examples/random-sequence.ipynb

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    "## Analyzing a random sequence\n",
    "\n",
    "In this notebook, we will implement and query the following probabilistic model of a random sequence in SPPL.\n",
    "1. For $i = 0$, $X[0]$ is a standard normal.\n",
    "2. For $i > 0$\n",
    " - if $X[i-1]$ is negative, then $X[i]$ is a standard normal\n",
    " - otherwise, let $W[i]$ be the outcome of a fair coin:\n",
    "    - if $W[i] = 0$, then $X[i] = 0.5*(X[i-1])^2 + X[i-1]$\n",
    "    - othewrise, $X[i]$ is a standard normal."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext sppl.magics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Defining the model in SPPL.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%sppl random_sequence\n",
    "from fractions import Fraction\n",
    "\n",
    "X = array(3)\n",
    "W = array(3)\n",
    "\n",
    "X[0] ~= norm()\n",
    "for i in range(1, 3):\n",
    "    if X[i-1] > 0:\n",
    "        X[i] ~= norm(loc=0, scale=1)\n",
    "        W[i] ~= atomic(loc=0)\n",
    "    else:\n",
    "        W[i] ~= bernoulli(p=0.5)\n",
    "        if W[i] == 0:\n",
    "            X[i] ~= Fraction(1,2)*X[i-1]**2 + X[i-1]\n",
    "        else:\n",
    "            X[i] ~= norm(loc=0, scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**SPML programs are represented as sum-product expressions.<br/>\n",
    "We can view the SPE that corresponds to the `random_sequence` model.**"
   ]
  },
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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     "execution_count": 3,
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    }
   ],
   "source": [
    "%sppl_to_graph random_sequence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is informative to consider the distribution encoded by the graphical structure above.\n",
    "\n",
    "1. `+` (sum) nodes indicate a probabilistic mixture model over its children, with weights along the edges.\n",
    "2. `x` (proudct) nodes indicate that the children are independent.\n",
    "3. Leaf nodes indicate names and distributions of the base random variables in the model.\n",
    " - Dashed edges and shaded nodes indicate deterministic transforms of random variables, with the symbol shown along the edge and the expression shown in the node label."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Executing the model returns a namespace `ns` that contains all the variables in the model.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "ns = %sppl_get_namespace random_sequence\n",
    "random_sequence = ns.random_sequence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Suppose we observe that X[1] > 0.<br/>\n",
    "What is the posterior probability that X[0] > 0?**\n",
    "\n",
    "We can use `condition` and `logprob` queries from **SPQL** to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pr(X[0]>0)\t\t= 0.50\n",
      "Pr(X[0]>0 | X[1]>0)\t= 0.647\n"
     ]
    }
   ],
   "source": [
    "# Condition `random_sequence` on the observation X[1] > 0.\n",
    "random_sequence_given_X1 = random_sequence.condition(ns.X[1] > 0)\n",
    "\n",
    "# Print the prior and posterior probabilities of X[0] > 0.\n",
    "print('Pr(X[0]>0)\\t\\t= %1.2f' % (random_sequence.prob(ns.X[0]>0),))\n",
    "print('Pr(X[0]>0 | X[1]>0)\\t= %1.3f' % (random_sequence_given_X1.prob(ns.X[0]>0),))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Suppose we observe that X[2] > 0.<br/>\n",
    "What is the posterior probability that X[0] > 0?**\n",
    "\n",
    "We can use `condition` and `logprob` queries from **SPQL** to answer this question."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pr(X[0]>0)\t\t= 0.50\n",
      "Pr(X[0]>0 | X[2]>0)\t= 0.544\n"
     ]
    }
   ],
   "source": [
    "# Condition `random_sequence` on the observation X[2] > 0.\n",
    "random_sequence_given_X2 = random_sequence.condition(ns.X[2] > 0)\n",
    "\n",
    "# Print the prior and posterior probabilities of X[0] > 0.\n",
    "print('Pr(X[0]>0)\\t\\t= %1.2f' % (random_sequence.prob(ns.X[0]>0),))\n",
    "print('Pr(X[0]>0 | X[2]>0)\\t= %1.3f' % (random_sequence_given_X2.prob(ns.X[0]>0),))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Notice that observing (X[1]>0) shifts the probability of (X[0]>0) from 0.5 to 0.647,<br/>\n",
    "whereas observing (X[2]>0) shifts the probability of (X[0]>0) from 0.5 to 0.544.**\n",
    "\n",
    "Intuitively, X[1] contains 'more information' about X[0] as compared to X[2], since X[2] occurs later in the randoms sequence.<br/>\n",
    "We can use `mutual_infomration` to make this intuition quantitative.<br/><br/>\n",
    "The observation X[1]>0 has 10x more information about X[0]>0 as compared to the observation X[2]>0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "MI(X[0]>0 : X[1]>0)\t= 0.028\n",
      "MI(X[0]>0 : X[2]>0)\t= 0.002\n"
     ]
    }
   ],
   "source": [
    "print('MI(X[0]>0 : X[1]>0)\\t= %1.3f' % (random_sequence.mutual_information(ns.X[0]>0, ns.X[1]>0),))\n",
    "print('MI(X[0]>0 : X[2]>0)\\t= %1.3f' % (random_sequence.mutual_information(ns.X[0]>0, ns.X[2]>0),))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note that the output of `random_sequence.condition(X[1] > 0)` is just another SPE.**\n",
    "\n",
    "Look how much smaller the conditioned SPE is!<br/>\n",
    "In effect, we have analytically (symbolically) eliminated all the exections of `random_sequence` where X[1] < 0."
   ]
  },
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   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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    "%sppl_to_graph random_sequence_given_X1"
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    "**Similarly, the output of `random_sequence.condition(X[2] > 0)` is also another SPE.**\n",
    "\n",
    "In this case the SPE has the same structure, but the numeric probabiliites on the edges have changed.<br/>\n",
    "In general `.condition` can grow or shrink the underlying SPE, depending on the complexity of the query and SPE."
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