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What does the phrase conjugate family of distribution mean?
The 2019 Stack Overflow Developer Survey Results Are InI want to have a positive uniform prior for $mu$ in a Poisson distribution, what $gamma(alpha, beta)$ do I use?What does it mean to divide by the standard deviation?What do these arrows mean? (Froda's Thm)Why is the area under the pdf for the Von Mises distribution not one?How to calculate a population mean for a normal distributionwhat does `ensemble average` mean?Getting a feel for the Normal-Inverse-Wishart conjugate prior to multivariate normal distributionSimulation in statisticsin a lattice, does the GLB and LUB of each element need to be contained in the lattice itself?Sampling distribution of the mean confusionConjugate prior of a normal distribution with unknown mean
$begingroup$
I'm doing some self study where I encountered the phrase "conjugate family of distribution".
I tried to good it and look thought online posts and Wikipedia, but I'm still confused.
What is a family of distribution? What does family mean?
What's a conjugate family? Why call it conjugate?
statistics definition
asked Mar 23 at 17:47
user9976437user9976437
17710
$endgroup$
add a comment |
$begingroup$
I'm doing some self study where I encountered the phrase "conjugate family of distribution".
I tried to good it and look thought online posts and Wikipedia, but I'm still confused.
What is a family of distribution? What does family mean?
What's a conjugate family? Why call it conjugate?
statistics definition
asked Mar 23 at 17:47
user9976437user9976437
17710
$endgroup$
add a comment |
$begingroup$
I'm doing some self study where I encountered the phrase "conjugate family of distribution".
I tried to good it and look thought online posts and Wikipedia, but I'm still confused.
What is a family of distribution? What does family mean?
What's a conjugate family? Why call it conjugate?
statistics definition
asked Mar 23 at 17:47
user9976437user9976437
17710
$endgroup$
I'm doing some self study where I encountered the phrase "conjugate family of distribution".
I tried to good it and look thought online posts and Wikipedia, but I'm still confused.
What is a family of distribution? What does family mean?
What's a conjugate family? Why call it conjugate?
statistics definition
statistics definition
asked Mar 23 at 17:47
user9976437user9976437
17710
asked Mar 23 at 17:47
user9976437user9976437
17710
asked Mar 23 at 17:47
user9976437user9976437
17710
asked Mar 23 at 17:47
user9976437user9976437
17710
asked Mar 23 at 17:47
user9976437user9976437
17710
17710
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Family: A family of distributions is a collection of distributions with a similar formula for the PDF, in which different choices of constant parameter values are used to specify various members of the family.
Three important examples of continuous families are $mathsfNorm(textmean=mu,, textSD=sigma),,$ $mathsfGamma(textshape=alpha,, textrate=lambda),$
and $mathsfBeta(textshape=alpha, textshape=beta).$
Two important examples of discrete families are $mathsfBinom(n,p)$
and $mathsfPoisson(lambda).$
Conjugacy: This terminology is mainly used in Bayesian statistics to mean 'mathematically compatible' in such a way that certain relationships are simple to show.
For example, a beta prior distribution is said to be 'conjugate' to binomial likelihood, because the posterior distribution (found by multiplying) is easily seen to be a beta distribution. (Similarly, we say that a gamma prior is conjugate to a Poisson likelihood function.)
Example Consider the prior distribution $mathsfBeta(2,3)$ and a binomial likelihood function based on observing $x$ successes in $n$ trials. The 'kernel' of the beta posterior has the form
$$theta^x + 2-1(1 - theta)^n - x + 3-1 propto
theta^2-1(1 - theta)^3-1 times theta^2(1-theta)^n-x.$$
Here the success probability is modeled as the random variable $theta$ and the symbol $propto$ is read "proportional to." The kernel of a density or likelihood
function omits the norming constant multiple that makes a density integrate to $1.$
In this example the mathematical compatibility of the beta and binomial distributions allow us to recognize that the kernel of the posterior is that of the distribution
$mathsfBeta(x+2, n-x+3).$ This 'conjugacy' makes it possible to identify the
posteriar distribution without having to integrate the denominator in the general form of Bayes' Theorem.
In particular, if the prior distribution is $theta sim mathsfBeta(2,3)$ and we observe $x=10$ Successes in $n=30$ trials, the posterior distribution of $theta$
is $mathsfBeta(12, 23)$ and a 95% posterior interval estimate for $theta$ is
$(0.1975, 0.5053)$, as computed using R.
qbeta(c(.025,.975), 12, 23)
## 0.1974586 0.5052653
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
Family: A family of distributions is a collection of distributions with a similar formula for the PDF, in which different choices of constant parameter values are used to specify various members of the family.
Three important examples of continuous families are $mathsfNorm(textmean=mu,, textSD=sigma),,$ $mathsfGamma(textshape=alpha,, textrate=lambda),$
and $mathsfBeta(textshape=alpha, textshape=beta).$
Two important examples of discrete families are $mathsfBinom(n,p)$
and $mathsfPoisson(lambda).$
Conjugacy: This terminology is mainly used in Bayesian statistics to mean 'mathematically compatible' in such a way that certain relationships are simple to show.
For example, a beta prior distribution is said to be 'conjugate' to binomial likelihood, because the posterior distribution (found by multiplying) is easily seen to be a beta distribution. (Similarly, we say that a gamma prior is conjugate to a Poisson likelihood function.)
Example Consider the prior distribution $mathsfBeta(2,3)$ and a binomial likelihood function based on observing $x$ successes in $n$ trials. The 'kernel' of the beta posterior has the form
$$theta^x + 2-1(1 - theta)^n - x + 3-1 propto
theta^2-1(1 - theta)^3-1 times theta^2(1-theta)^n-x.$$
Here the success probability is modeled as the random variable $theta$ and the symbol $propto$ is read "proportional to." The kernel of a density or likelihood
function omits the norming constant multiple that makes a density integrate to $1.$
In this example the mathematical compatibility of the beta and binomial distributions allow us to recognize that the kernel of the posterior is that of the distribution
$mathsfBeta(x+2, n-x+3).$ This 'conjugacy' makes it possible to identify the
posteriar distribution without having to integrate the denominator in the general form of Bayes' Theorem.
In particular, if the prior distribution is $theta sim mathsfBeta(2,3)$ and we observe $x=10$ Successes in $n=30$ trials, the posterior distribution of $theta$
is $mathsfBeta(12, 23)$ and a 95% posterior interval estimate for $theta$ is
$(0.1975, 0.5053)$, as computed using R.
qbeta(c(.025,.975), 12, 23)
## 0.1974586 0.5052653
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
$endgroup$
add a comment |
$begingroup$
Family: A family of distributions is a collection of distributions with a similar formula for the PDF, in which different choices of constant parameter values are used to specify various members of the family.
Three important examples of continuous families are $mathsfNorm(textmean=mu,, textSD=sigma),,$ $mathsfGamma(textshape=alpha,, textrate=lambda),$
and $mathsfBeta(textshape=alpha, textshape=beta).$
Two important examples of discrete families are $mathsfBinom(n,p)$
and $mathsfPoisson(lambda).$
Conjugacy: This terminology is mainly used in Bayesian statistics to mean 'mathematically compatible' in such a way that certain relationships are simple to show.
For example, a beta prior distribution is said to be 'conjugate' to binomial likelihood, because the posterior distribution (found by multiplying) is easily seen to be a beta distribution. (Similarly, we say that a gamma prior is conjugate to a Poisson likelihood function.)
Example Consider the prior distribution $mathsfBeta(2,3)$ and a binomial likelihood function based on observing $x$ successes in $n$ trials. The 'kernel' of the beta posterior has the form
$$theta^x + 2-1(1 - theta)^n - x + 3-1 propto
theta^2-1(1 - theta)^3-1 times theta^2(1-theta)^n-x.$$
Here the success probability is modeled as the random variable $theta$ and the symbol $propto$ is read "proportional to." The kernel of a density or likelihood
function omits the norming constant multiple that makes a density integrate to $1.$
In this example the mathematical compatibility of the beta and binomial distributions allow us to recognize that the kernel of the posterior is that of the distribution
$mathsfBeta(x+2, n-x+3).$ This 'conjugacy' makes it possible to identify the
posteriar distribution without having to integrate the denominator in the general form of Bayes' Theorem.
In particular, if the prior distribution is $theta sim mathsfBeta(2,3)$ and we observe $x=10$ Successes in $n=30$ trials, the posterior distribution of $theta$
is $mathsfBeta(12, 23)$ and a 95% posterior interval estimate for $theta$ is
$(0.1975, 0.5053)$, as computed using R.
qbeta(c(.025,.975), 12, 23)
## 0.1974586 0.5052653
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
$endgroup$
add a comment |
$begingroup$
Family: A family of distributions is a collection of distributions with a similar formula for the PDF, in which different choices of constant parameter values are used to specify various members of the family.
Three important examples of continuous families are $mathsfNorm(textmean=mu,, textSD=sigma),,$ $mathsfGamma(textshape=alpha,, textrate=lambda),$
and $mathsfBeta(textshape=alpha, textshape=beta).$
Two important examples of discrete families are $mathsfBinom(n,p)$
and $mathsfPoisson(lambda).$
Conjugacy: This terminology is mainly used in Bayesian statistics to mean 'mathematically compatible' in such a way that certain relationships are simple to show.
For example, a beta prior distribution is said to be 'conjugate' to binomial likelihood, because the posterior distribution (found by multiplying) is easily seen to be a beta distribution. (Similarly, we say that a gamma prior is conjugate to a Poisson likelihood function.)
Example Consider the prior distribution $mathsfBeta(2,3)$ and a binomial likelihood function based on observing $x$ successes in $n$ trials. The 'kernel' of the beta posterior has the form
$$theta^x + 2-1(1 - theta)^n - x + 3-1 propto
theta^2-1(1 - theta)^3-1 times theta^2(1-theta)^n-x.$$
Here the success probability is modeled as the random variable $theta$ and the symbol $propto$ is read "proportional to." The kernel of a density or likelihood
function omits the norming constant multiple that makes a density integrate to $1.$
In this example the mathematical compatibility of the beta and binomial distributions allow us to recognize that the kernel of the posterior is that of the distribution
$mathsfBeta(x+2, n-x+3).$ This 'conjugacy' makes it possible to identify the
posteriar distribution without having to integrate the denominator in the general form of Bayes' Theorem.
In particular, if the prior distribution is $theta sim mathsfBeta(2,3)$ and we observe $x=10$ Successes in $n=30$ trials, the posterior distribution of $theta$
is $mathsfBeta(12, 23)$ and a 95% posterior interval estimate for $theta$ is
$(0.1975, 0.5053)$, as computed using R.
qbeta(c(.025,.975), 12, 23)
## 0.1974586 0.5052653
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
$endgroup$
Family: A family of distributions is a collection of distributions with a similar formula for the PDF, in which different choices of constant parameter values are used to specify various members of the family.
Three important examples of continuous families are $mathsfNorm(textmean=mu,, textSD=sigma),,$ $mathsfGamma(textshape=alpha,, textrate=lambda),$
and $mathsfBeta(textshape=alpha, textshape=beta).$
Two important examples of discrete families are $mathsfBinom(n,p)$
and $mathsfPoisson(lambda).$
Conjugacy: This terminology is mainly used in Bayesian statistics to mean 'mathematically compatible' in such a way that certain relationships are simple to show.
For example, a beta prior distribution is said to be 'conjugate' to binomial likelihood, because the posterior distribution (found by multiplying) is easily seen to be a beta distribution. (Similarly, we say that a gamma prior is conjugate to a Poisson likelihood function.)
Example Consider the prior distribution $mathsfBeta(2,3)$ and a binomial likelihood function based on observing $x$ successes in $n$ trials. The 'kernel' of the beta posterior has the form
$$theta^x + 2-1(1 - theta)^n - x + 3-1 propto
theta^2-1(1 - theta)^3-1 times theta^2(1-theta)^n-x.$$
Here the success probability is modeled as the random variable $theta$ and the symbol $propto$ is read "proportional to." The kernel of a density or likelihood
function omits the norming constant multiple that makes a density integrate to $1.$
In this example the mathematical compatibility of the beta and binomial distributions allow us to recognize that the kernel of the posterior is that of the distribution
$mathsfBeta(x+2, n-x+3).$ This 'conjugacy' makes it possible to identify the
posteriar distribution without having to integrate the denominator in the general form of Bayes' Theorem.
In particular, if the prior distribution is $theta sim mathsfBeta(2,3)$ and we observe $x=10$ Successes in $n=30$ trials, the posterior distribution of $theta$
is $mathsfBeta(12, 23)$ and a 95% posterior interval estimate for $theta$ is
$(0.1975, 0.5053)$, as computed using R.
qbeta(c(.025,.975), 12, 23)
## 0.1974586 0.5052653
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
edited Mar 23 at 21:24
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
answered Mar 23 at 21:05
BruceETBruceET
36.4k71540
36.4k71540
add a comment |
add a comment |
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