diff --git a/probability.ipynb b/probability.ipynb index 2fd1c9dae..365039874 100644 --- a/probability.ipynb +++ b/probability.ipynb @@ -11,21 +11,19 @@ }, { "cell_type": "code", - "execution_count": 1, + "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from probability import *\n", - "from notebook import psource" + "from notebook import *" ] }, { "cell_type": "markdown", - "metadata": { - "collapsed": true - }, + "metadata": {}, "source": [ "## Probability Distribution\n", "\n", @@ -34,7 +32,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 34, "metadata": { "collapsed": true }, @@ -45,7 +43,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -54,7 +52,7 @@ "0.75" ] }, - "execution_count": 2, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -255,9 +253,7 @@ }, { "cell_type": "markdown", - "metadata": { - "collapsed": true - }, + "metadata": {}, "source": [ "_A probability model is completely determined by the joint distribution for all of the random variables._ (**Section 13.3**) The probability module implements these as the class **JointProbDist** which inherits from the **ProbDist** class. This class specifies a discrete probability distribute over a set of variables. " ] @@ -512,9 +508,124 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 5, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\n", + " \n", + " \n", + " \n", + "\n", + "\n", + "

\n", + "\n", + "

def enumerate_joint_ask(X, e, P):\n",
+       "    """Return a probability distribution over the values of the variable X,\n",
+       "    given the {var:val} observations e, in the JointProbDist P. [Section 13.3]\n",
+       "    >>> P = JointProbDist(['X', 'Y'])\n",
+       "    >>> P[0,0] = 0.25; P[0,1] = 0.5; P[1,1] = P[2,1] = 0.125\n",
+       "    >>> enumerate_joint_ask('X', dict(Y=1), P).show_approx()\n",
+       "    '0: 0.667, 1: 0.167, 2: 0.167'\n",
+       "    """\n",
+       "    assert X not in e, "Query variable must be distinct from evidence"\n",
+       "    Q = ProbDist(X)  # probability distribution for X, initially empty\n",
+       "    Y = [v for v in P.variables if v != X and v not in e]  # hidden variables.\n",
+       "    for xi in P.values(X):\n",
+       "        Q[xi] = enumerate_joint(Y, extend(e, X, xi), P)\n",
+       "    return Q.normalize()\n",
+       "

\n", + "\n", + "\n" + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "psource(enumerate_joint_ask)" ] @@ -792,7 +903,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 26, "metadata": {}, "outputs": [], "source": [ @@ -1178,7 +1289,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 32, "metadata": { "collapsed": true }, @@ -1418,21 +1529,8 @@ ] }, { - "cell_type": "code", - "execution_count": 45, + "cell_type": "markdown", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "'False: 0.184, True: 0.816'" - ] - }, - "execution_count": 45, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ "likelihood_weighting('Cloudy', dict(Rain=True), sprinkler, 200).show_approx()" ] @@ -1450,7 +1548,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 27, "metadata": { "collapsed": true }, @@ -1485,6 +1583,245 @@ "source": [ "gibbs_ask('Cloudy', dict(Rain=True), sprinkler, 200).show_approx()" ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Inference in Temporal Models" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Before we start, it will be helpful to understand the structure of a temporal model. We will use the example of the book with the guard and the umbrella. In this example, the state $\\textbf{X}$ is whether it is a rainy day (`X = True`) or not (`X = False`) at Day $\\textbf{t}$. In the sensor or observation model, the observation or evidence $\\textbf{U}$ is whether the professor holds an umbrella (`U = True`) or not (`U = False`) on **Day** $\\textbf{t}$. Based on that, the transition model is \n", + "\n", + "| $X_{t-1}$ | $X_{t}$ | **P**$(X_{t}| X_{t-1})$| \n", + "| ------------- |------------- | ----------------------------------|\n", + "| ***${False}$*** | ***${False}$*** | 0.7 |\n", + "| ***${False}$*** | ***${True}$*** | 0.3 |\n", + "| ***${True}$*** | ***${False}$*** | 0.3 |\n", + "| ***${True}$*** | ***${True}$*** | 0.7 |\n", + "\n", + "And the the sensor model will be,\n", + "\n", + "| $X_{t}$ | $U_{t}$ | **P**$(U_{t}|X_{t})$| \n", + "| :-------------: |:-------------: | :------------------------:|\n", + "| ***${False}$*** | ***${True}$*** | 0.2 |\n", + "| ***${False}$*** | ***${False}$*** | 0.8 |\n", + "| ***${True}$*** | ***${True}$*** | 0.9 |\n", + "| ***${True}$*** | ***${False}$*** | 0.1 |\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the filtering task we are given evidence **U** in each time **t** and we want to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$. \n", + "We can think of it as a three step process:\n", + "1. In every step we start with the current belief $P(X_{t}|e_{1:t})$\n", + "2. We update it for time\n", + "3. We update it for evidence\n", + "\n", + "The forward algorithm performs the step 2 and 3 at once. It updates, or better say reweights, the initial belief using the transition and the sensor model. Let's see the umbrella example. On **Day 0** no observation is available, and for that reason we will assume that we have equal possibilities to rain or not. In the **`HiddenMarkovModel`** class, the prior probabilities for **Day 0** are by default [0.5, 0.5]. " + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [], + "source": [ + "%psource HiddenMarkovModel" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We instantiate the object **`hmm`** of the class using a list of lists for both the transition and the sensor model." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "umbrella_transition_model = [[0.7, 0.3], [0.3, 0.7]]\n", + "umbrella_sensor_model = [[0.9, 0.2], [0.1, 0.8]]\n", + "hmm = HiddenMarkovModel(umbrella_transition_model, umbrella_sensor_model)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The **`sensor_dist()`** method returns a list with the conditional probabilities of the sensor model." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[0.9, 0.2]" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "hmm.sensor_dist(ev=True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The observation update is calculated with the **`forward()`** function. Basically, we update our belief using the observation model. The function returns a list with the probabilities of **raining or not** on **Day 1**." + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [], + "source": [ + "psource(forward)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The probability of raining on day 1 is 0.82\n" + ] + } + ], + "source": [ + "belief_day_1 = forward(hmm, umbrella_prior, ev=True)\n", + "print ('The probability of raining on day 1 is {:.2f}'.format(belief_day_1[0]))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In **Day 2** our initial belief is the updated belief of **Day 1**. Again using the **`forward()`** function we can compute the probability of raining in **Day 2**" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The probability of raining in day 2 is 0.88\n" + ] + } + ], + "source": [ + "belief_day_2 = forward(hmm, belief_day_1, ev=True)\n", + "print ('The probability of raining in day 2 is {:.2f}'.format(belief_day_2[0]))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the smoothing part we are interested in computing the distribution over past states given evidence up to the present. Assume that we want to compute the distribution for the time **k**, for $0\\leq k