{"id":1191,"date":"2013-12-07T13:16:23","date_gmt":"2013-12-07T07:46:23","guid":{"rendered":"http:\/\/ucanalytics.com\/blogs\/?p=1191"},"modified":"2015-04-04T10:21:15","modified_gmt":"2015-04-04T04:51:15","slug":"logical-rigor","status":"publish","type":"post","link":"https:\/\/ucanalytics.com\/blogs\/logical-rigor\/","title":{"rendered":"Murder Cases, Evidence and Logical Rigor"},"content":{"rendered":"<hr \/>\n<h2><span style=\"color: #99ccff; font-size: 18px;\">Aarushi Murder Case &amp; Logical Rigor<\/span><\/h2>\n<div id=\"attachment_1194\" style=\"width: 219px\" class=\"wp-caption alignright\"><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg\"><img aria-describedby=\"caption-attachment-1194\" data-attachment-id=\"1194\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/logical-rigor\/photo-1\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?fit=285%2C409&amp;ssl=1\" data-orig-size=\"285,409\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Two Face &#8211; by Roopam\" data-image-description=\"&lt;p&gt;Two Face &#8211; by Roopam&lt;\/p&gt;\n\" data-image-caption=\"&lt;p&gt;Two Face &#8211; by Roopam&lt;\/p&gt;\n\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?fit=209%2C300&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?fit=285%2C409&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\"size-medium wp-image-1194\" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?resize=209%2C300\" alt=\"Two Face - by Roopam\" width=\"209\" height=\"300\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?resize=209%2C300&amp;ssl=1 209w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/11\/photo-1.jpg?w=285&amp;ssl=1 285w\" sizes=\"(max-width: 209px) 100vw, 209px\" data-recalc-dims=\"1\" \/><\/a><p id=\"caption-attachment-1194\" class=\"wp-caption-text\">Two Face &#8211; by Roopam<\/p><\/div>\n<p>Last week, on 25th November 2013, Ghaziabad High Court has given the verdict in the Aarushi murder case. The media frenzy in India has not allowed the public interest in the case to die down for the last five and a half year. The verdict as expected was all over the tube. However, I see this as a missed opportunity for media to inform the general public about the process of law to arrive at meaningful conclusions through evidence and logical rigor. The process, I feel, is common in judiciary, science and business analytics. Before I jump to explain my point let me quickly give the gist of the case.<\/p>\n<p>Aarushi Talwar, a 14 year old girl, used to live with her parents Dr. Rajesh and Dr. Nupur Talwar. Both parents are dentists and were at the peak of their careers when Aarushi was murdered in their house on the night of 15th May 2008. She was found lying in a pool of blood with her throat slit.\u00a0 The Talwar family\u2019s 45 year old male domestic help, Hemraj, was the main suspect for the crime. However, the plot thickened when Hemraj was found dead in a similar fashion as Aarushi on the terrace of the Talwars\u2019 house the next day i.e. 17th May 2008. Forensic revealed that both Aarushi and Hemraj were killed around the same time in a similar fashion. Additionally, both parents were present in the house when the crime took place.<\/p>\n<p>The High court has found the Talwar couple guilty of double murder and sentenced them with life imprisonment. People\u2019s opinion about the verdict is completely divided. This is a good case for the media to explain the reason the high court has found the couple guilty. Some people sympathize with the mourning parents, others see them as monsters. This is where media could have explained the case, evidence and the logical progress of the verdict better.\u00a0 However, the media took the easy route and made it into a Bollywood drama. The following are some of the statements I saw repeatedly flashing on the news channels, apparently taken from the verdict.<\/p>\n<ul>\n<li><span style=\"text-align: justify;\">Rajesh Talwar was a social drinker and must have drunk on the night of the double murder (so?)<\/span><\/li>\n<li>Parents were not crying when they saw their daughters body (they could be shocked?)<\/li>\n<li>They did not hug their daughter\u2019s dead body since their clothes were not blood stained (so?)<\/li>\n<\/ul>\n<p>These seem more like moral judgement rather than the process of law. I am not sure how the high court has arrived at the conclusion but I hope they have firmer arguments for their verdict. Let me also discuss another high profile murder case from the mid-nineties that could be easily termed as the most publicized criminal trial in the American history.<\/p>\n<h2><span style=\"color: #99ccff; font-size: 18px;\">O J Simpson Case and Bayes Statistics<\/span><\/h2>\n<p>Nicole Brown Simpson and her friend Ronald Goldman were found murdered on Brown&#8217;s Bundy Drive condo in the Brentwood area of Los Angeles on 13<sup>th<\/sup> June 1994. \u00a0Nicole\u2019s ex-husband O J Simpson, also an ex-American football player and actor, was the prime suspect for the murders. The trial started with prosecution proving that O J Simpson has a history of physical violence against Nicole. Additionally, they argued that this is a run-up to homicide. To this the defense lawyer, Alan Dershowitz, argued \u201cWe knew we could prove, if we had to, that an infinitesimal percentage \u2014 certainly fewer than 1 of 2,500 \u2014 of men who slap or beat their domestic partners go on to murder them.\u201d Before we analyse this statement by Alan Dershowitz, let me introduce four conditions necessary with their abbreviations (A &amp;B) to understand the problem better.<\/p>\n<pre style=\"text-align: justify;\"><span style=\"font-family: georgia, palatino; font-size: 14px;\"><b>   A     <\/b>:  Husband has a history of physical violence against his wife\r\n<b>~A     <\/b>: Husband does not have a history of physical violence against his wife\r\n<b>   B     <\/b>: Wife got murdered\r\n<b>~B     <\/b>: Wife does not get murdered<\/span><\/pre>\n<p>If we consider Alan Dershowitz\u2019s argument, the probability or chance of O J Simpson killing Nicole is certainly really low. We are asking chances of wife getting killed if the husband has a history of physical violence against her. This statement can be written probabilistically as<\/p>\n<pre style=\"text-align: justify;\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P+%28B%7CA%29+%3D%5Cfrac%7B1%7D%7B2500%7D%3D0.04%5C%25++&#038;bg=ffffff&#038;fg=000&#038;s=1&#038;c=20201002\" alt=\"P (B|A) =&#92;frac{1}{2500}=0.04&#92;%  \" class=\"latex\" \/>\r\n\r\nThis is conditional probability where we are asking chances of B when A has happened.<\/pre>\n<p>However, this is a misinterpretation of facts, a clever trick to fool the non-mathematical audience and the jury. In the end it did work as O J Simpson was acquitted of the charges and set free contrary to the popular belief. The defense lawyer had confused the jury with a wrong question. He asked, how many wife beaters eventually kill their wives? The answer to this question is 0.04%, as mentioned earlier.\u00a0 However, since, the fact available with the court is that the wife is killed; hence the right question to ask over here is \u2018how many killed women are murdered by their wife beater husbands?\u2019 This is a completely different question with a completely different answer. To answer this we need to find the conditional probability P (A|B) (Notice this is a flip of P(B|A) that Alan Dershowitz asked). Let us try to answer this question.<\/p>\n<p>According to Alan Dershowitz\u2019s statement 1 in 2500 woman was getting killed by their husband with the history of domestic violence. This is another way of saying that 40 out of every hundred thousand women were getting killed by their husband with history of physical violence against them. Additionally, in the United States in the mid-nineties, the murder rate for women was 1 in 20,000. This means additional 5 in hundred thousand women were getting killed by someone else. Hence the answer to the above question is<\/p>\n<pre style=\"text-align: justify;\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28A%7CB%29%3D%5Cfrac%7B40%7D%7B%2840%2B5%29%7D%3D88.9%5C%25%5Csimeq+90%5C%25++&#038;bg=ffffff&#038;fg=000&#038;s=1&#038;c=20201002\" alt=\"P(A|B)=&#92;frac{40}{(40+5)}=88.9&#92;%&#92;simeq 90&#92;%  \" class=\"latex\" \/><\/pre>\n<p>This probability is staggeringly higher than 0.04% in the\u00a0defense\u00a0lawyer\u2019s argument. We could\u00a0arrive\u00a0at the same answer using the following schematic<\/p>\n<div id=\"attachment_1245\" style=\"width: 527px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg\"><img aria-describedby=\"caption-attachment-1245\" data-attachment-id=\"1245\" data-permalink=\"https:\/\/ucanalytics.com\/blogs\/logical-rigor\/bayes\/\" data-orig-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?fit=738%2C564&amp;ssl=1\" data-orig-size=\"738,564\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Bayesian statistics for Logical Rigor\" data-image-description=\"&lt;p&gt;Bayesian statistics for Logical Rigor&lt;\/p&gt;\n\" data-image-caption=\"&lt;p&gt;Bayesian statistics for Logical Rigor&lt;\/p&gt;\n\" data-medium-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?fit=300%2C229&amp;ssl=1\" data-large-file=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?fit=640%2C489&amp;ssl=1\" decoding=\"async\" loading=\"lazy\" class=\" wp-image-1245 \" src=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?resize=517%2C395\" alt=\"Bayesian statistics for Logical Rigor\" width=\"517\" height=\"395\" srcset=\"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?w=738&amp;ssl=1 738w, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2013\/12\/bayes.jpg?resize=300%2C229&amp;ssl=1 300w\" sizes=\"(max-width: 517px) 100vw, 517px\" data-recalc-dims=\"1\" \/><\/a><p id=\"caption-attachment-1245\" class=\"wp-caption-text\">Bayesian statistics for Logical Rigor<\/p><\/div>\n<p>Nicole Brown is killed and we are asking her chances of getting killed by her husband O J Simpson who also had the history of physical violence against Nicole. Since B has already happened, In this case the two probabilities marked in red are useless to answer the posterior probability P(A|B) . The final Bayesian probability is given as<\/p>\n<pre style=\"text-align: justify;\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28A%7CB%29%3D%5Cfrac%7BP%28B%7CA%29%7D%7B%28P%28B%7CA%29%2BP%28B%7C%5Csim+A%29%29%7D%3D%5Cfrac%7B0.04%5C%25%7D%7B%280.04%5C%25%2B0.005%5C%25%29%7D%3D+88.9%5C%25%5Csimeq+90%5C%25++&#038;bg=ffffff&#038;fg=000&#038;s=1&#038;c=20201002\" alt=\"P(A|B)=&#92;frac{P(B|A)}{(P(B|A)+P(B|&#92;sim A))}=&#92;frac{0.04&#92;%}{(0.04&#92;%+0.005&#92;%)}= 88.9&#92;%&#92;simeq 90&#92;%  \" class=\"latex\" \/><\/pre>\n<p>This probability is the same as the probability we have found using frequency method above. This is an example of logical rigor missing in the O J Simpson murder trial. This error was sighted by mathematicians much after the verdict was announced. I believe that the judicial system across the globe could be improved through transparent analysis of verdicts in public forums. The media has a particularly important role to play in this.<\/p>\n<h4><span style=\"color: #99ccff; font-size: 16px;\">Sign-off Note<\/span><\/h4>\n<p>In our <a href=\"http:\/\/ucanalytics.com\/blogs\/customer-segmentation-cluster-analysis-telecom-case-study\/\" target=\"_blank\">last article<\/a> we have started with a case study from the telecom sector. We learned about cluster analysis using black holes as an analogy. We have not lost the case study in a black hole rather we will continue with the case study in the subsequent articles. See you soon!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aarushi Murder Case &amp; Logical Rigor Last week, on 25th November 2013, Ghaziabad High Court has given the verdict in the Aarushi murder case. The media frenzy in India has not allowed the public interest in the case to die down for the last five and a half year. The verdict as expected was all<\/p>\n<p><a class=\"excerpt-more blog-excerpt\" href=\"https:\/\/ucanalytics.com\/blogs\/logical-rigor\/\">Read More&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":1245,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_newsletter_tier_id":0,"jetpack_publicize_message":"","jetpack_is_tweetstorm":false,"jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[64,65],"tags":[33,7,71,6,10],"jetpack_publicize_connections":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Murder Cases, Evidence and Logical Rigor &ndash; YOU CANalytics |<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ucanalytics.com\/blogs\/logical-rigor\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Murder Cases, Evidence and Logical Rigor &ndash; YOU CANalytics |\" \/>\n<meta property=\"og:description\" content=\"Aarushi Murder Case &amp; Logical Rigor Last week, on 25th November 2013, Ghaziabad High Court has given the verdict in the Aarushi murder case. 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The media frenzy in India has not allowed the public interest in the case to die down for the last five and a half year. 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