• Upon closer inspection, many scientific questions from various fields concern variances

• Economics & Archeology:
  • Do products become more homogeneous with increased specialization? (Kvamme et al., 1996)

• Legal Studies:
  • Which interventions best reduce unwanted variability in civil damage awards? (Sasks et al., 1997)

• Genetics:
  • What are the genetic effects on the variance of quantitative traits? (Pare et al., 2010)

• Psychology:
  • Do males show higher variance on intelligence? (Johnson et al., 2008)
  • Do males show higher variance on personality traits? (Borkenau et al., 2013)
  • Does the variance of mathematical ability increase across school grades? (Aunola et al., 2004)

• Let group \(k\) consist of \(n_k\) observations \(\mathbf{x}_k = \{x_{k1}, \ldots, x_{kn_1}\}\). We assume that

\[ x_{ki} \overset{\tiny{\text{i.i.d.}}}{\sim} \mathcal{N}\left(\mu_k, \sigma^2_k\right) \enspace , \]

• for all \(k \in \{1, \ldots, K\}\) and \(i \in \{1, \ldots, n_k\}\)
• We restrict our focus to the \(K = 2\) case. Our aim is to test the hypotheses

\[ \begin{aligned} \mathcal{H}_0 &: \sigma^2_1 = \sigma^2_2 \\[.5em] \mathcal{H}_1 &: \sigma^2_1 \neq \sigma^2_2 \enspace . \end{aligned} \]

• Bayesian inference provides a predictive perspective
• We require models that instantiate these hypotheses and that make predictions
• Without priors, no predictions!

From https://fabiandablander.com/r/Law-of-Practice.html

• Let \(\mathbf{d} = (\mathbf{x}_1, \mathbf{x}_2)\)
• We focus here on the prior predictive perspective
• The ratio of two marginal likelihoods is called the Bayes factor
• Quantifies how much better one model predicts the data compared to another model

\[ \underbrace{\frac{p(\mathcal{M}_1 \mid \mathbf{d})}{p(\mathcal{M}_0 \mid \mathbf{d})}}_{\text{Posterior odds}} = \underbrace{\frac{p(\mathbf{d} \mid \mathcal{M}_1)}{p(\mathbf{d} \mid \mathcal{M}_0)}}_{\text{Bayes factor}} \hspace{.25em} \underbrace{\frac{p(\mathcal{M}_1)}{p( \mathcal{M}_0)}}_{\text{Prior odds}} \]

• We focus on the \(K = 2\) case and derive a default Bayes factor
• Next steps: Assigning priors to all parameters and integrating them out

• Let \(\phi = (\mu_1, \mu_2)\) denote the test-irrelevant parameters
• Under \(\mathcal{H}_1\) we specify

\[ \begin{aligned} \pi(\phi) &\propto 1 \\[.5em] \pi(\sigma_1^2) &\sim \text{Inverse-Gamma}(\alpha, \beta) \\[.5em] \pi(\sigma_2^2) &\sim \text{Inverse-Gamma}(\alpha, \beta) \\[.5em] \end{aligned} \]

• Under \(\mathcal{H}_0\) we specify

\[ \begin{aligned} \pi(\phi) &\propto 1 \\[.5em] \pi(\sigma^2) &\sim \text{Inverse-Gamma}(\alpha, \beta) \end{aligned} \]

• where \(\sigma^2 = \sigma_1^2 = \sigma_2^2\)

• The resulting Bayes factor is given by

\[ \begin{aligned} \text{BF}_{10} &= \frac{p(\mathbf{d} \mid \mathcal{M}_1)}{p(\mathbf{d} \mid \mathcal{M}_0)} \\[.5em] &= \frac{\int_{\sigma_1}\int_{\sigma_2}\int_{\phi} f(\mathbf{d} \mid \phi, \sigma_1, \sigma_2) \, \pi(\phi, \sigma_1, \sigma_2) \, \mathrm{d}\phi \, \mathrm{d}\sigma_1 \mathrm{d} \sigma_2}{\int_{\sigma}\int_{\phi}f(\mathbf{d} \mid \phi, \sigma) \, \pi(\phi, \sigma) \, \mathrm{d}\phi \, \mathrm{d}\sigma} \\[.5em] &= \frac{\frac{1}{2} \frac{\beta^\alpha}{\Gamma(\alpha)} \Gamma\left(\frac{n_1}{2} + \alpha\right) \Gamma\left(\frac{n_2}{2} + \alpha\right) \left(\beta + \frac{n_1 s_1^2}{2} \right)^{-\frac{n_1}{2} - \alpha} \left(\beta + \frac{n_2 s_2^2}{2} \right)^{-\frac{n_2}{2} - \alpha}}{\Gamma\left(\frac{n}{2} + \alpha\right) \left(\beta + \frac{n_1 s_1^2 + n_2 s_2^2}{2} \right)^{-\frac{n}{2} - \alpha}} \enspace , \end{aligned} \]

• where \(n = n_1 + n_2\) and \(n_k s_k = \sum_{i=1}^{n_k} (x_{ik} - \bar{x}_k)^2\)
• This Bayes factor is not measurement invariant nor predictively matched

• Select priors such that certain desiderata are fulfilled:

• Measurement invariance:
  • Bayes factor does not depend on the unit of measurements

• Predictive matching:
  • Bayes factor equals 1 for uninformative data

• Model selection consistent:
  • Bayes factor in favour of true model goes to \(\infty\) as \((n_1, n_2) \rightarrow \infty\)

• Information consistent:
  • Bayes factor in favour of true model goes to \(\infty\) as (finite) data provide infinite support

• Limit consistent:
  • If \(n_2\) fixed but \(n_1 \rightarrow \infty\), Bayes factor remains bounded

• We call the resulting Bayes factor a default Bayes factor (Jeffreys, 1961; Bayarri et al. 2012; Ly, 2018)

• We borrow an idea from Jeffreys (1939) and write for \(\rho \in [0, 1]\)

\[ \begin{aligned} \sigma_1^2 &= \rho \sigma^2 \\[.5em] \sigma_2^2 &= (1 - \rho) \sigma^2 \end{aligned} \]

• such that \(\sigma_1^2 + \sigma_2^2 = \rho \sigma^2 + (1 - \rho)\sigma^2 = \sigma^2\)
• Observe that \(\rho = \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2}\)
• We can rewrite the hypotheses know in terms of \(\rho\):

\[ \begin{aligned} \mathcal{H}_0 &: \rho = 0.50 \\[.5em] \mathcal{H}_1 &: \rho \neq 0.50 \end{aligned} \]

• This leads to \(p(\mathbf{y} \mid \mathcal{M}_0) = p(\mathbf{y} \mid \rho = 0.50, \mathcal{M}_1)\)

• Let \(\phi = (\mu_1, \mu_2, \sigma)\) be the set of test-irrelevant parameters. Then

\[ \text{BF}_{10} = \frac{p(\mathbf{d} \mid \mathcal{M}_1)}{p(\mathbf{d} \mid \mathcal{M}_0)} = \frac{\int_{\rho} \int_{\phi} f(\mathbf{d} \mid \phi, \rho) \, \pi(\phi) \, \pi(\rho) \, \mathrm{d}\phi \, \mathrm{d}\rho}{\int_{\phi} f(\mathbf{d} \mid \phi, \rho = 0.50) \, \pi(\phi) \, \mathrm{d}\phi}\enspace . \]

• We can use improper priors \(\pi(\phi) = 1 \cdot 1 \cdot \sigma^{-1}\) because the normalizing constants cancel
• Key question: What prior do we assign the test-relevant parameter \(\rho\)?

• Principled approach of choosing priors
  • Measurement invariance \(\checkmark\)
  • Predictive matching \(\checkmark\)
  • Model selection consistent \(\checkmark\)
  • Information consistent \(\checkmark\)
  • Limit consistent \(\checkmark\)

• A class of Beta priors fulfills these criteria

\[ \rho = \frac{\sigma_1^2}{\sigma_1^2 + \sigma_2^2} \sim \text{Beta}(\alpha, \alpha) \]

• This induces a Generalized Beta prime distribution on

\[ \delta \equiv \frac{\sigma_1}{\sigma_2} \sim \text{GeneralizedBetaPrime}(\alpha, 2, 1) \]

• \(\alpha = 4.50\) implies that \(p(\delta \in [0.50, 2]) = 0.95\)

• The Bayes factor in favour of \(\mathcal{H}_1\) is given by: \[ \text{BF}_{10} = \frac{\text{B}\left({\frac{n_2 - 1}{2} + \alpha},\ {\frac{n_1 - 1}{2} + \alpha}\right) \,_2F_1\left({\frac{n - 2}{2}};{\frac{n_2 - 1}{2} + \alpha};{\frac{n - 2}{2} + 2\alpha};{1 - \frac{n_2 s_2^2}{n_1 s_1^2}}\right)}{\text{B}\left({\alpha},\ {\alpha}\right) \left(1 + \frac{n_2 s_2^2}{n_1 s_1^2}\right)^{\frac{2 - n}{2}}} \]

• where \(n_k s_k^2 = \sum_{i=1}^{n_k} (x_{ik} - \bar{x}_{ik})^2\) and \(n = n_1 + n_2\)

• Note that the data only enter through the ratio \(\frac{n_2 s_2^2}{n_1 s_1^2}\)

• Borkenau et al. (2013) study the variability in personality traits in four countries
• We re-analyze the Estonian sample: \(s_f^2 = 15.6\), \(s_m^2 = 19.9\), \(n_f = 969\), \(n_m = 716\)

• There are alternative ways of specifying a prior distribution for testing
• One might sacrifice part of the data to specify minimally informative priors
  • Partial Bayes factors (O’Hagan, 1991)
  • Intrinsic Bayes factors (Berger & Pericchi, 1996)
  • Fractional Bayes factors (O’Hagan, 1995)

• Böing-Messing & Mulder (2018) propose an Automatic Fractional Bayes factor (AFBF) for testing hypotheses on variances
• It turns out that this is a special case of our Bayes factor with \(\alpha = 0.50\)

• We again write

\[ x_{ki} \overset{\tiny{\text{i.i.d.}}}{\sim} \mathcal{N}(\mu_k, \rho_k \sigma^2) \]

• for all \(k \in \{1, \ldots, K\}\) and \(i \in \{1, \ldots, n_k\}\) where \(\sum_{k=1}^K \rho_k = 1\)
• Standard \(\mathcal{H}_0\) and \(\mathcal{H}_1\) are

\[ \begin{align*} \mathcal{H}_0&: \rho_k = \frac{1}{k} \hspace{1em} \forall k \in \{1, \ldots, k\} \\ \mathcal{H}_1&: \mathbf{\rho} \sim \text{Dirichlet}(\alpha_1, \ldots, \alpha_k) \end{align*} \]

• However, we can flexibly test any hypotheses with mixed constraints such as

\[ \mathcal{H}_r: \rho_1 = \rho_2 > \rho_3, \rho_4, \rho_5 = \rho_6 > \rho_7 \]

• by constraining the prior and computing the marginal likelihood in Stan using bridgesampling (Meng & Wong, 1996; Gronau et al., 2017)

• Aunola et al. (2004) find that the variance in mathematical ability increases across school grade
• We replicate this finding using large-scale data from MathGarden (\(n = 41801\))

\[ \begin{align*} \mathcal{H}_0&: \rho_i = \rho_j \hspace{1em} \forall (i, j) \hspace{5em} p(\mathcal{M}_0 \mid y) = 0.00\\ \mathcal{H}_f&: \rho_i \neq \rho_j \hspace{1em} \forall (i, j) \hspace{5em} p(\mathcal{M}_f \mid y) = 0.00\\ \mathcal{H}_r&: \rho_i < \rho_j \hspace{1em} \forall (i < j) \hspace{4.25em} p(\mathcal{M}_r \mid y) = 1.00\\ \end{align*} \]

• Many scientific questions from various fields concern variances

• Derived a default Bayes factor for the \(K = 2\) case
  • Fulfills a number of desiderata
  • Limit consistency yields \(K = 1\) group Bayes factor
  • Can extend point null hypothesis to a null-region

• Extended it to \(K > 2\) groups
  • Compute marginal likelihood for mixed hypotheses using bridgesampling

• Generalizes the ‘automatic’ Bayes factor proposed by Böing-Messing & Mulder (2018)
  • Allows sensitivity analyses and incorporation of prior information

• You can find me online @fdabl and fabiandablander.com where I write about
  • Love affairs and linear differential equations
  • Spurious correlations and random walks
  • Bayesian modeling in Stan: A case study
  • …

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