(upper[i] + innov,low_bnd),up_bnd)\n",
      "            lower[i+1] = min(max(lower[i] + innov,low_bnd),up_bnd)\n",
      "    return upper,lower"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def perfect_sampling(M):\n",
      "    l = M\n",
      "    noises = generate_noise(l)\n",
      "    while True:\n",
      "        low = run(noises,0,M,0)\n",
      "        up  = run(noises,0,M,M)\n",
      "        if low[l] == up[l]: return low[l];\n",
      "        noises = np.concatenate((generate_noise(l),noises),axis=0)\n",
      "        l = 2*l\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N = 10000\n",
      "samples = np.zeros(N)\n",
      "for i in range(N):\n",
      "    samples[i] = perfect_sampling(10)\n",
      "plt.hist(samples,bins=11)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "(array([ 910.,  872.,  957.,  886.,  919.,  908.,  893.,  888.,  941.,\n",
        "        931.,  895.]),\n",
        " array([  0.        ,   0.90909091,   1.81818182,   2.72727273,\n",
        "         3.63636364,   4.54545455,   5.45454545,   6.36363636,\n",
        "         7.27272727,   8.18181818,   9.09090909,  10.        ]),\n",
        " <a list of 11 Patch objects>)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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p1+tpa2vLgQMHUqlUkiSrV6/O7t27M2vWrHR2dqatrS1tbW1ZtGhRDh48mK9//esTuwYX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\n/wOr5MGYVwJ5zAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x109172ed0>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Fantastic! CFTP indeed works! It should be noted that, here it was important to \n",
      "\n",
      "* use common random numbers for all paths\n",
      "* extend the common path backward (not forward) until coupling takes place\n",
      "* re-use the random numbers to run the backward processes\n",
      "* sample at time 0\n",
      "\n",
      "At first glance, however, it may not be clear why these are important and why CFTP works. Let's see what can go wrong. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "What if we do not re-use random numbers? "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The code below illustrates what happens if we generate new random noise rather than reusing the old random noise when we go back to the further past. The only differences between \"perfect_sampling\" above and \"without_reuse_sampling\" below are the last two lines."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def without_reuse_sampling(M):\n",
      "    l = M;\n",
      "    noises = generate_noise(l)\n",
      "    while 'true':\n",
      "        up,low = cycle(noises,0,M)\n",
      "        if low[l] == up[l]:\n",
      "            return low[l];\n",
      "        l = 2*l\n",
      "        noises = generate_noise(l)\n",
      "\n",
      "        \n",
      "N = 10000\n",
      "samples = np.zeros(N)\n",
      "for i in range(N):\n",
      "    samples[i] = without_reuse_sampling(10)\n",
      "plt.hist(samples,bins=11)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "(array([ 1314.,  1027.,   878.,   741.,   672.,   626.,   700.,   783.,\n",
        "         867.,  1032.,  1360.]),\n",
        " array([  0.        ,   0.90909091,   1.81818182,   2.72727273,\n",
        "         3.63636364,   4.54545455,   5.45454545,   6.36363636,\n",
        "         7.27272727,   8.18181818,   9.09090909,  10.        ]),\n",
        " <a list of 11 Patch objects>)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a60c9d0>"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This looks quite different from the uniform distribution. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "What if we run the process forward?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "One may wonder \"what if we do the same thing, but search for the coalescence in forward direction?\" "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def forward_sampling(M):\n",
      "    l = M;\n",
      "    innovs = generate_noise(l)\n",
      "    while 'true':\n",
      "        up,low = cycle(innovs,0,M)\n",
      "        if low[l] == up[l]:\n",
      "            return low[l];\n",
      "        l = 2*l\n",
      "        innovs = innovs = np.concatenate((innovs,generate_noise(l)),axis=0)\n",
      "\n",
      "        \n",
      "N = 10000\n",
      "samples = np.zeros(N)\n",
      "for i in range(N):\n",
      "    samples[i] = forward_sampling(10)\n",
      "plt.hist(samples,bins=11)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": [
        "(array([ 1343.,  1091.,   827.,   720.,   707.,   650.,   663.,   713.,\n",
        "         895.,  1039.,  1352.]),\n",
        " array([  0.        ,   0.90909091,   1.81818182,   2.72727273,\n",
        "         3.63636364,   4.54545455,   5.45454545,   6.36363636,\n",
        "         7.27272727,   8.18181818,   9.09090909,  10.        ]),\n",
        " <a list of 11 Patch objects>)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x10a63e610>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The answer is no."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Why does CFTP work?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In general, a Markov chain can be written as a recursion. For example, in case of our random walk example, the recursion looks as follows:\n",
      "\n",
      "$$ x^{(n+1)} = r(x^{(n)}, \\xi^{(n+1)})$$\n",
      "\n",
      "where $r(s,t) = \\min\\big( \\max\\big( s+t, 0\\big), M\\big)$.\n",
      "\n",
      "If we denote $r(\\cdot, \\xi^{(n)})$ with $\\varphi_i(\\cdot)$, then each $\\varphi_i$ is a random function, and $x^{(n)}$ can be written as an iterated random functions. \n",
      "\n",
      "\\begin{align}\n",
      "x^{(1)} &= r(x^{(0)}, \\xi^{(1)}) = \\varphi_1(x^{(0)})\\\\\n",
      "x^{(2)} &= r(x^{(1)}, \\xi^{(2)}) = \\varphi_2(x^{(1)}) = \\varphi_2\\circ \\varphi_1(x^{(0)})\\\\\n",
      "&\\vdots\\\\\n",
      "x^{(n)} &= \\varphi_n\\circ \\cdots \\circ \\varphi_1(x^{(0)})\n",
      "\\end{align}\n",
      "\n",
      "Note that since $\\xi^{(i)}$-s are iid, $\\varphi_i(\\cdot) = r(\\cdot, \\xi^{(i)})$ are iid, and hence,\n",
      "\n",
      "\\begin{align*}\n",
      "x^{(n)} \n",
      "&= \\varphi_n\\circ \\cdots \\circ \\varphi_1(x^{(0)})\\\\\n",
      "&\\stackrel{\\mathcal D}{=} \\varphi_1\\circ \\cdots \\circ\\varphi_n(x^{(0)})\\\\\n",
      "&\\stackrel{\\mathcal D}{=} \\varphi_{-1}\\circ \\cdots \\circ \\varphi_{-n}(x^{(0)})\\\\\n",
      "&\\triangleq \\tilde x^{(n)}\n",
      "\\end{align*}\n",
      "\n",
      "where $\\tilde x^{(n)}$ is the \"backward process.\" This implies that\n",
      "\n",
      "\\begin{align*}\n",
      "x^{(\\infty)} \\stackrel{\\mathcal D}{=} \\tilde x^{(\\infty)}\n",
      "\\end{align*}\n",
      "\n",
      "and hence if we can generate exact samples of $\\tilde x^{(\\infty)}$, then we are generating samples from $\\{x^{(n)}\\}$'s exact equilibrium distribution. \n",
      "\n",
      "At this point, it might not be clear yet why sampling from $\\tilde x^{(\\infty)}$ is easier than sampling from $x^{(\\infty)}$. It is worthwhile noticing a couple of important differences between the forward process and the backward process:\n",
      "\n",
      "* $\\tilde x^{(n)}$ is not a Markov chain. We can not compute $\\tilde x^{(n)}$ from $\\tilde x^{(n-1)}$ and $\\varphi_{-n}$ alone. To compute $\\tilde x^{(n)}$ we need all the previous $\\varphi_{i}$-s (and the starting point, of cousre).\n",
      "* If $\\varphi_{-1}\\circ\\cdots\\circ\\varphi_{-k}$ is constant, $\\tilde x^{(k)} = \\tilde x^{(k+1)} = \\cdots = \\tilde x^{(\\infty)}$ for all $i\\geq 0$.\n",
      "\n",
      "The second property makes it possible to generate $\\tilde x^{(\\infty)}$ in finite time (that is, $k$). In other words, if we let \n",
      "\n",
      "$$T \\triangleq \\sup\\{s\\leq 0: \\varphi_{-1}\\circ\\cdots\\circ\\varphi_{-k}(\\cdot) \\text{ is constant}\\},$$ \n",
      "\n",
      "then $\\tilde x^{(T)} = \\tilde x^{(\\infty)}$ w.p. 1.\n",
      "\n",
      "Therefore, if \n",
      "\n",
      "* we can construct a probability space where $\\varphi_{-1}\\circ\\varphi_{-2}\\circ\\cdots\\circ\\varphi_{-n}$ becomes constant (coalescence) with probability 1;\n",
      "* we can detect such coalescence,\n",
      "\n",
      "then, CFTP algorithm should work.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Challenges for Routine Applications"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Although it was deceptively easy in case of our random walk example, constructing a coupling so that all paths coalesce at reasonable time is not a trivial task. Even for a very simple Markov chain with two states, we can construct a coupling (a joint probability space) where two paths from the two starting points never coalesce. Suppose that the state space $S = \\{0,1\\}$, and the perturbations $\\xi_i$ are iid Bernoulli with success probability 1/2. Let \n",
      "\n",
      "$$\\varphi_i(s) = r(s,\\xi_i) = \\delta_{s\\xi_i}.$$\n",
      "\n",
      "where $\\delta_{st}$ is the Kronecker delta function. Then, \n",
      "\n",
      "$$\\varphi_i(s) \\neq \\varphi_i(t) \\quad \\text{iff} \\quad s \\neq t$$ \n",
      "\n",
      "and hence \n",
      "\n",
      "$$\\varphi_1\\circ \\cdots \\circ \\varphi_n(0) \\neq \\varphi_1\\circ \\cdots \\circ \\varphi_{n}(1)$$\n",
      "\n",
      "w.p. 1 for all $n$. That is, for $x^{(n)}(s) = \\varphi_n\\circ \\cdots \\circ \\varphi_1(s)$ which has stationary distribution $\\pi(0) = \\pi(1) = 1/2$, the backward chain does not converge in pathwise sense.\n",
      "\n",
      "\n",
      "Even if we are successful in constructing a reasonable coupling, detecting the coalescence can be challenging. For example, if the state space is extremely large (e.g., Ising model) which is, the norm rather than the exception in perfect simulation context&mdash;for small state spaces we can easily solve for the steady state equation $\\pi P = \\pi$ to directly compute the equilibrium distribution. The problem becomes even harder if we consider (coutably) infinite state spaces, or even continuous state spaces. As we have seen in the previous implementation of our random walk example, the monotonicity can help a lot, and there are other clever ways developed to detect the coalescence even without monotonicity. In fact, the class of processes for which we can construct CFTP algorithm has been identified."
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "\n",
      "Necessary and Sufficient Condition for Coupling from the Past\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Foss and Tweedie proved that successful CFTP algorithms can be constructed if and only if the chain is uniformly ergodic. The sufficient condition can be proved by following the construction of multigamma coupling.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Multi-gamma Sampler"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If a Markov chain on a state space $S$ with transition kernel $p(x,dy)$ is uniformly ergodic, then there exists a measure $\\nu$ and constants $m\\geq 1$ and $0< \\beta\\leq 1$ such that\n",
      "\n",
      "$$P^m(s,\\cdot) \\geq \\beta \\nu(\\cdot), \\qquad \\forall s \\in S$$\n",
      "\n",
      "This effectively means that we can construct a probability space where $\\varphi_i\\circ\\varphi_{i+1}\\circ\\cdots\\circ\\varphi_{i+m-1}(\\cdot)$ is constant with positive probability for each $i$. In view of what we have discussed in section \"Why does CFTP work?,\" we can see why uniform ergodicity is a sufficient condition for successful CFTP algorithm. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Variations of CFTP and Other Perfect Sampling Algorithms"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There are many variants of CFTP and other perfect sampling algorithms such as \n",
      "\n",
      "* Fill's Algorithm (Fill 1998; Fill, Machida, Murdock 1999)\n",
      "* Read-Once CFTP (Wilson 2000)\n",
      "* Perfect Slice Sampling (Mira, Moller, Roberts 2001)\n",
      "* Dominated CFTP (Kendall 2004; Conor, Kendall 2007)\n",
      "* Stationarity Detection (Asmussen, Glynn, Thorisson 1992)\n",
      "\n",
      "just name a few. David Wilson (one of the original inventors of CFTP) maintains extensive online bibliography regarding CFTP\n",
      "(http://research.microsoft.com/~dbwilson/exact/)."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}