Skip to content

Commit 5bdcfb3

Browse files
reachtarunherereachtarunhere
committed
Added section on Bayesian Networks
1 parent 5c0f23c commit 5bdcfb3

File tree

2 files changed

+184
-0
lines changed

2 files changed

+184
-0
lines changed

‎images/bayesnet.png

79.1 KB
Loading

‎probability.ipynb

Lines changed: 184 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -425,6 +425,190 @@
425425
"source": [
426426
"You can verify that the first value is the same as we obtained earlier by manual calculation."
427427
]
428+
},
429+
{
430+
"cell_type": "markdown",
431+
"metadata": {},
432+
"source": [
433+
"## Bayesian Networks\n",
434+
"\n",
435+
"A Bayesian network is a representation of the joint probability distribution encoding a collection of conditional independence statements.\n",
436+
"\n",
437+
"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",
438+
"\n",
439+
"Let us dive into the **BayesNode** implementation."
440+
]
441+
},
442+
{
443+
"cell_type": "code",
444+
"execution_count": null,
445+
"metadata": {
446+
"collapsed": false
447+
},
448+
"outputs": [],
449+
"source": [
450+
"%psource BayesNode"
451+
]
452+
},
453+
{
454+
"cell_type": "markdown",
455+
"metadata": {},
456+
"source": [
457+
"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",
458+
"\n",
459+
"The example below where we implement the network shown in **Figure 14.3** of the book will make this more clear.\n",
460+
"\n",
461+
"<img src=\"files/images/bayesnet.png\">\n",
462+
"\n",
463+
"The alarm node can be made as follows: "
464+
]
465+
},
466+
{
467+
"cell_type": "code",
468+
"execution_count": null,
469+
"metadata": {
470+
"collapsed": true
471+
},
472+
"outputs": [],
473+
"source": [
474+
"alarm_node = BayesNode('Alarm', ['Burglary', 'Earthquake'], \n",
475+
" {(True, True): 0.95,(True, False): 0.94, (False, True): 0.29, (False, False): 0.001})"
476+
]
477+
},
478+
{
479+
"cell_type": "markdown",
480+
"metadata": {},
481+
"source": [
482+
"It is possible to avoid using a tuple when there is only a single parent. So an alternative format for the **cpt** is"
483+
]
484+
},
485+
{
486+
"cell_type": "code",
487+
"execution_count": null,
488+
"metadata": {
489+
"collapsed": true
490+
},
491+
"outputs": [],
492+
"source": [
493+
"john_node = BayesNode('JohnCalls', ['Alarm'], {True: 0.90, False: 0.05})\n",
494+
"mary_node = BayesNode('MaryCalls', 'Alarm', {(True, ): 0.70, (False, ): 0.01}) # Using string for parents.\n",
495+
"# Equvivalant to john_node definition. "
496+
]
497+
},
498+
{
499+
"cell_type": "markdown",
500+
"metadata": {},
501+
"source": [
502+
"The general format used for the alarm node always holds. For nodes with no parents we can also use. "
503+
]
504+
},
505+
{
506+
"cell_type": "code",
507+
"execution_count": null,
508+
"metadata": {
509+
"collapsed": true
510+
},
511+
"outputs": [],
512+
"source": [
513+
"burglary_node = BayesNode('Burglary', '', 0.001)\n",
514+
"earthquake_node = BayesNode('Earthquake', '', 0.002)"
515+
]
516+
},
517+
{
518+
"cell_type": "markdown",
519+
"metadata": {},
520+
"source": [
521+
"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.)"
522+
]
523+
},
524+
{
525+
"cell_type": "code",
526+
"execution_count": null,
527+
"metadata": {
528+
"collapsed": false
529+
},
530+
"outputs": [],
531+
"source": [
532+
"john_node.p(False, {'Alarm': True, 'Burglary': True}) # P(JohnCalls=False | Alarm=True)"
533+
]
534+
},
535+
{
536+
"cell_type": "markdown",
537+
"metadata": {},
538+
"source": [
539+
"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."
540+
]
541+
},
542+
{
543+
"cell_type": "code",
544+
"execution_count": null,
545+
"metadata": {
546+
"collapsed": true
547+
},
548+
"outputs": [],
549+
"source": [
550+
"%psource BayesNet"
551+
]
552+
},
553+
{
554+
"cell_type": "markdown",
555+
"metadata": {},
556+
"source": [
557+
"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",
558+
"\n",
559+
"**burglary** global is an instance of **BayesNet** corresponding to the above example.\n",
560+
"\n",
561+
" T, F = True, False\n",
562+
"\n",
563+
" burglary = BayesNet([\n",
564+
" ('Burglary', '', 0.001),\n",
565+
" ('Earthquake', '', 0.002),\n",
566+
" ('Alarm', 'Burglary Earthquake',\n",
567+
" {(T, T): 0.95, (T, F): 0.94, (F, T): 0.29, (F, F): 0.001}),\n",
568+
" ('JohnCalls', 'Alarm', {T: 0.90, F: 0.05}),\n",
569+
" ('MaryCalls', 'Alarm', {T: 0.70, F: 0.01})\n",
570+
" ])"
571+
]
572+
},
573+
{
574+
"cell_type": "code",
575+
"execution_count": null,
576+
"metadata": {
577+
"collapsed": false
578+
},
579+
"outputs": [],
580+
"source": [
581+
"burglary"
582+
]
583+
},
584+
{
585+
"cell_type": "markdown",
586+
"metadata": {},
587+
"source": [
588+
"**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."
589+
]
590+
},
591+
{
592+
"cell_type": "code",
593+
"execution_count": null,
594+
"metadata": {
595+
"collapsed": false
596+
},
597+
"outputs": [],
598+
"source": [
599+
"type(burglary.variable_node('Alarm'))"
600+
]
601+
},
602+
{
603+
"cell_type": "code",
604+
"execution_count": null,
605+
"metadata": {
606+
"collapsed": false
607+
},
608+
"outputs": [],
609+
"source": [
610+
"burglary.variable_node('Alarm').cpt"
611+
]
428612
}
429613
],
430614
"metadata": {

0 commit comments

Comments
 (0)
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:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy