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 pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.

Alternative Proxies: