Price 2.0 - Revision history

http://evolutionandgames.nl/wiki/index.php?action=history&feed=atom&title=Price_2.0 Price 2.0 - Revision history 2026-10-22T11:29:33Z Revision history for this page on the wiki MediaWiki 1.15.3 http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=438&oldid=prev Matthijs.van.veelen: /* What would a statistician do? */ 2010-09-20T11:06:53Z

<p><span class="autocomment">What would a statistician do?</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:06, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 183:</td> <td colspan="2" class="diff-lineno">Line 183:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test the assumption of fair meiosis</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* Test the assumption of fair meiosis</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>For this test we need the meiosis term that is often discarded. Any test would involve this part of the formula, because if it remains far away from 0 even for large populations, that  <del class="diffchange diffchange-inline">wewould </del>refute fair meiosis. So here is yet another reason why it would be just wrong to throw this quantity away, or just assume that it is 0.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>For this test we need the meiosis term that is often discarded. Any test would involve this part of the formula, because if it remains far away from 0 even for large populations, that  <ins class="diffchange diffchange-inline">we would </ins>refute fair meiosis. So here is yet another reason why it would be just wrong to throw this quantity away, or just assume that it is 0.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== So how does the Price equation help here? ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== So how does the Price equation help here? ==</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It doesn't. And it should worry Price Equationists that even with a relatively simple bit of modelling content, the Price equation is not of any use. Maybe things get better when the models are a bit more complicated? Well, no, of course not. As a matter of fact, things get even worse.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>It doesn't. And it should worry Price Equationists that even with a relatively simple bit of modelling content, the Price equation is not of any use. Maybe things get better when the models are a bit more complicated? Well, no, of course not. As a matter of fact, things get even worse.</div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=437&oldid=prev Matthijs.van.veelen: /* What would a proper statistician do? */ 2010-09-20T11:06:41Z

<p><span class="autocomment">What would a proper statistician do?</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:06, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 175:</td> <td colspan="2" class="diff-lineno">Line 175:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>All of this can be done without the Price equation. In fact, it is better done without, because the Price equation just does not fit the purpose.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>All of this can be done without the Price equation. In fact, it is better done without, because the Price equation just does not fit the purpose.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== What would a <del class="diffchange diffchange-inline">proper </del>statistician do? ==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== What would a statistician do? ==</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A statistician would try to find out what the true model is, given data. That implies that she would:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A statistician would try to find out what the true model is, given data. That implies that she would:</div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=436&oldid=prev Matthijs.van.veelen: /* What would a proper probability theorist do? */ 2010-09-20T11:06:30Z

<p><span class="autocomment">What would a proper probability theorist do?</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:06, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 163:</td> <td colspan="2" class="diff-lineno">Line 163:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If you tried a few times, you may have found out that <math>\triangle Q</math> can attain different values; the next generation is a random draw, which implies that the frequency <math>Q</math> can go up or down by a variety of amounts. The right hand side just tags along with <math>\triangle Q</math>. This can create an illusion of understanding, but whatever happens, the terms on the right hand side of the identity just change along with the outcome of the chance experiment. In other words: the data explain the data. Moreover, crossing of a term that typically is not 0 turns the identity into a false statement.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>If you tried a few times, you may have found out that <math>\triangle Q</math> can attain different values; the next generation is a random draw, which implies that the frequency <math>Q</math> can go up or down by a variety of amounts. The right hand side just tags along with <math>\triangle Q</math>. This can create an illusion of understanding, but whatever happens, the terms on the right hand side of the identity just change along with the outcome of the chance experiment. In other words: the data explain the data. Moreover, crossing of a term that typically is not 0 turns the identity into a false statement.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>== What would a <del class="diffchange diffchange-inline">proper </del>probability theorist do? ==</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>== What would a probability theorist do? ==</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Try to figure out the dynamic properties of the suggested model. With the transition probabilities (see above) we can compute the expected change in frequency, and the fixation probability of ''A'' and ''a'' given the current state. Or given any starting state.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Try to figure out the dynamic properties of the suggested model. With the transition probabilities (see above) we can compute the expected change in frequency, and the fixation probability of ''A'' and ''a'' given the current state. Or given any starting state.  </div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=435&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T11:05:52Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:05, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 61:</td> <td colspan="2" class="diff-lineno">Line 61:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><toggledisplay></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><toggledisplay></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{E}\left[ X\right] =\sum_{x=0,\frac{1}{2},1}x\mathbb{P}\left(X=x\right) = </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{E}\left[ X\right] =\sum_{x=0,\frac{1}{2},1}x\mathbb{P}\left(X=x\right) = </math></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>=\frac{1}{2}\mathbb{P}\left( X=\frac{1}{2}\right) +\mathbb{P}\left(X=1\right) = </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>=\frac{1}{2}\mathbb{P}\left( X=\frac{1}{2}\right) +\mathbb{P}\left(X=1\right) = </math></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>=\frac{1}{2}\cdot \frac{1}{2}+\frac{1}{4}=\frac{1}{2}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>=\frac{1}{2}\cdot \frac{1}{2}+\frac{1}{4}=\frac{1}{2}</math></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></toggledisplay></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></toggledisplay></div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=434&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T11:02:48Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:02, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 40:</td> <td colspan="2" class="diff-lineno">Line 40:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>transition) and then draw the next generation (this is the actual random thing that happens in the transition). The random variable <math>X</math> is again defined as the genotype of the parent. For our given starting point, that is  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>transition) and then draw the next generation (this is the actual random thing that happens in the transition). The random variable <math>X</math> is again defined as the genotype of the parent. For our given starting point, that is  </div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then we define the random variable <math>Y</math> as the number of successful  gametes that this parent produces. This is a slightly more  complicated random variable, because we need to draw 8 parents for 4 offspring and because the chances depend on which of the four the (candidate) parent is that was drawn in step 1. So we get a pretty long list of conditional probabilities.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then we define the random variable <math>Y</math> as the number of successful  gametes that this parent produces. This is a slightly more  complicated random variable, because we need to draw 8 parents for 4 offspring and because the chances depend on which of the four the (candidate) parent is that was drawn in step 1. So we get a pretty long list of conditional probabilities.</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( Y=k|X=0\right) =\binom{8}{k}\left( p_{0}\right)^{k}\left( 1-p_{0}\right) ^{8-k},\ k=1,..,8 </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( Y=k|X=0\right) =\binom{8}{k}\left( p_{0}\right)^{k}\left( 1-p_{0}\right) ^{8-k},\ k=1,..,8 </math></div></td></tr> <tr><td colspan="2" class="diff-lineno">Line 53:</td> <td colspan="2" class="diff-lineno">Line 56:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is well defined again, so we can also compute the covariance of these two actual random variables. Let us start with expectations...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>This is well defined again, so we can also compute the covariance of these two actual random variables. Let us start with expectations...</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{E}\left[ X\right] =\frac{1}{2}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{E}\left[ X\right] =\frac{1}{2}</math></div></td></tr> <tr><td colspan="2" class="diff-lineno">Line 63:</td> <td colspan="2" class="diff-lineno">Line 67:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For symmetry reasons (the total number of succesful gametes is 8, and either of the 4 could be the parent)...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For symmetry reasons (the total number of succesful gametes is 8, and either of the 4 could be the parent)...</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=433&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T11:02:05Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:02, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 43:</td> <td colspan="2" class="diff-lineno">Line 43:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Then we define the random variable <math>Y</math> as the number of successful  gametes that this parent produces. This is a slightly more  <del class="diffchange diffchange-inline">omplicated </del>random variable, because we need to draw 8 parents for 4 offspring and because the chances depend on which of the four the (candidate) parent is that was drawn in step 1. So we get a pretty long list of conditional probabilities.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Then we define the random variable <math>Y</math> as the number of successful  gametes that this parent produces. This is a slightly more  <ins class="diffchange diffchange-inline">complicated </ins>random variable, because we need to draw 8 parents for 4 offspring and because the chances depend on which of the four the (candidate) parent is that was drawn in step 1. So we get a pretty long list of conditional probabilities.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( Y=k|X=0\right) =\binom{8}{k}\left( p_{0}\right)^{k}\left( 1-p_{0}\right) ^{8-k},\ k=1,..,8 </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( Y=k|X=0\right) =\binom{8}{k}\left( p_{0}\right)^{k}\left( 1-p_{0}\right) ^{8-k},\ k=1,..,8 </math></div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=432&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T11:01:46Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:01, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 39:</td> <td colspan="2" class="diff-lineno">Line 39:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>transition) and then draw the next generation (this is the actual random thing that happens in <del class="diffchange diffchange-inline">the </del>the transition). The random variable <math>X</math> is again defined as the genotype of the parent. For our given starting point, that is  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>transition) and then draw the next generation (this is the actual random thing that happens in the transition). The random variable <math>X</math> is again defined as the genotype of the parent. For our given starting point, that is  </div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\mathbb{P}\left( X=1\right) =\mathbb{P}\left( X=0\right) =\frac{1}{4}</math> and <math>\mathbb{P}\left( X=\frac{1}{2}\right) =\frac{1}{2}</math>.  </div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=431&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T11:00:19Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 11:00, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 36:</td> <td colspan="2" class="diff-lineno">Line 36:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>These formulas <del class="diffchange diffchange-inline">tells </del>us that Assumption 1) is equivalent to saying that <math>p_{\frac{1}{2}}</math> is the average of <math>p_{1}</math> and <math>p_{0}</math>. The second assumption just means that the probability of either locus of the parent to be chosen for passing on is 1/2.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>These formulas <ins class="diffchange diffchange-inline">tell </ins>us that Assumption 1) is equivalent to saying that <math>p_{\frac{1}{2}}</math> is the average of <math>p_{1}</math> and <math>p_{0}</math>. The second assumption just means that the probability of either locus of the parent to be chosen for passing on is 1/2.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Again, we first draw one of the individuals from the parent population (this is a hypothetical random variable that has nothing to do with the actual</div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=430&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T10:59:55Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 10:59, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 17:</td> <td colspan="2" class="diff-lineno">Line 17:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Both assumptions concern probabilities in this update step. The first concerns the probabilities with which an individual is chosen for parenthood. We assume that the probability with which individual <math>i</math> is chosen is</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Both assumptions concern probabilities in this update step. The first concerns the probabilities with which an individual is chosen for parenthood. We assume that the probability with which individual <math>i</math> is chosen is</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mathbb{P}\left( i\text{ is chosen}\right) =\frac{\alpha q_{i}+\beta }{\sum_{j=1}^{n}\left( \alpha q_{j}+\beta \right) }  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mathbb{P}\left( i\text{ is chosen}\right) =\frac{\alpha q_{i}+\beta }{\sum_{j=1}^{n}\left( \alpha q_{j}+\beta \right) }  </div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div></math></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With our starting point, <math>q_{1}=1</math>, <math>q_{2}=q_{3}=1/2</math> and <math>q_{4}=0</math>. which  gives us the following probabilities...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>With our starting point, <math>q_{1}=1</math>, <math>q_{2}=q_{3}=1/2</math> and <math>q_{4}=0</math>. which  gives us the following probabilities...</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>p_{1} =\mathbb{P}\left( 1\text{ is chosen}\right) =\frac{\alpha +\beta }{2\alpha +4\beta } </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>p_{1} =\mathbb{P}\left( 1\text{ is chosen}\right) =\frac{\alpha +\beta }{2\alpha +4\beta } </math></div></td></tr>  </table>

Matthijs.van.veelen http://evolutionandgames.nl/wiki/index.php?title=Price_2.0&diff=429&oldid=prev Matthijs.van.veelen: /* A model with sexual reproduction */ 2010-09-20T10:59:29Z

<p><span class="autocomment">A model with sexual reproduction</span></p> <table style="background-color: white; color:black;"> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr valign='top'> <td colspan='2' style="background-color: white; color:black;">← Older revision</td> <td colspan='2' style="background-color: white; color:black;">Revision as of 10:59, 20 September 2010</td> </tr> <tr><td colspan="2" class="diff-lineno">Line 12:</td> <td colspan="2" class="diff-lineno">Line 12:</td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We will make two crucial assumptions here.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>We will make two crucial assumptions here.</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>1) ''A'' is dominant nor recessive in fitness terms</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>1) ''A'' is <ins class="diffchange diffchange-inline">neither </ins>dominant nor recessive in fitness terms</div></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr> <tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2) Fair meosis</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>2) Fair meosis</div></td></tr>  </table>

Matthijs.van.veelen