rigliztetbf

2022-06-26

I'm reading about Bayesian data analysis by Gelman et al. and I'm having big trouble interpreting the following part in the book (note, the rat tumor rate $\theta $ in the following text has:$\theta \sim Beta(\alpha ,\beta )$

Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution.

Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\alpha ,\beta )$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\alpha}{\alpha +\beta})=\mathrm{log}(\frac{\alpha}{\beta})$ and $\mathrm{log}(\alpha +\beta )$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta $. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on $(-\mathrm{\infty},\mathrm{\infty})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\alpha +\beta )\to \mathrm{\infty}$, and so this particular prior density cannot be used here.

In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,

$p(\alpha ,\beta )\propto (\alpha +\beta {)}^{-5/2},$

and on the natural transformed scale:

$p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\propto \alpha \beta (\alpha +\beta {)}^{-5/2}.$

My problem is especially the bolded parts in the text.

Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$

Question (2): What is the appropriate Jacobian?

Question (3): How does the author arrive into the original and transformed scale priors?

To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.

P.S. if you need more information, or me to clarify my questions please let me know.

Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution.

Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\alpha ,\beta )$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\alpha}{\alpha +\beta})=\mathrm{log}(\frac{\alpha}{\beta})$ and $\mathrm{log}(\alpha +\beta )$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta $. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on $(-\mathrm{\infty},\mathrm{\infty})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\alpha +\beta )\to \mathrm{\infty}$, and so this particular prior density cannot be used here.

In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,

$p(\alpha ,\beta )\propto (\alpha +\beta {)}^{-5/2},$

and on the natural transformed scale:

$p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\propto \alpha \beta (\alpha +\beta {)}^{-5/2}.$

My problem is especially the bolded parts in the text.

Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\alpha}{\alpha +\beta},(\alpha +\beta {)}^{-1/2})$

Question (2): What is the appropriate Jacobian?

Question (3): How does the author arrive into the original and transformed scale priors?

To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.

P.S. if you need more information, or me to clarify my questions please let me know.

Raven Higgins

Beginner2022-06-27Added 17 answers

If anyone runs into the similar section in Gelman's book, I'm going to give my own solution that I came up with below (pages 110-111).

By this, the author only implies that

$p(\frac{\alpha}{\alpha +\beta},\phantom{\rule{thickmathspace}{0ex}}(\alpha +\beta {)}^{-1/2})=\text{constant}\propto 1.$

Answer (2):

When the author refers to a "appropriate Jacobian," he is referring to the Jacobian matrix's determinant in the formula for density functions with changed variables:

$p(\varphi )=p(\theta )\phantom{\rule{thinmathspace}{0ex}}{det\left(\frac{d\theta}{d\varphi}\right)}$

Answer (3):

Simply said, the author uses the change of variables formula twice. The fact that

$p(\gamma ,\delta )=p(\gamma (\alpha ,\beta ),\phantom{\rule{thinmathspace}{0ex}}\delta (\alpha ,\beta ))=p(\frac{\alpha}{\alpha +\beta},\phantom{\rule{thickmathspace}{0ex}}(\alpha +\beta {)}^{-1/2})=\text{constant}\propto 1.$

If we denote$\theta =(\gamma ,\delta )$ and $\varphi =(\alpha ,\beta )$), then:

$det\left(\frac{d\theta}{d\varphi}\right)=\left|\begin{array}{cc}\frac{d\gamma}{d\alpha}& \frac{d\gamma}{d\beta}\\ \frac{d\delta}{d\alpha}& \frac{d\delta}{d\beta}\end{array}\right|=\left|\begin{array}{cc}\frac{\beta}{(\alpha +\beta {)}^{2}}& -\frac{\alpha}{(\alpha +\beta {)}^{2}}\\ -\frac{1}{2(\alpha +\beta {)}^{3/2}}& -\frac{1}{2(\alpha +\beta {)}^{3/2}}\end{array}\right|=-\frac{1}{2(\alpha +\beta {)}^{5/2}}.$

From the formula for variable change, we obtain:

$p(\alpha ,\beta )=\underset{\text{= constant}\propto \phantom{\rule{thickmathspace}{0ex}}1}{\underset{\u23df}{p(\frac{\alpha}{\alpha +\beta},\phantom{\rule{thickmathspace}{0ex}}(\alpha +\beta {)}^{-1/2})}}\cdot (-\frac{1}{2(\alpha +\beta {)}^{5/2}})\propto (\alpha +\beta {)}^{-5/2},$

and there it is, i.e. the prior in original scale.

By using the exact same modification of variables for the alternate scale:

$p(\alpha ,\beta )=p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\phantom{\rule{thinmathspace}{0ex}}det\left(\frac{d\theta}{d\varphi}\right),$

where this time $\gamma (\alpha ,\beta )=\mathrm{log}\left(\frac{\alpha}{\beta}\right)$and $\delta (\alpha ,\beta )=\mathrm{log}(\alpha +\beta )$ We obtain: for the Jacobian determinant.

$det\left(\frac{d\theta}{d\varphi}\right)=\left|\begin{array}{cc}\frac{d\gamma}{d\alpha}& \frac{d\gamma}{d\beta}\\ \frac{d\delta}{d\alpha}& \frac{d\delta}{d\beta}\end{array}\right|=\left|\begin{array}{cc}1/\alpha & -1/\beta \\ (\alpha +\beta {)}^{-1}& (\alpha +\beta {)}^{-1}\end{array}\right|=\frac{1}{\alpha \beta},$

so we get:

$\underset{\propto \phantom{\rule{thinmathspace}{0ex}}(\alpha +\beta {)}^{-5/2}}{\underset{\u23df}{p(\alpha ,\beta )}}=p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\phantom{\rule{thinmathspace}{0ex}}\frac{1}{\alpha \beta},$

or

$p(\mathrm{log}\left(\frac{\alpha}{\beta}\right),\mathrm{log}(\alpha +\beta ))\propto \alpha \beta (\alpha +\beta {)}^{-5/2}.$

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