Price 1.0 - Revision history

Matthijs.van.veelen: /* What would a probability theorist do? */

Matthijs.van.veelen — Mon, 27 Sep 2010 13:19:30 GMT

What would a probability theorist do?

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	==What would a probability theorist do?==		==What would a probability theorist do?==

-	A probability theorist would try to figure out the transition probabilities for the suggested model. That is exactly what we did in the [[A very simple model \| description of the model]], where we determined the transition probabilities and <math>Cov(X,Y)</math>.	+	A probability theorist would try to figure out the transition probabilities for the suggested model. That is exactly what we did in the [[#A very simple model \| description of the model]], where we determined the transition probabilities and <math>Cov(X,Y)</math>.

	== What would a statistician do? ==		== What would a statistician do? ==

Matthijs.van.veelen: /* Estimation of p */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:54:54 GMT

Estimation of p

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	<math>\mathbb{E}\left[ ^{\prime \prime }Cov\left( z,q\right) ^{\prime \prime }\right] =p-\frac{1}{2}</math>.		<math>\mathbb{E}\left[ ^{\prime \prime }Cov\left( z,q\right) ^{\prime \prime }\right] =p-\frac{1}{2}</math>.

-	The quantity <math>^{\prime\prime }Cov\left( z,q\right) ^{\prime \prime }</math> is also regularly referred to as the ''sample'' covariance~~, which~~ is not the true covariance (the covariance is a fixed property of two random variables, the sample covariance is a random variable itself).	+	The quantity <math>^{\prime\prime }Cov\left( z,q\right) ^{\prime \prime }</math> is also regularly referred to as the ''sample'' covariance. It is very important to realise that this is not the true covariance; the covariance is a fixed property of two random variables, while the sample covariance is a random variable itself.

	===Testing properties of p ===		===Testing properties of p ===

Matthijs.van.veelen: /* What would a statistician do? */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:52:09 GMT

What would a statistician do?

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	== What would a statistician do? ==		== What would a statistician do? ==

-	If you are interested in statistics, which is only natural if you care for evolution, and you do not know the true model that generates the transitions, then you would want to try to answer the question what ''p'' is. Is ''p>1/2''? Then having the gene is good for reproduction. Is ''p<1/2''? Then not.	+	If you are interested in statistics, which is only natural if you care for evolution, and you do not know the true model that generates the transitions, then you would want to try to answer the question what ''p'' is. Is ''p>1/2''? Then having the gene is good for reproduction. Is ''p<1/2''? Then not. Is ''p=1/2''? Then it is good nor bad. So how do we find out? Well, the thing with statistics is that you will never find out for sure. All we can do is try to find ways to estimate this ''p'', or design tests on if and how ''p'' differs from ''1/2'', that will help us make statements about the true model with some confidence. Here are a few such statements.
-		+
-	Is ''p=1/2''? Then it is good nor bad. So how do we find out? Well, the thing with statistics is that you will never find out for sure. All we can do is try to find ways to estimate this ''p'', or design tests on if and how ''p'' differs from ''1/2'', that will help us make statements about the true model with some confidence. Here are a few such statements.	+

	===Estimation of p===		===Estimation of p===

Matthijs.van.veelen: /* What would a probability theorist do? */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:51:03 GMT

What would a probability theorist do?

← Older revision		Revision as of 08:51, 27 September 2010
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	==What would a probability theorist do?==		==What would a probability theorist do?==

-	A probability theorist would try to figure out the transition probabilities for the suggested model. That is exactly what we did in the description of the model, where we determined the ~~trasition~~ probabilities and.	+	A probability theorist would try to figure out the transition probabilities for the suggested model. That is exactly what we did in the [[A very simple model \| description of the model]], where we determined the transition probabilities and <math>Cov(X,Y)</math>.

	== What would a statistician do? ==		== What would a statistician do? ==

Matthijs.van.veelen: /* What would a probability theorist do? */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:46:30 GMT

What would a probability theorist do?

← Older revision		Revision as of 08:46, 27 September 2010
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	==What would a probability theorist do?==		==What would a probability theorist do?==

-	A probability theorist would try to figure out the transition probabilities for the suggested model. ~~It seems that you are a proper probability theorist, because that~~ is what ~~you just~~ did. ~~Congrats!~~	+	A probability theorist would try to figure out the transition probabilities for the suggested model. That is exactly what we did in the description of the model, where we determined the trasition probabilities and.

	== What would a statistician do? ==		== What would a statistician do? ==

Matthijs.van.veelen: /* What would a probability theorist do? */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:24:22 GMT

What would a probability theorist do?

← Older revision		Revision as of 08:24, 27 September 2010
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	==What would a probability theorist do?==		==What would a probability theorist do?==

-	~~Try~~ to figure out the transition probabilities for the suggested model. It seems that you are a proper probability theorist, because that is what you just did. Congrats!	+	A probability theorist would try to figure out the transition probabilities for the suggested model. It seems that you are a proper probability theorist, because that is what you just did. Congrats!

	== What would a statistician do? ==		== What would a statistician do? ==

Matthijs.van.veelen: /* A very simple model */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:20:09 GMT

A very simple model

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	</toggledisplay>		</toggledisplay>

-	For obvious symmetry reasons; the total number of offspring is 2, and either one could be the parent, so ...	+	For obvious symmetry reasons, the expectation of <math>Y</math> is 1; the total number of offspring is 2, and either one could be the parent, so ...

	<math>		<math>

Matthijs.van.veelen: /* A very simple model */

Matthijs.van.veelen — Mon, 27 Sep 2010 08:17:02 GMT

A very simple model

← Older revision		Revision as of 08:17, 27 September 2010
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-	What the new generation will be is a random variable. What the current generation is, ~~'''~~is not~~'''~~; the current generation is simply given. We can however think of a ''hypothetical'' chance experiment such that, in combination with the actual random transition, we get two random variables, between which we can define a covariance. So here is the recipe.	+	What the new generation will be is a random variable. What the current generation is, is not; the current generation is simply given. We can however think of a ''hypothetical'' chance experiment such that, in combination with the actual random transition, we get two random variables, between which we can define a covariance. So here is the recipe.

	1) Draw one of the individuals from the parent population (this is the hypothetical random variable).		1) Draw one of the individuals from the parent population (this is the hypothetical random variable).

Matthijs.van.veelen: /* So how does the Price equation help here? */

Matthijs.van.veelen — Mon, 20 Sep 2010 10:56:19 GMT

So how does the Price equation help here?

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	Not. For determining the properties of the model, the Price equation did not help, and in fact cannot even be of use. We found the transition		Not. For determining the properties of the model, the Price equation did not help, and in fact cannot even be of use. We found the transition
	probabilities just by doing probability calculations. For doing statistics the Price equation did not help either. And yes, the Price equation applied to one actual draw does have the sample covariance of that draw in it, and the sample covariance is a useful thing, but we do not need the Price		probabilities just by doing probability calculations. For doing statistics the Price equation did not help either. And yes, the Price equation applied to one actual draw does have the sample covariance of that draw in it, and the sample covariance is a useful thing, but we do not need the Price
-	equation for knowing that the sample covariance is useful. The sample covariance has been around ~~forever;~~ it was there well before George Price	+	equation for knowing that the sample covariance is useful. The sample covariance has been around since the 1880's, when Karl Pearson (1857 – 1936) introduced it, along with the distinction between covariance and sample covariance. It was there well before George Price
-	was born, and it is used in parameter ~~estmation~~ and hypothesis testing in all other fields of science, where no one has ever heard of G.R. Price. And, most importantly, using the sample covariance is obviously not the same as using the Price equation. It is using the sample covariance.	+	was born, and it is used in parameter estimation and hypothesis testing in all other fields of science, where no one has ever heard of G.R. Price. And, most importantly, using the sample covariance is obviously not the same as using the Price equation. It is using the sample covariance.

Matthijs.van.veelen: /* Estimation of p */

Matthijs.van.veelen — Mon, 20 Sep 2010 10:52:28 GMT

Estimation of p

← Older revision		Revision as of 10:52, 20 September 2010
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	<math>\mathbb{E}\left[ ^{\prime \prime }Cov\left( z,q\right) ^{\prime \prime }\right] =p-\frac{1}{2}</math>.		<math>\mathbb{E}\left[ ^{\prime \prime }Cov\left( z,q\right) ^{\prime \prime }\right] =p-\frac{1}{2}</math>.

-	The quantity <math>^{\prime\prime }Cov\left( z,q\right) ^{\prime \prime }</math> is also regularly referred to as the ''sample'' covariance, which is not the true covariance (~~true~~ covariance = fixed property of two random variables, sample covariance ~~= itself~~ a random variable).	+	The quantity <math>^{\prime\prime }Cov\left( z,q\right) ^{\prime \prime }</math> is also regularly referred to as the ''sample'' covariance, which is not the true covariance (the covariance is a fixed property of two random variables, the sample covariance is a random variable itself).

	===Testing properties of p ===		===Testing properties of p ===