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Added section on Bayesian Networks
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‎images/bayesnet.png

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‎probability.ipynb

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"source": [
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"You can verify that the first value is the same as we obtained earlier by manual calculation."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Bayesian Networks\n",
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"\n",
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"A Bayesian network is a representation of the joint probability distribution encoding a collection of conditional independence statements.\n",
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"\n",
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"A Bayes Network is implemented as the class **BayesNet**. It consisits of a collection of nodes implemented by the class **BayesNode**. The implementation in the above mentioned classes focuses only on boolean variables. Each node is associated with a variable and it contains a **conditional probabilty table (cpt)**. The **cpt** represents the probability distribution of the variable conditioned on its parents **P(X | parents)**.\n",
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"\n",
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"Let us dive into the **BayesNode** implementation."
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]
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"%psource BayesNode"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The constructor takes in the name of **variable**, **parents** and **cpt**. Here **variable** is a the name of the variable like 'Earthquake'. **parents** should a list or space separate string with variable names of parents. The conditional probability table is a dict {(v1, v2, ...): p, ...}, the distribution P(X=true | parent1=v1, parent2=v2, ...) = p. Here the keys are combination of boolean values that the parents take. The length and order of the values in keys should be same as the supplied **parent** list/string. In all cases the probability of X being false is left implicit, since it follows from P(X=true).\n",
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"\n",
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"The example below where we implement the network shown in **Figure 14.3** of the book will make this more clear.\n",
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"\n",
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"<img src=\"files/images/bayesnet.png\">\n",
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"\n",
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"The alarm node can be made as follows: "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"alarm_node = BayesNode('Alarm', ['Burglary', 'Earthquake'], \n",
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" {(True, True): 0.95,(True, False): 0.94, (False, True): 0.29, (False, False): 0.001})"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"It is possible to avoid using a tuple when there is only a single parent. So an alternative format for the **cpt** is"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"john_node = BayesNode('JohnCalls', ['Alarm'], {True: 0.90, False: 0.05})\n",
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"mary_node = BayesNode('MaryCalls', 'Alarm', {(True, ): 0.70, (False, ): 0.01}) # Using string for parents.\n",
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"# Equvivalant to john_node definition. "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The general format used for the alarm node always holds. For nodes with no parents we can also use. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"burglary_node = BayesNode('Burglary', '', 0.001)\n",
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"earthquake_node = BayesNode('Earthquake', '', 0.002)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"It is possible to use the node for lookup function using the **p** method. The method takes in two arguments **value** and **event**. Event must be a dict of the type {variable:values, ..} The value corresponds to the value of the variable we are interested in (False or True).The method returns the conditional probability **P(X=value | parents=parent_values)**, where parent_values are the values of parents in event. (event must assign each parent a value.)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"john_node.p(False, {'Alarm': True, 'Burglary': True}) # P(JohnCalls=False | Alarm=True)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"With all the information about nodes present it is possible to construct a Bayes Network using **BayesNet**. The **BayesNet** class does not take in nodes as input but instead takes a list of **node_specs**. An entry in **node_specs** is a tuple of the parameters we use to construct a **BayesNode** namely **(X, parents, cpt)**. **node_specs** must be ordered with parents before children."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": [
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"%psource BayesNet"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The constructor of **BayesNet** takes each item in **node_specs** and adds a **BayesNode** to its **nodes** object variable by calling the **add** method. **add** in turn adds node to the net. Its parents must already be in the net, and its variable must not. Thus add allows us to grow a **BayesNet** given its parents are already present.\n",
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"\n",
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"**burglary** global is an instance of **BayesNet** corresponding to the above example.\n",
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"\n",
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" T, F = True, False\n",
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"\n",
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" burglary = BayesNet([\n",
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" ('Burglary', '', 0.001),\n",
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" ('Earthquake', '', 0.002),\n",
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" ('Alarm', 'Burglary Earthquake',\n",
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" {(T, T): 0.95, (T, F): 0.94, (F, T): 0.29, (F, F): 0.001}),\n",
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" ('JohnCalls', 'Alarm', {T: 0.90, F: 0.05}),\n",
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" ('MaryCalls', 'Alarm', {T: 0.70, F: 0.01})\n",
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" ])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"burglary"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**BayesNet** method **variable_node** allows to reach **BayesNode** instances inside a Bayes Net. It is possible to modify the **cpt** of the nodes directly using this method."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"type(burglary.variable_node('Alarm'))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"burglary.variable_node('Alarm').cpt"
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]
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}
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],
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"metadata": {

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