Conservapedia - User contributions [en]

Talk:Integral

2009-03-02T21:39:30Z

Qwestor:

This page is in need of serious revision. Someone who isn't already familiar with integral calculus would find this section extremely opaque. Someone with a solid understanding of calculus should come in and explain, in clearer language, the geometric meaning behind the integral and provide a picture of a Riemann sum. I can't do everything here.--[[User:Mathoreilly|Mathoreilly]] 13:40, 2 July 2008 (EDT)

This page should be merged it seems with the Reimann Integral page. --[[User:Qwestor|Qwestor]] 16:40, 2 March 2009 (EST)

Bayesian Probability

2009-02-28T19:40:56Z

Qwestor:

'''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of [[Jacob Bernoulli|Bernoulli]], [[Rev Thomas Bayes|Bayes]], [[Pierre Simon de Laplace|Laplace]] (and more recently of [[Sir Harold Jeffreys|Jeffreys]], [[Richard T. Cox|Cox]], and [[Edwin T. Jaynes|Jaynes]]) in which probability is interpreted as the degree that a proposition/hypothesis/model is true ranging from complete certainty to complete certainty of its falsehood and all intermediate values.

[[Category:Probability]]

Bayesian Probability

2009-02-28T19:39:21Z

Qwestor:

'''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of [[Jacob Bernoulli|Bernoulli]], [[Rev Thomas Bayes|Bayes]], [[Pierre Simon de Laplace|Laplace]] (and more recently of [[Sir Harold Jeffreys|Jeffreys]],[[Richard T. Cox|Cox]], and [[Edwin T. Jaynes|Jaynes]]) in which probability is interpreted as the degree that a proposition/hypothesis/model is true ranging from complete certainty to complete certainty of its falsehood and all intermediate values.

[[Category:Probability]]

Statistic

2009-02-28T18:04:42Z

Qwestor:

A '''statistic''' is a function of a distributed variable. Notionally, it is a calculation made on the basis of a set numbers typically derived as a sample from some presumed underlying [[probability distribution]], and usually used in order to estimate something about the distribution from which the sample is taken. The use of a '''statistic''' to characterize a set of observations is generally justified on the basis of its [[asymptote|asymptotic]] behavior, that is, a given '''statistic''' accurately characterizes the underlying phenomena only probabilistically (this consideration is the genesis of [[confidence interval]]s in [[classical statistics]]) and is considered to be accurate only in the limit as the number of observations increases without bounds. It should be noted however that the use of [[confidence interval]]s is somewhat problematic since their calculations are based on certain presumptions about the nature of the underlying true distribution, which may or may not prove to be good.

For example, suppose a [[random sample]] of three children is chosen from a particular class, and their heights measured as 1.42cm., 1.54cm., and 1.48cm; then the [[arithmetic mean]] of these heights is 1.48cm. We might then go on to use this value of 1.48cm to represent the [[average]] height of a child in that class.

Clearly the [[validity]] and [[reliability]] of such [[estimation]]s will depend enormously on a range of factors such as the type of distributions, the number in the sample, and on sampling methods used.

===Formal Definition:===

Let X1, X2, X3, ...., Xn be a random sample of size n from some distribution. A statistic calculated on the sample is defined to be any [[function]] of the set of values X1, X2, X3, ...., Xn, involving no unknown quantities <ref> Francis, A. (2005) Advanced Level Statistics, Stanley Thornes </ref>

The point of this definition is to ensure that the process results in an actual numerical value, rather than a formula involving variables.

===Examples of Statistics:===

* [[Arithmetic mean]]
* [[Median]]
* [[Standard deviation]]
* [[Pearson's measure of skewness]] '' = 3*(mean - median)/standard deviation

===References===

<references/>

[[Category:Science]]
[[Category:Statistics]]

Statistic

2009-02-28T18:02:58Z

Qwestor:

A '''statistic''' is a function of a distributed variable. Notionally, it is a calculation made on the basis of a set numbers typically derived as a sample from some presumed underlying [[probability distribution]], and usually used in order to estimate something about the distribution from which the sample is taken. The use of a '''statistic''' to characterize a set of observations is generally justified on the basis of its [[asymptote|asymptotic]] behavior, that is, a given '''statistic''' accurately characterizes the underlying phenomena only probabilistically (this consideration is the genesis of [[confidence interval]]s in [[classical statistics]]) and is considered to be accurate only in the limit as the number of observations increases without bounds. It should be noted however that the use of [[confidence interval]]s is somewhat problematic since their calculations are based on certain presumptions about the nature of the underlying true distribution, which may or may not prove to be good. [[Bayesian Probability]] does not use '''statistics'''.

For example, suppose a [[random sample]] of three children is chosen from a particular class, and their heights measured as 1.42cm., 1.54cm., and 1.48cm; then the [[arithmetic mean]] of these heights is 1.48cm. We might then go on to use this value of 1.48cm to represent the [[average]] height of a child in that class.

Clearly the [[validity]] and [[reliability]] of such [[estimation]]s will depend enormously on a range of factors such as the type of distributions, the number in the sample, and on sampling methods used.

===Formal Definition:===

Let X1, X2, X3, ...., Xn be a random sample of size n from some distribution. A statistic calculated on the sample is defined to be any [[function]] of the set of values X1, X2, X3, ...., Xn, involving no unknown quantities <ref> Francis, A. (2005) Advanced Level Statistics, Stanley Thornes </ref>

The point of this definition is to ensure that the process results in an actual numerical value, rather than a formula involving variables.

===Examples of Statistics:===

* [[Arithmetic mean]]
* [[Median]]
* [[Standard deviation]]
* [[Pearson's measure of skewness]] '' = 3*(mean - median)/standard deviation

===References===

<references/>

[[Category:Science]]
[[Category:Statistics]]

Parameter estimation

2009-02-27T23:28:33Z

Qwestor:

In probability, once a model class has been selected it is then necessary to estimate the values of its associated ''parameters'' in order to specify that particular model out of all members of the model class which differ only on the basis of the values of these parameters.

In [[Bayesian Probability]], the parameters are estimated from data by computing the joint [[posterior distribution]] of the parameters and the data using [[Bayes Rule]].

[[category:mathematics]]

Random sampling

2009-02-27T23:16:13Z

Qwestor:

'''Random sampling''' is a statistical technique which gives everyone in the target group the same opportunity to participate in a [[poll]].

An example is all eligible voters being equally likely to be included in a poll. This method does not ensure accuracy, but increases its likelihood.

In statistical studies, '''random sampling''' refers to the extraction of values (''sampling'') of a designated input [[probability distribution]] for ensuing statistical calculations, such that the relative [[probability]] of extraction is that indicated by the input [[probability density function]] in the case of a continuous distribution or the [[probability mass function]] in the case of a discrete distribution.

[[Category:United States Government]]
[[Category:Probability]]

Expectation (math)

2009-02-27T23:10:23Z

Qwestor:

The mathematical '''expectation''' of a continuously distributed random variable <math>X</math> with [[probability density function]]
<math>f(x)</math> is
:<math>
\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.
</math>
The expectation is also known as the [[mean]] of <math>X</math>.

The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is
:<math>
\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.
</math>

For a discretely distributed random variable <math>X</math> with [[probability mass function]]
<math>p_{k}</math> it is
:<math>
\mbox{E}[X]=\sum_{k} p_{k}x_{k}.
</math>

[[category:Probability and Statistics]]

Standard deviation

2009-02-27T23:09:20Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance:Probability and Statistics|variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>

(where <math>\bar X = {\sum_n X_n \over N}</math> is the [[sample mean]]).

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this [[statistical estimator]] itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Bayesian Probability

2009-02-22T00:27:49Z

Qwestor:

'''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of [[Jacob Bernoulli|Bernoulli]], [[Rev Thomas Bayes|Bayes]], [[Pierre Simon de Laplace|Laplace]] (and more recently of [[Sir Harold Jeffreys|Jeffreys]]) in which probability is interpreted as the degree that a proposition/hypothesis/model is true ranging from complete certainty to complete certainty of its falsehood and all intermediate values.

[[Category:Probability]]

Talk:Exact sciences

2009-02-21T16:46:33Z

Qwestor:

==Witnessing Evolution==

Wasn't the whole point of the Lenski Experiment that it provided conclusive evidence of bacteria evolving a useful trait over thousands of generations, and the specimens that inherited the trait thrived compared to those that did not? That would be a contemporary witnessing of evolution, I'd think. --[[User:DinsdaleP|DinsdaleP]] 10:16, 21 February 2009 (EST)

:By evolution, I mean the [[theory of evolution through natural selection]], which posits that significant new species have come into being without intelligent intervention.

:Whenever we discuss evolution, we need to cut through the mental fog and specify which [[aspects of evolution]] we're talking about. Most evolution opponents grant that a species can develop new characteristics in the wild (see [[microevolution]]). --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 10:26, 21 February 2009 (EST)

:Perhaps it was just a contemporary witnessing of [[adaptation]] - an ability it's designer gave it to deal with differing environmental contexts. --[[User:Qwestor|Qwestor]] [[User talk:Qwestor|Talk]] 11:39, 21 February 2009 (EST)

Talk:Exact sciences

2009-02-21T16:39:50Z

Qwestor:

==Witnessing Evolution==

Wasn't the whole point of the Lenski Experiment that it provided conclusive evidence of bacteria evolving a useful trait over thousands of generations, and the specimens that inherited the trait thrived compared to those that did not? That would be a contemporary witnessing of evolution, I'd think. --[[User:DinsdaleP|DinsdaleP]] 10:16, 21 February 2009 (EST)

:By evolution, I mean the [[theory of evolution through natural selection]], which posits that significant new species have come into being without intelligent intervention.

:Whenever we discuss evolution, we need to cut through the mental fog and specify which [[aspects of evolution]] we're talking about. Most evolution opponents grant that a species can develop new characteristics in the wild (see [[microevolution]]). --[[User:Ed Poor|Ed Poor]] [[User talk:Ed Poor|Talk]] 10:26, 21 February 2009 (EST)

:Perhaps it was just a contemporary witnessing of adaptation - an ability it's designer gave it to deal with differing environmental contexts. --[[User:Qwestor|Qwestor]] [[User talk:Qwestor|Talk]] 11:39, 21 February 2009 (EST)

Standard deviation

2009-01-19T15:34:25Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance:Probability and Statistics|variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>

(where <math>\bar X = {\sum_n X_n \over N}</math> is the [[sample mean]]).

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Variance:Probability and Statistics

2009-01-19T15:32:03Z

Qwestor:

{{math-h}}

'''Variance''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

The formula for variance must not be confused with the formula

:<math>S_{n}^2 = {\sum_n(X_n - \bar X)^2 \over n - 1}</math>

(where <math>\bar X = {\sum_n X_n \over N}</math> is the [[sample mean]]).

which is the formula for a [[point estimate]] of the true variance from a sample of size ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Variance:Probability and Statistics

2009-01-19T15:06:59Z

Qwestor: New page: {{math-h}} '''Variance''' is a measure in statistics of the dispersion of a set of values (represented as <math>X</math>). It is defined as :<math>\sigma^2 = \operat...

{{math-h}}

'''Variance''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

The formula for variance must not be confused with the formula

:<math>S_{n}^2 = {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true variance from a sample of size ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Standard deviation

2009-01-19T15:06:42Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance:Probability and Statistics|variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Talk:Expectation (math)

2009-01-19T15:03:58Z

Qwestor:

== Rename ==

I think that this article should be moved to Expected Value and create a redirect from expectation (math) and expectation value to this page. Does this seem reasonable? In my experience expected value is the far more common term. [[User:AndyJM|AndyJM]] 09:59, 19 January 2009 (EST)

Both terms occur frequently in all kinds of literature. "Mathematical Expectation" is perhaps the more official math term, but "Expected Value", "Expectation Value" all occur frequently. Not really sure what the best way to handle this is. Perhaps wiki supports something like aliases? [[User:Qwestor|Qwestor]] 10:02, 19 January 2009 (EST)

Talk:Expectation (math)

2009-01-19T15:02:05Z

Qwestor:

Probability and Statistics

2009-01-19T14:59:29Z

Qwestor: New page: {{math-h}} '''Variance''' is a measure in statistics of the dispersion of a set of values (represented as <math>X</math>). It is defined as :<math>\sigma^2 = \operat...

{{math-h}}

'''Variance''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

The formula for variance must not be confused with the formula

:<math>S_{n}^2 = {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true variance from a sample of size ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Standard deviation

2009-01-19T14:56:01Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[Probability and Statistics|variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Standard deviation

2009-01-19T14:54:00Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation (math)|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Expectation (math)

2009-01-19T14:44:08Z

Qwestor:

The mathematical '''expectation''' of a continuously distributed variable <math>X</math> with [[probability density function]]
<math>f(x)</math> is
:<math>
\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.
</math>

For a discretely distributed variable <math>X</math> with [[probability mass function]]
<math>p_{k}</math> it is
:<math>
\mbox{E}[X] = \sum_{k} p_{k}x_{k}
</math>

The expectation is also the [[mean]] of a distributed variable <math>X</math>.

The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is
:<math>
\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.
</math>

[[category:Probability and Statistics]]

Expectation (math)

2009-01-19T14:43:06Z

Qwestor:

The '''expectation''' of continuously distributed variable <math>X</math> with [[probability density function]]
<math>f(x)</math> is
:<math>
\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.
</math>

For a discretely distributed variable <math>X</math> with [[probability mass function]]
<math>p_{k}</math> it is
:<math>
\mbox{E}[X] = \sum_{k} p_{k}x_{k}
</math>

The expectation is also the [[mean]] of a distributed variable <math>X</math>.

The expectation with respect to some function <math>g(X)</math> where <math>X</math> is distributed according to <math>f(x)</math> is
:<math>
\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.
</math>

[[category:Probability and Statistics]]

Expectation (math)

2009-01-19T14:42:45Z

Qwestor:

Standard deviation

2009-01-19T14:41:27Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:Probability and Statistics]]
[[Category:Mathematics]]

Expectation (math)

2009-01-19T14:40:37Z

Qwestor:

Standard deviation

2009-01-19T13:45:16Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:statistics]]

Expectation (math)

2009-01-19T13:40:47Z

Qwestor:

Standard deviation

2009-01-19T13:36:53Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum(x - \bar x)^2 \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:statistics]]

Expectation (math)

2009-01-19T00:24:14Z

Qwestor:

The '''expectation''' of continuously distributed variable <math>X</math> with [[probability density function]]
<math>f(x)</math> is
:<math>
\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.
</math>

For a discretely distributed variable <math>X</math> with [[probability mass function]]
<math>p_{k}</math> it is
:<math>
\mbox{E}[X] = \sum_{k} p_{k}x_{k}
</math>

The expectation is also the [[mean]] of a distributed variable <math>X</math>.

The expectation with respect to some function <math>g(X)</math> is
:<math>
\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.
</math>

[[category:Probability and Statistics]]

Standard deviation

2009-01-19T00:21:11Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum(x - \bar x) \over n - 1}</math>

which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:statistics]]

Standard deviation

2009-01-19T00:18:17Z

Qwestor:

{{math-h}}

'''Standard deviation''' is a measure in [[Statistics|statistics]] of the [[dispersion]] of a set of values (represented as <math>X</math>). It is defined as the square root of the [[variance]] of these values, where variance is defined as

:<math>\sigma^2 = \operatorname{E}[(X-\operatorname{E}[X])^2] = \operatorname{E}[X^2] - (\operatorname{E}[X])^2</math>

where the [[expectation|expected value]] of ''X'' is E(''X'').

Thus the standard deviation is

:<math>\sigma = \sqrt{\operatorname{E}[(X-\operatorname{E}[X])^2]} = \sqrt{\operatorname{E}[X^2] - (\operatorname{E}[X])^2}</math>

The formula for standard deviation must not be confused with the formula

:<math>S_{n} = \sqrt {\sum(x - \bar x) \over n - 1}</math>

which is the formula for an [[estimator]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.

[[category:statistics]]

Expectation (math)

2009-01-18T23:53:07Z

Qwestor:

The '''expectation''' of continuously distributed variable <math>X</math> with [[probability density function]]
<math>f(x)</math> is
:<math>
\mbox{E}[X]
=\int\limits_{-\infty}^\infty
x
f(x)dx.
</math>

For a discretely distributed variable <math>X</math> with [[probability mass function]]
<math>p_{k}</math> it is
:<math>
\mbox{E}[X] = \sum_{k} p_{k}x_{k}
</math>

The expectation is also the [[mean]] of random variable <math>X</math>.

The expectation with respect to some function <math>g(X)</math> is
:<math>
\mbox{E}[g(X)]
=\int\limits_{-\infty}^\infty
g(x)
f(x)dx.
</math>

[[category:Probability and Statistics]]

Dot product

2009-01-17T15:16:05Z

Qwestor:

The '''dot product''' is defined for two [[Vector|vectors]] X and Y to be:

: <math>X \cdot Y = |X||Y|\cos\theta</math>

where |x| is the norm and theta is the angle between the vectors.

Unlike the [[cross product]], the dot product is a [[scalar]], not a vector, and has no direction.

It follows from the above definition that the dot product of X and Y is 0 if X is perpendicular to Y.

Alternatively, one can describe the dot product as the length of the geometric projection of X onto Y times the length of Y, when the tails of the two vectors are placed at the same point. It must be remembered though, that the dot product is positive if the angle between the two vectors is less than 90 degrees, negative if the angle is between 90 and 180 degrees (it is in this sense that the algebraic sign of the dot product does give some limited directional information).

===See also===
[[Cross product]]

[[category:Linear algebra]]
[[category:physics]]

Bayesian model selection

2009-01-17T15:11:48Z

Qwestor:

In [[Bayesian Probability]], '''Bayesian model selection''' is a method for choosing the best [[hypothesis]] or ''model class'' or [[mathematical model]] posed as a [[probabilistic likelihood model]] out of a set of competing model classes (loosely ''models'') which best explains some observed data. Best here is measured by the [[Bayesian posterior]] [[Bayes odds|odds]] ratio of the winner compared against all other candidates in the competition.

The posterior [[Bayes odds|odds]] ratio is the product of the [[Bayes Factor]] and the [[Bayes prior]] odds ratio. In short one is comparing the [[marginal likelihood]] of entire families of models (i.e., parameterized model classes) by [[marginal distribution|marginalizing]] over their associated parameter values. After the model class has been so selected, one can then go on to do [[parameter estimation]] to determine the best inference as to the values of the parameters of that particular model class.

[[category:Probability]]

Bijection

2009-01-17T15:00:40Z

Qwestor:

A '''bijection''' is a [[injectivity|one-to-one]], [[surjectivity|onto]] [[mapping]] between two [[set]]s. In other words, a bijection between sets A and B is a mapping such that every element in set A is mapped to a distinct element in set B, and every element in set B has a distinct element in the set A mapped to it. For example, one bijection between the sets {A, B, C} and {1, 2, 3}, maps: A to 1, B to 2, and 3 to C.

If there is a bijection between two sets, then we say that they have the same [[cardinality]], or size.

Bijections between [[infinite]] sets are particularly interesting, as they produce some counterintuitive results. For instance, there is a bijection between the integers and the even integers that maps every integer x to the even integer 2x. Notice that this satisfies the definition of bijection, even though it would then seem to imply that the integers and the even integers have the same size. However, since both sets are of infinite size, it merely illustrates the idea that "infinity times two is infinity".

The existence of a bijection from one set to the other also implies that a converse bijection exists.

[[Category:Mathematics]]
[[Category:Set theory]]

Covariance

2009-01-17T14:46:56Z

Qwestor:

'''Covariance''' is a measure of the linear dependence of two [[variable]]s. If two variables tend to vary in the same direction, then they have a positive covariance. If they tend to vary in opposite directions, then they have a negative covariance.

The covariance between two random variables ''X'' and ''Y'', having [[expected value]]s <math>\mu</math> and <math>\nu</math> respectively, is as follows:

: <math>\operatorname{Cov}(X, Y) = \operatorname{E}[(X - \mu) (Y - \nu)], \,</math>

where E is the operator for the [[expected value]].

If ''X'' and ''Y'' are completely [[statistically independent]] from each other, then they have zero covariance.

Note that if ''X'' and ''Y'' have covariance zero, they are un[[correlated]] but are not necessarily independent.

[[category:probability and Statistics]]

Covariance

2009-01-17T14:45:47Z

Qwestor:

'''Covariance''' is a measure of the linear dependence of two [[random variable]]s. If two variables tend to vary in the same direction, then they have a positive covariance. If they tend to vary in opposite directions, then they have a negative covariance.

The covariance between two random variables ''X'' and ''Y'', having [[expected value]]s <math>\mu</math> and <math>\nu</math> respectively, is as follows:

: <math>\operatorname{Cov}(X, Y) = \operatorname{E}[(X - \mu) (Y - \nu)], \,</math>

where E is the operator for the [[expected value]].

If ''X'' and ''Y'' are completely statistically independent from each other, then they have zero covariance.

Note that if ''X'' and ''Y'' have covariance zero, they are un[[correlated]] but are not necessarily independent.

[[category:probability and Statistics]]

Origins debate

2008-12-31T13:14:34Z

Qwestor:

The '''origins debate''' is the age-old conflict of ideas about how the [[Earth]] was formed and [[life]] came into being.
It also concerns the timing and process whereby the various kinds of [[animals]] and [[plants]] came into being, and
how human beings appeared on the [[Earth]].

==Major views==

Most people throughout [[history]] have ascribed origins to a [[first cause]] - [[God]] or various [[gods]]. Materialist views, i.e. that our creation happened due to natural circumstances, have also been put forth in this debate, some as early as [[ancient Greece]].

[[Judaism]] and [[Christianity]] posit that "In the beginning, God created the heavens and the earth." ([[Genesis]] 1:1) Then He created plants ([[Genesis]] 1:11-13), animals ([[Genesis]] 1:20-23), and finally people (Genesis 1:26-27). Other major religions have similar creation stories.

[[Materialism|Materialistic]] views are many and various.
The prevailing contemporary view in the natural [[science|sciences]] (i.e., [[astronomy]], [[geology]] and [[biology]]) among those who hold to a materialist [[worldview]] is that the [[Big Bang]] theory explains cosmic origins and the [[Theory of Evolution]] (in this case necessarily including [[abiogenesis]]) explains the emergence of plants, animals and people.

[[Teleology|Teleological]] views of origins recognize purpose and therefore design in nature and are not in general in conflict with [[science]]; rather they are in conflict with any [[worldview]] that posits that no references to a creator, purpose, or design is necessary.

There is considerable debate over whether there is any viable middle ground between the materialist views and the [[creationism|creatio hold to a onist]] views. Some would hold, though, that a viable middle position could be found if both sides recognized how miraculous any theory of creation is, whether it is perfect conditions for development or a Prime Mover such as God.

A 2001 Gallup poll showed that 37% of Americans believed that human beings "evolved" in a process guided by God, 45% believed God created human beings in roughly their present form, and 12% believed a process of natural selection over millions of years developed humans completely independent of God. <ref>[http://www.unl.edu/rhames/courses/current/creation/evol-poll.htm] </ref>

==Notes==
<references/>

[[Category:Origins Debate]]

Four Last Things

2008-01-13T23:43:50Z

Qwestor:

The '''Four Last Things''' in [[Roman Catholicism]] are death, judgment, [[Heaven]] and [[Hell]]. This denies there is a reincarnation, as believed in [[Hinduism]]. Rather, in [[Christianity]] everyone is judged immediately after death.

The [[Latin]] term for this is ''novissima'', meaning "the newest things."
[[category:Christianity]]

Four Last Things

2008-01-13T13:15:22Z

Qwestor:

The '''Four Last Things''' in [[Roman Cathlocism]] are death, judgment, [[Heaven]] and [[Hell]]. This denies there is a reincarnation, as believed in [[Hinduism]]. Rather, in [[Christianity]] everyone is judged immediately after death.

The [[Latin]] term for this is ''novissima'', meaning "the newest things."
[[category:Christianity]]

Bayesian Probability

2008-01-07T00:19:04Z

Qwestor:

'''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of [[Jacob Bernoulli|Bernoulli]], [[Rev Thomas Bayes|Bayes]], [[Pierre Simon de Laplace|Laplace]] (and more recently of [[Sir Harold Jeffreys|Jeffreys]]) in which probability is interpreted as the degree that a proposition/hypothesis/model is true ranging from certainty to uncertainty and all intermediate values.

[[Category:Probability]]

Bayesian Probability

2008-01-07T00:09:38Z

Qwestor: New page: '''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of Bernoulli, Bayes, [[Pierre Simon de Lapl...

'''Bayesian probability''' is a particular calculus of inductive plausible reasoning based on the works of [[Jacob Bernoulli|Bernoulli]], [[Rev Thomas Bayes|Bayes]], [[Pierre Simon de Laplace|Laplace]] (and more recently of [[Sir Harold Jeffreys|Jeffreys]]) in which probability is interpreted as the degree that a proposition/hypothesis/model is true ranging from certainty to uncertainty and all intermediate values.

Random sampling

2008-01-06T01:48:18Z

Qwestor:

'''Random sampling''' is a statistical technique which gives everyone in the target group the same opportunity to participate in a [[poll]].

An example is all eligible voters.

In statistical studies, '''random sampling''' refers to the extraction of values (''sampling'') of a designated input probability distribution for ensuing statistical calculations.

[[Category:United States Government]]
[[Category:Probability]]

Random sampling

2008-01-06T01:47:37Z

Qwestor:

Parameter estimation

2008-01-06T01:43:36Z

Qwestor:

In probability, once a distribution has been selected it is then necessary to estimate the values of its associated ''parameters'' in order to specify the particular distribution out of the family or class of distributions which differ only on the basis of the values of these parameters.

In [[Bayesian Probability]], parameters are estimated by computing the joint [[posterior distribution]] of the parameter set using [[Bayes Rule]].

Parameter estimation

2008-01-06T01:37:09Z

Qwestor: New page: In probability, once a distribution has been selected it is then necessary to estimate the values of its associated ''parameters'' in order to specify the particular distribution out of th...

Cumulative distribution function

2008-01-03T19:18:49Z

Qwestor:

In [[probability theory]], a ''cumulative distribution function'' ''F(x)'' of a [[probability density function]] say ''f(x)'' is a real valued and continuous function whose value is the proportion of probability values of a variable which occur on the part of the real line up and including the value of that variable; i.e.,

:<math> F(x) = \int_{-\infty}^x \,f(\lambda)\,d\lambda</math>

Considering this definition in light of the fundamental theorem of the [[Integral Calculus]] yields:

:<math> f(x) = \frac{dF(x)}{dx}</math>

Due to the properties of the [[probability density function]] ''f(x)'', the cumulative distribution function ''F(x)'' will have the following properties:

(1) <math> F(x) \geq 0 </math> <math> \forall x </math> inside its domain of support.

(2) <math> \lim_{x \to \infty}F(x) = 1 </math>, i.e., finitely convergent (to unity by convention).

(3) <math> F(a) \leq F(b) </math> for a<b, i.e., is non-decreasing

If the [[domain]] of the variable is [[finite]], then the upper limit in equation (2) above should be the upper bound of the variables [[domain of support]].

[[Category:mathematics]]

Universal common ancestry

2008-01-03T15:35:14Z

Qwestor:

The idea of '''universal common ancestry''' is the idea that all life on Earth is related via a single [[family tree]]. Accordingly, it supposes that human beings, as well as animals, plants and every other form of life on Earth are related. People and monkeys are thus distant cousins.

While universal common ancestry (UCA) is often said to be a keystone of the naturalistic [[theory of evolution]] (NTOE), this can hardly be true, since that theory does not predict UCA; there is rather a consilience between the two. UCA is also accepted by some [[Intelligent design]] advocates, including [[Michael Behe]].<ref>http://www.evolutionnews.org/2007/05/_most_people_including.html</ref>

[[Creationism]] vigorously rejects this idea, based on their belief that the scientific evidence makes universal common ancestry appear highly improbable or impossible. They believe instead that life originated in a number of unrelated forms, as understood by [[Baraminology]]. They also reject universal common ancestry because it contradicts a straightforward reading of [[Genesis]], which reports that God created each living "according to its kinds," and that God created mankind ''in His Image'', having a superior position and dominion over the rest of God's creation.

== History ==
Universal common ancestry is an ancient idea. Many ancient [[pagan]]s held that humans were related to animals. [[Anaxagoras]], for instance, argued that humans are related to fish, but became humans because the water became too crowded, and the fish kicked humans out of the water. Many ancient American Indian cultures also believed humans were descended from animals, and some commemorated these beliefs through [[totem pole]]s.<ref>http://www.hud.gov/local/shared/working/r10/nwonap/totems.pdf</ref>

More recently, in the 18th and 19th centuries, the idea of universal common ancestry regained popularity in [[Europe]]. In 1745, French mathematician and scientist Pierre-Louis Moreau de Maupertuis (1698-1759) wrote in his book ''Venus Physique'':

:"Could one not say that, in the fortuitous combinations of the productions of nature, as there must be some characterized by a certain relation of fitness which are able to subsist, it is not to be wondered at that this fitness is present in all the species that are currently in existence? Chance, one would say, produced an innumerable multitude of individuals; a small number found themselves constructed in such a manner that the parts of the animal were able to satisfy its needs; in another infinitely greater number, there was neither fitness nor order: all of these latter have perished. Animals lacking a mouth could not live; others lacking reproductive organs could not perpetuate themselves... The species we see today are but the smallest part of what blind destiny has produced..."

[[Charles Darwin]] further popularized the idea with his book, ''[[Origin of Species]]'', in which he wrote:

:"[P]robably all of the organic beings which have ever lived on this earth have descended from some one primordial form, into which life was first breathed."
:"The whole history of the world, as at present known, ... will hereafter be recognised as a mere fragment of time, compared with the ages which have elapsed since the first creature, the progenitor of innumerable extinct and living descendants, was created."
:"When I view all beings not as special creations, but as the lineal descendants of some few beings which lived long before the first bed of the Silurian system was deposited, they seem to me to become ennobled."

== Bases for belief ==
=== Similarities ===
Belief in common descent is largely derived from the similarity among forms of life. For example:

*All [[cell]]s are bound by a [[membrane]]
*All cells store heritable information as [[DNA]].
*All cells adhere to the same basic mechanism of [[gene expression]].
*All cells utilize the same [[metabolic]] pathways.
*All cells have [[ribosome]]s to produce [[protein]].

From these similarities, an evolutionist infers that all life is related through one original life form.

==== Criticism ====
Creationists argue that the inference of common ancestry cannot reasonably be drawn from similarity, for two reasons:
* Similarity is not meaningful evidence of common ancestry
* Differences which cannot be explained by evolutionary mechanisms are powerful evidence ''against'' common ancestry.

To illustrate the first objection, it stands to reason that if a designer were creating several lifeforms, he would probably use similar designs, because the designs are ''functional''. Just as all cars have design similarities, including four tires, an internal combustion engine, a steering wheel, but obviously are not related; diverse species may simply share ''common design characteristics''. Thus creationists argue that just because humans and plants both have cell membranes is no more evidence of ''common ancestry'' than it is of ''common design''.

The second objection relies on incidents of [[irreducible complexity]] and [[specified complexity]] to point to the unlikelihood that all life forms diversified through variation and selection alone. They point to issues such as:
* The transition from non-self-reproducing chemicals to self-replicating cells: the process of self-replication requires an enormous number of components be present in the cell for it to function at all, and none of the components required for self-replication serve any purpose unless all are present simultaneously; therefore it is unreasonable to conclude that [[abiogenesis]] occurred without [[design]].
* The transition from asexual reproduction to sexual reproduction: male reproductive organs are incredibly complex, and serve no purpose unless female reproductive organs (which are equally, if not more, complex) already exist. Therefore the sexes could not have evolved in a step-wise fashion, because neither set of reproductive organs would serve any purpose unless the other set already existed.

For the above reasons, creationists conclude that similarity does not provide meaningful evidence of common ancestry.

=== Shared endogenous retroviruses ===
A [[Retrovirus]] is a virus that incorporates itself into an host cell's DNA, and programs the host cell to produce copies of the virus. [[Endogenous retrovirus]]es (ERVs) are retroviruses that infected an organism's [[germ cell]]s, and were thus passed to the next generation, and incorporated into the organism's genetic line over time.

Researchers estimate that 1% of the human genome is occupied by ERVs, constituting approximately 30,000 different ERVs.<ref>Sverdlov, E. D. (2000) "Retroviruses and primate evolution." BioEssays 22: 161-171. [http://www.ncbi.nlm.nih.gov/sites/entrez?cmd=Retrieve&db=PubMed&list_uids=10655035&dopt=AbstractPlus]</ref> Some of these ERVs are shared by humans and apes, and are found in the same place in the respective genomes. <ref>Lebedev, Y. B., Belonovitch, O. S., Zybrova, N. V, Khil, P. P., Kurdyukov, S. G., Vinogradova, T. V., Hunsmann, G., and Sverdlov, E. D. (2000) "Differences in HERV-K LTR insertions in orthologous loci of humans and great apes." Gene 247: 265-277.[http://www.ncbi.nlm.nih.gov/sites/entrez?cmd=Retrieve&db=PubMed&list_uids=10773466&dopt=AbstractPlus]</ref>

From these facts, evolutionists argue that these ERVs are shared by humans and primates because they were inherited by a common ancestor.

==== Criticism ====
Evolutionists gloss over two important facts when making this argument: First, many retroviruses can infect both humans and apes. The most notable of these is [[HIV]], which is widely believed to have originated as [[SIV]] in chimpanzees, but can also infect humans, apes, and monkeys.<ref>http://news.bbc.co.uk/1/hi/health/5012268.stm</ref> It is entirely possible, therefore, that humans and apes were ''independently infected'' with the same virus. Second, some retroviruses have been shown to have highly targeted insertion points, meaning that the virus selects very specific segments of the genome for insertion.<ref>Citation: (2004) Retroviral Gene Vectors Show Clear Target Preferences. PLoS Biol 2(12): e443 doi:10.1371/journal.pbio.0020443 [http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pbio.0020443&ct=1]</ref> Consequently, it is entirely possible that the same virus infected both humans and apes, and targeted the same location. This seems especially plausible in light of the fact that humans and apes have ''tens of thousands'' of endogenous retroviruses in their respective genomes -- at least a ''few'' of the retroviruses are likely to infect both humans and apes at the same location.

Additionally, scientists have determined that some endogenous retroviruses are indispensable to a species' life or reproduction.<ref>"Researchers Discover that Sheep Need Retroviruses for Reproduction," Texas A&M AgNews, 11 Sep 06.[http://agnews.tamu.edu/dailynews/stories/ANSC/Sep1106b.htm]</ref> If the retrovirus is advantageous in some way, that would explain how the retrovirus spread to the entire species. It also presents the possibility that retroviruses were used as part of the creative process, as [[retrovirus]]es are often used in genetic engineering today, to introduce new genetic material to cells.

=== Common DNA-based genes in diverse species ===
The most persuasive argument for common ancestry is existence of common DNA-based genes in diverse species. But there are at least three other plausible explanations -- [[parallel evolution]], [[horizontal gene transfer]], and [[creationism]].

* '''Parallel evolution''': Some people who believe in evolution deny universal common ancestry. They point out that there is still no good theory for how DNA-based life could have arisen from non-life. If it happened once, it may have happened many times. Some people say that separate evolution might have had an entirely different genetic code, but no one knows. It may well be that DNA-based life evolved from non-life many times, and the descendants of that life are living today.

* '''Horizonal gene transfer''': It is also known that genes sometime migrate somehow from one species to another species, so having a common gene is not proof of common ancestry. Maybe there is no universal common ancestor, and important genes were spread by horizontal gene transfer.

* '''Creationism''': Creationists argue that common genes in diverse species is no more evidence of ''common descent'' as it is of ''common design''.

== References ==
<References/>
[[Category:Evolution]]
[[Category:Creationism]]

Bayes Factor

2008-01-02T17:30:04Z

Qwestor:

The ''Bayes Factor'' of model class M1 to model class M2 for some set of observations ''X'' is the ratio of their associated marginal class likelihoods. The marginal class likelihood of some set of observations is the [[marginal likelihood]] of the observations for the model class obtained by marginalizing the [[joint probability distribution]] of the observations and the model class parameters by treating the model class parameters <math>\theta</math> as [[nuisance parameters]]. I.e.,

::<math>m_{\theta}(X|M) = \int_\theta p(X,\theta|M) \, d\theta = \int_\theta p(X|\theta , M) \, p(\theta|M) \, d\theta </math>

illustrating that it may be calculated directly using the [[joint probability distribution]] or from the [[conditional probability distribution]] where the second form of the integral was obtained using the [[Bayesian Product Rule]]. Thus the ''Bayes factor'' for model class M1 with parameters <math>\theta_1</math> to model class M2 with parameters <math>\theta_2</math> is

::<math>BF(X,M_1,M_2) = \frac{m_{\theta_1}(X|M_1)}{m_{\theta_2}(X|M_2)} </math>

By integrating out the model class parameters for comparative classes of likelihood models, one is effectively comparing the weighted strengths of the predictions of the two model classes concerning the observed data. In particular, in determining which model class has the largest [[posterior odds ratio]] (of which the ''Bayes factor'' is a principle component) when compared against all others, the model class or [[mathematical model]] which best explains the data among the competition is determined. It is then left to determine the best inference as to the values of that models parameters via [[parameter estimation]].

[[Category:Probability]]