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	<title>Bayes Theorem &#8211; FootballPerspective.com</title>
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		<title>Quick Thoughts on the Saints 0-2 Start</title>
		<link>http://www.footballperspective.com/quick-thoughts-on-the-saints-0-2-start/</link>
					<comments>http://www.footballperspective.com/quick-thoughts-on-the-saints-0-2-start/#comments</comments>
		
		<dc:creator><![CDATA[Chase Stuart]]></dc:creator>
		<pubDate>Tue, 16 Sep 2014 14:45:38 +0000</pubDate>
				<category><![CDATA[Current Events]]></category>
		<category><![CDATA[Bayes Theorem]]></category>
		<category><![CDATA[Saints]]></category>
		<guid isPermaLink="false">http://www.footballperspective.com/?p=21750</guid>

					<description><![CDATA[No team wants to start the season 0-2. By now you&#8217;ve heard the statistic that since 1990, only 12% of teams to start 0-2 have made the playoffs. While that&#8217;s true, that&#8217;s just one way &#8212; and not the only way &#8212; to examine the Saints start. That analysis is based on the following idea: [&#8230;]]]></description>
										<content:encoded><![CDATA[<p>No team wants to start the season 0-2.  By now you&#8217;ve heard the statistic that since 1990, only 12% of teams to start 0-2 have made the playoffs.  While that&#8217;s true, that&#8217;s just one way &#8212; and not the only way &#8212; to examine the Saints start.  That analysis is based on the following idea:</p>
<blockquote><p>Look at group of teams with the same start &#8211;> see how they finish the year</p></blockquote>
<p>But there&#8217;s another way to consider New Orleans&#8217; early season woes.  The Saints lost both games on the road.  So while New Orleans is 0-2, the team still has 8 home games remaining.  Based on the Saints history under <a href="http://www.pro-football-reference.com/coaches/PaytSe0.htm" target="_blank">Sean Payton</a>, projecting a a 7-1 home record doesn&#8217;t seem unreasonable.  And while the team lost both games so far, note that Saints opponents have already kicked three game-winning or game-tying field goals at the end of regulation or overtime already. <span class="footnote_referrer"><a role="button" tabindex="0" onclick="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_1');" onkeypress="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_1');" ><sup id="footnote_plugin_tooltip_21750_2_1" class="footnote_plugin_tooltip_text">[1]</sup></a><span id="footnote_plugin_tooltip_text_21750_2_1" class="footnote_tooltip"><a rel="nofollow" target="_blank" href="http://www.pro-football-reference.com/players/b/bryanmat01.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Matt  Bryant</a> forced overtime with a 51-yard field goal as time ran out in the 4th quarter, and then won the game for Atlanta in week 1 with a 52-yarder.</span></span><script type="text/javascript"> jQuery('#footnote_plugin_tooltip_21750_2_1').tooltip({ tip: '#footnote_plugin_tooltip_text_21750_2_1', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top center', relative: true, offset: [-7, 0], });</script>  That&#8217;s an amazing feat to have occurred after just two games; from a predictive standpoint, the Saints could just as easily be 2-0.  And from a predictive standpoint, a 3-3 finish in road games the rest of the way doesn&#8217;t seem unreasonable, either.  That would give the team a 10-6 record, and probably a playoff berth.<span id="more-21750"></span></p>
<p>Now, attack this problem from a different perspective: if you assumed before the season that the Saints would be a playoff team, how likely would you be to revise your prediction if you knew that <em>at some point</em> in the year, the Saints would lose two straight road games?  In other words:</p>
<blockquote><p>Look at all playoff teams &#8211;> how often did they lose two straight road games</p></blockquote>
<p>As it turns out, <strong>63% of all playoff teams since 1990 have lost at least two road games in a row.</strong> <span class="footnote_referrer"><a role="button" tabindex="0" onclick="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_2');" onkeypress="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_2');" ><sup id="footnote_plugin_tooltip_21750_2_2" class="footnote_plugin_tooltip_text">[2]</sup></a><span id="footnote_plugin_tooltip_text_21750_2_2" class="footnote_tooltip">If a team lost a road game, won a home game, and then lost a road game, that counts as two road games in a row.</span></span><script type="text/javascript"> jQuery('#footnote_plugin_tooltip_21750_2_2').tooltip({ tip: '#footnote_plugin_tooltip_text_21750_2_2', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top center', relative: true, offset: [-7, 0], });</script>  Given the Saints&#8217; unusual home/road proclivities, we might view the Saints as even more likely than most playoff teams to drop a pair of consecutive road games.  In fact, New Orleans did just that in 2011 when the <a href="http://www.pro-football-reference.com/teams/nor/2011.htm" target="_blank">team went 13-3</a>, <span class="footnote_referrer"><a role="button" tabindex="0" onclick="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_3');" onkeypress="footnote_moveToReference_21750_2('footnote_plugin_reference_21750_2_3');" ><sup id="footnote_plugin_tooltip_21750_2_3" class="footnote_plugin_tooltip_text">[3]</sup></a><span id="footnote_plugin_tooltip_text_21750_2_3" class="footnote_tooltip">That includes a road loss to the Rams, who went 2-14 that year.</span></span><script type="text/javascript"> jQuery('#footnote_plugin_tooltip_21750_2_3').tooltip({ tip: '#footnote_plugin_tooltip_text_21750_2_3', tipClass: 'footnote_tooltip', effect: 'fade', predelay: 0, fadeInSpeed: 200, delay: 400, fadeOutSpeed: 200, position: 'top center', relative: true, offset: [-7, 0], });</script> and the Saints lost 5 out of 6 road games in one stretch just last season.</p>
<p>To be fair, it&#8217;s not as though the Saints have lost just any two road games in a row; they&#8217;ve lost <em>the only two road games they&#8217;ve had</em>.  From a Bayes Theorem standpoint &#8212; and if I have time, I plan to revisit this issue later in the week &#8212; we would have to downgrade the Saints after this start.  Whatever we thought of the Saints, there&#8217;s some new information that would cause us to reevaluate them.  Yes, there&#8217;s a possibility (probability?) that New Orleans is exactly who we thought they were, and if these games were in weeks 7 and 13, nobody would notice.  But there&#8217;s also the possibility that the Saints are just not a very good team, and these two road losses are the first two signs of that.  That&#8217;s one of the questions Bayes Theorem is really good at answering.</p>
<p>With <a rel="nofollow" target="_blank" href="http://www.pro-football-reference.com/players/B/BreeDr00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Drew  Brees</a> and <a rel="nofollow" target="_blank" href="http://www.pro-football-reference.com/players/G/GrahJi00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Jimmy  Graham</a>, though, I&#8217;m betting against that.  And I know the Saints have a much higher odds of making the playoffs than 12%.  And with upcoming home games against the Vikings and Bucs sandwiched around a road trip to Dallas, a 3-2 start heading into the bye doesn&#8217;t seem too far-fetched, either.</p>
<div class="speaker-mute footnotes_reference_container"> <div class="footnote_container_prepare"><p><span role="button" tabindex="0" class="footnote_reference_container_label pointer" onclick="footnote_expand_collapse_reference_container_21750_2();">References</span><span role="button" tabindex="0" class="footnote_reference_container_collapse_button" style="display: none;" onclick="footnote_expand_collapse_reference_container_21750_2();">[<a id="footnote_reference_container_collapse_button_21750_2">+</a>]</span></p></div> <div id="footnote_references_container_21750_2" style=""><table class="footnotes_table footnote-reference-container"><caption class="accessibility">References</caption> <tbody> 

<tr class="footnotes_plugin_reference_row"> <th scope="row" class="footnote_plugin_index_combi pointer"  onclick="footnote_moveToAnchor_21750_2('footnote_plugin_tooltip_21750_2_1');"><a id="footnote_plugin_reference_21750_2_1" class="footnote_backlink"><span class="footnote_index_arrow">&#8593;</span>1</a></th> <td class="footnote_plugin_text"><a rel="nofollow" target="_blank" href="http://www.pro-football-reference.com/players/b/bryanmat01.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Matt  Bryant</a> forced overtime with a 51-yard field goal as time ran out in the 4th quarter, and then won the game for Atlanta in week 1 with a 52-yarder.</td></tr>

<tr class="footnotes_plugin_reference_row"> <th scope="row" class="footnote_plugin_index_combi pointer"  onclick="footnote_moveToAnchor_21750_2('footnote_plugin_tooltip_21750_2_2');"><a id="footnote_plugin_reference_21750_2_2" class="footnote_backlink"><span class="footnote_index_arrow">&#8593;</span>2</a></th> <td class="footnote_plugin_text">If a team lost a road game, won a home game, and then lost a road game, that counts as two road games in a row.</td></tr>

<tr class="footnotes_plugin_reference_row"> <th scope="row" class="footnote_plugin_index_combi pointer"  onclick="footnote_moveToAnchor_21750_2('footnote_plugin_tooltip_21750_2_3');"><a id="footnote_plugin_reference_21750_2_3" class="footnote_backlink"><span class="footnote_index_arrow">&#8593;</span>3</a></th> <td class="footnote_plugin_text">That includes a road loss to the Rams, who went 2-14 that year.</td></tr>

 </tbody> </table> </div></div><script type="text/javascript"> function footnote_expand_reference_container_21750_2() { jQuery('#footnote_references_container_21750_2').show(); jQuery('#footnote_reference_container_collapse_button_21750_2').text('−'); } function footnote_collapse_reference_container_21750_2() { jQuery('#footnote_references_container_21750_2').hide(); jQuery('#footnote_reference_container_collapse_button_21750_2').text('+'); } function footnote_expand_collapse_reference_container_21750_2() { if (jQuery('#footnote_references_container_21750_2').is(':hidden')) { footnote_expand_reference_container_21750_2(); } else { footnote_collapse_reference_container_21750_2(); } } function footnote_moveToReference_21750_2(p_str_TargetID) { footnote_expand_reference_container_21750_2(); var l_obj_Target = jQuery('#' + p_str_TargetID); if (l_obj_Target.length) { jQuery( 'html, body' ).delay( 0 ); jQuery('html, body').animate({ scrollTop: l_obj_Target.offset().top - window.innerHeight * 0.2 }, 380); } } function footnote_moveToAnchor_21750_2(p_str_TargetID) { footnote_expand_reference_container_21750_2(); var l_obj_Target = jQuery('#' + p_str_TargetID); if (l_obj_Target.length) { jQuery( 'html, body' ).delay( 0 ); jQuery('html, body').animate({ scrollTop: l_obj_Target.offset().top - window.innerHeight * 0.2 }, 380); } }</script>]]></content:encoded>
					
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			</item>
		<item>
		<title>Denver&#8217;s Scoring Pace and Manning/Brady Update</title>
		<link>http://www.footballperspective.com/denvers-scoring-pace-and-manningbrady-update/</link>
					<comments>http://www.footballperspective.com/denvers-scoring-pace-and-manningbrady-update/#comments</comments>
		
		<dc:creator><![CDATA[Chase Stuart]]></dc:creator>
		<pubDate>Sat, 23 Nov 2013 04:01:49 +0000</pubDate>
				<category><![CDATA[History]]></category>
		<category><![CDATA[Statistics]]></category>
		<category><![CDATA[Trivia]]></category>
		<category><![CDATA[Bayes Theorem]]></category>
		<category><![CDATA[Broncos]]></category>
		<category><![CDATA[Peyton Manning]]></category>
		<guid isPermaLink="false">http://www.footballperspective.com/?p=15433</guid>

					<description><![CDATA[Today&#8217;s insane statistic comes courtesy of RJ Bell: the difference between Denver and the #2 team in points per game is larger than the difference between the #2 and #31 teams. The Broncos are averaging 39.8 points per game this season, 11.6 points more than the (surprisingly second-ranked) Bears. And Chicago is averaging just 9.9 [&#8230;]]]></description>
										<content:encoded><![CDATA[<p><div id="attachment_14341" style="width: 310px" class="wp-caption alignright"><a href="http://www.footballperspective.com/wp-content/uploads/2013/10/Manning-Welker.jpg"><img fetchpriority="high" decoding="async" aria-describedby="caption-attachment-14341" src="http://www.footballperspective.com/wp-content/uploads/2013/10/Manning-Welker-300x168.jpg" alt="Denver has scored at a historic rate" width="300" height="168" class="size-medium wp-image-14341" srcset="http://www.footballperspective.com/wp-content/uploads/2013/10/Manning-Welker-300x168.jpg 300w, http://www.footballperspective.com/wp-content/uploads/2013/10/Manning-Welker.jpg 400w" sizes="(max-width: 300px) 100vw, 300px" /></a> <p id="caption-attachment-14341" class="wp-caption-text">Denver has scored at a historic rate.</p></div>Today&#8217;s insane statistic comes courtesy of <a href="https://twitter.com/RJinVegas/status/403916916818849793" target="_blank">RJ Bell</a>: the difference between Denver and the #2 team in points per game <strong>is larger than the difference between the #2 and #31 teams</strong>. The Broncos are averaging 39.8 points per game this season, 11.6 points more than the (<a href="http://www.footballperspective.com/new-york-times-post-week-11-2013/" target="_blank">surprisingly second-ranked</a>) Bears. And Chicago is averaging just 9.9 more points per game than the Jets, the #31 ranked scoring team.</p>
<p>That is, well, crazy. The record for points per game in a season is 38.8, set by the <a href="http://www.pro-football-reference.com/teams/ram/1950.htm" target="_blank">1950 Rams</a>. The <a href="http://www.pro-football-reference.com/teams/nwe/2007.htm" target="_blank">2007 Patriots</a> are second at 36.8, and both of those teams scored <a href="http://www.pro-football-reference.com/play-index/tgl_finder.cgi?request=1&amp;match=single&amp;year_min=1940&amp;year_max=2013&amp;game_type=R&amp;playoff_round=&amp;game_num_min=1&amp;game_num_max=10&amp;week_num_min=0&amp;week_num_max=99&amp;game_day_of_week=&amp;game_time=&amp;time_zone=&amp;game_location=&amp;surface=&amp;roof=&amp;game_result=&amp;overtime=&amp;league_id=&amp;team_id=&amp;team_conf_id=&amp;team_div_id=&amp;opp_id=&amp;opp_conf_id=&amp;opp_div_id=&amp;team_off_scheme=&amp;team_def_align=&amp;opp_off_scheme=&amp;opp_def_align=&amp;conference_game=&amp;division_game=&amp;tm_is_playoff=&amp;opp_is_playoff=&amp;tm_is_winning=&amp;opp_is_winning=&amp;tm_scored_first=&amp;tm_led=&amp;tm_trailed=&amp;c1stat=&amp;c1comp=gt&amp;c1val=&amp;c2stat=&amp;c2comp=gt&amp;c2val=&amp;c3stat=&amp;c3comp=gt&amp;c3val=&amp;c4stat=&amp;c4comp=gt&amp;c4val=&amp;order_by=points" target="_blank">slightly more points</a> through ten games than the 2013 Broncos. So while Denver is on pace to break the scoring record, some regression to the mean over the final six games should be expected.</p>
<p>If the Broncos want to set the record for most points scored relative to the <em>second</em> highest scoring team in the league, <a target="_blank" href="http://www.pro-football-reference.com/players/M/MannPe00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Peyton  Manning</a> and company have some work to do. That mark is held by the <a href="http://www.pro-football-reference.com/teams/chi/1941.htm" target="_blank">&#8217;41 Bears</a>, who averaged 36.0 points per game, 12.5 more than the Packers that year. Second and third on that list are the &#8217;07 Patriots (8.4) and &#8217;50 Rams (8.3), so Denver has a realistic shot of setting the modern record. </p>
<p>I&#8217;ll be honest: as dominant as the Broncos offense has been, I&#8217;m a little surprised to see them so far ahead of the competition in points scored. After all, consider:</p>
<ul>
<li>The Eagles have just 19 fewer yards than the Broncos, and <a target="_blank" href="http://www.pro-football-reference.com/players/F/FoleNi00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Nick  Foles</a> actually leads Manning in both passer rating and Adjusted Net Yards per Attempt;</li>
<li>In <a href="http://www.pro-football-reference.com/years/2013/" target="_blank">PFR&#8217;s Expected Points Added</a>, the Broncos offense is at 14.7 EPA-added per game, while the Saints offense is at 11.3.  That&#8217;s a relatively small difference considering the fact that Denver has scored 12.1 more points per game than New Orleans.</li>
<li>The Chargers have a higher completion percentage than the Broncos and six fewer turnovers, but have averaged 17 fewer points per game.</li>
<li>The Packers are actually a hair ahead of Denver in yards per play (6.3531 to 6.3529), but have scored two fewer touchdowns per game.</li>
</ul>
<p>So what&#8217;s going on? I&#8217;m perfectly fine with Denver being <a href="http://www.footballoutsiders.com/dvoa-ratings/2013/week-11-dvoa-ratings" target="_blank">general run-of-the-mill dominant</a>, but the team&#8217;s points scored numbers makes it seem like the Broncos might be the greatest offensive machine ever.  I think I&#8217;ve identified the two reasons to explain the gap:</p>
<p><strong>Red Zone success</strong></p>
<p>Philadelphia has scored a touchdown just 46% of the time the Eagles made it into the red zone, which ranks 28th in the league. San Diego isn&#8217;t much better at 50% (22nd). The Saints are at 52.5% (20th), and the Packers are down at 30th at 43%. So some excellent offenses are really struggling in the red zone, which gives them disproportionately low points per game averages. Oh, and Denver? They&#8217;re at 79.1%, by far the highest rate in the league. It&#8217;s not unusual for a great offense to dominate in the red zone &#8212; the &#8217;07 Pats were at 70% &#8212; but what is unusual is seeing the other top offenses struggle there.</p>
<p>I have red zone data going back to 1997, and the highest ever performance was set by Kansas City in 2003. The <a target="_blank" href="http://www.pro-football-reference.com/players/G/GreeTr00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Trent  Green</a>&#8211;<a target="_blank" href="http://www.pro-football-reference.com/players/H/HolmPr00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Priest  Holmes</a>&#8211;<a target="_blank" href="http://www.pro-football-reference.com/players/G/GonzTo00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Tony  Gonzalez</a> Chiefs scored a touchdown on 77.8% of all red zone opportunities (42 out of 54), so the Broncos (34 out of 43) could break that record this year.  More likely, though, is that the Broncos go from otherworldly in the red zone to just great, which would drop the team&#8217;s points per game average. </p>
<p><strong>Number of Drives</strong></p>
<p>The Broncos are averaging 2.85 points per drive, while the Saints are #2 at 2.46. That&#8217;s not a huge difference &#8212; the gap between #2 and #7 is slightly bigger. The difference, as you can deduce, is that the Broncos are averaging 13 drives per game while the Saints are at just 11.3 drives per game. Why is that? New Orleans&#8217; average drive takes 2:56 minutes, the third-longest in the league (and San Diego is #1 at 3:13), while the Broncos are in the bottom five at 2:17 (the Eagles are last at 2:02). That <a href="http://www.pro-football-reference.com/coaches/KellCh0.htm" target="_blank">Chip Kelly</a> edge is erased, though, because Philadelphia&#8217;s opponents average 2:48 per drive, the third <em>highest </em>rate in the league. Denver&#8217;s opponents take just 2:18 per drive, the third lowest (just a second ahead of Detroit and eight seconds longer than Kansas City).</p>
<p>The Broncos defense is not great, but it does rank 6th in completion percentage allowed. Combine that with the fact that Denver ranks 4th in percentage of opponent plays that are passes, and incomplete passes occur on 25% of all plays run by Broncos opponents, the second-highest rate in the league behind Kansas City. That&#8217;s not surprising for a team with such a high <a href="http://www.footballperspective.com/2013-game-scripts/" target="_blank">Game Script</a>, but it does stop the clock from running for long stretches, which gives Denver&#8217;s offense more possessions The Chargers are 28th in this statistic (18%), which is one reason why San Diego is dead last in offensive drives (10.2 per game).</p>
<p>But there&#8217;s another reason why Broncos&#8217; opponents tend to have short drives: Denver leads the league in 20+ yard plays allowed at 54. As a result, teams don&#8217;t end up with many clock-chewing drives against Denver: opponents tend to gain big yards quickly or throw incomplete passes. That increases the number of drives for the Broncos, which (one could argue) inflates the success of the team&#8217;s offense. It&#8217;s all relative, of course &#8212; Denver is still #1 in points per drive by a wide margin &#8212; but it&#8217;s worth recognizing that Denver has scored 75% more points per game than an average of the other 31 teams, but &#8220;just&#8221; 62% more on a per-drive basis. That accounts for about 3 points per game. Add in the insane success in the red zone, and the lack of success there by the other top teams, and you have the reasons for the crazy stat at the top of today&#8217;s post.<br />
<strong><br />
Manning Record Watch Update</strong></p>
<p>After six games, I analyzed how likely Manning was to <a href="http://www.footballperspective.com/how-likely-is-peyton-manning-to-break-the-single-season-td-record-part-ii/" target="_blank">break the single-season touchdown record</a>. At the time, he had 22 touchdowns, and the formula projected him to throw 2.99 TDs/G the rest of the way to finish with 52 touchdowns, narrowly breaking <a target="_blank" href="http://www.pro-football-reference.com/players/B/BradTo00.htm?utm_campaign=Linker&#038;utm_source=direct&#038;utm_medium=linker-www.footballperspective.com">Tom  Brady&#8217;s</a> record.</p>
<p>Now? Manning has 34 touchdowns, as his pace has only slightly declined. What does that mean? To calculate Manning’s odds using <a href="http://www.footballperspective.com/tag/bayes-theorem/" target="_blank">Bayes Theorem</a> we need to know four things:</p>
<p>1) His Bayesian prior mean (i.e., his historical average): <strong>2.38</strong>, as this number wouldn&#8217;t change from the original post.</p>
<p>2) His Bayesian prior variance (the variance surrounding his historical average): Again, no change here, so we use <strong>0.0986</strong>.</p>
<p>3) His observed mean: Instead of 3.667, we will use <strong>3.4</strong>.</p>
<p>4) His observed variance: This one involves just a little bit of work. What I suggested we do last time is calculate the number of passing touchdowns per game Manning averaged in the first six (now ten) games of each season since 2000, along with his average over the rest of the season (then, 8-10 games, now, 4-6 games). Then we take the difference of the variances of each column, as we did in step two.</p>
<div align="center">
<table id="tablepress-813" class="tablepress tablepress-id-813">
<thead>
<tr class="row-1">
	<th class="column-1">Year</th><th class="column-2">TD/G Thru 10</th><th class="column-3">ROY G</th><th class="column-4">TD/G ROY</th><th class="column-5">Diff</th>
</tr>
</thead>
<tbody class="row-striping row-hover">
<tr class="row-2">
	<td class="column-1">2000</td><td class="column-2">2.1</td><td class="column-3">6</td><td class="column-4">2</td><td class="column-5">0.1</td>
</tr>
<tr class="row-3">
	<td class="column-1">2001</td><td class="column-2">1.8</td><td class="column-3">6</td><td class="column-4">1.33</td><td class="column-5">0.47</td>
</tr>
<tr class="row-4">
	<td class="column-1">2002</td><td class="column-2">1.9</td><td class="column-3">6</td><td class="column-4">1.33</td><td class="column-5">0.57</td>
</tr>
<tr class="row-5">
	<td class="column-1">2003</td><td class="column-2">1.9</td><td class="column-3">6</td><td class="column-4">1.67</td><td class="column-5">0.23</td>
</tr>
<tr class="row-6">
	<td class="column-1">2004</td><td class="column-2">3.5</td><td class="column-3">5</td><td class="column-4">2.8</td><td class="column-5">0.7</td>
</tr>
<tr class="row-7">
	<td class="column-1">2005</td><td class="column-2">2</td><td class="column-3">4</td><td class="column-4">2</td><td class="column-5">0</td>
</tr>
<tr class="row-8">
	<td class="column-1">2006</td><td class="column-2">2</td><td class="column-3">6</td><td class="column-4">1.83</td><td class="column-5">0.17</td>
</tr>
<tr class="row-9">
	<td class="column-1">2007</td><td class="column-2">1.6</td><td class="column-3">5</td><td class="column-4">3</td><td class="column-5">-1.4</td>
</tr>
<tr class="row-10">
	<td class="column-1">2008</td><td class="column-2">1.7</td><td class="column-3">5</td><td class="column-4">1.8</td><td class="column-5">-0.1</td>
</tr>
<tr class="row-11">
	<td class="column-1">2009</td><td class="column-2">2.1</td><td class="column-3">4</td><td class="column-4">3</td><td class="column-5">-0.9</td>
</tr>
<tr class="row-12">
	<td class="column-1">2010</td><td class="column-2">2</td><td class="column-3">6</td><td class="column-4">2.17</td><td class="column-5">-0.17</td>
</tr>
<tr class="row-13">
	<td class="column-1">2012</td><td class="column-2">2.4</td><td class="column-3">6</td><td class="column-4">2.17</td><td class="column-5">0.23</td>
</tr>
<tr class="row-14">
	<td class="column-1">Variance</td><td class="column-2">0.22</td><td class="column-3"></td><td class="column-4">0.31</td><td class="column-5"></td>
</tr>
</tbody>
</table>
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<p>Manning’s variance over the rest of the season is 0.3052 TDs/G, while his variance through ten games is 0.2214; the differential there is 0.0838, which is the variance of our current mean.</p>
<p>Once you have your number for these four variables, then you substitute those numbers into this equation:</p>
<blockquote><p>Result_mean = [(prior_mean/prior_variance)+(observed_mean/observed_variance)]/[(1/prior_variance)+(1/observed_variance)]</p></blockquote>
<p>Or, using our numbers:</p>
<blockquote><p>[(2.38 /0.0986) + (3.4 / 0.0838)] / [(1/0.0986) + (1/0.0838)]</p></blockquote>
<p>which becomes</p>
<blockquote><p>[24.14 + 40.57] / (22.08) = 2.93
</p></blockquote>
<p><div id="attachment_2279" style="width: 310px" class="wp-caption alignright"><a href="http://www.footballperspective.com/wp-content/uploads/2012/08/BradyManning.png"><img decoding="async" aria-describedby="caption-attachment-2279" src="http://www.footballperspective.com/wp-content/uploads/2012/08/BradyManning-300x200.png" alt="This picture will never get old" width="300" height="200" class="size-medium wp-image-2279" srcset="http://www.footballperspective.com/wp-content/uploads/2012/08/BradyManning-300x200.png 300w, http://www.footballperspective.com/wp-content/uploads/2012/08/BradyManning.png 422w" sizes="(max-width: 300px) 100vw, 300px" /></a> <p id="caption-attachment-2279" class="wp-caption-text">This picture will never get old.</p></div>After averaging 3.667 TDs/G over 6 games, we projected Manning to average 2.99 TDs/G the rest of the year. Since he averaged &#8220;only&#8221; 3 touchdowns per game over his next four games, we downgrade him from 2.99 to 2.93.  Of course, we already had a significant regression factored into his future projection &#8212; we dropped him by 0.67 TDs/game from his average, which is the point of using Bayes Theorem.  So while he&#8217;s at &#8220;only&#8221; 3.4 TDs/G on the season after 10 games, since he&#8217;s played at that level for longer, he only loses about half a touchdown per game over his projection the rest of the way.</p>
<p>That gives Manning 17-18 touchdowns, which puts him at a season-ending projection of 51-52 touchdowns.  He&#8217;s still more likely than not to break the record, although obviously this analysis ignores lots of elements like strength of schedule. And with a visit to Kansas City and a game against the Titans (who have allowed a league-low 7 touchdowns through the air), perhaps he&#8217;s actually an underdog to even tie Brady at 50.</p>
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