1 Introduction↩︎

Student-\(t\) regression models are widely used for robust Bayesian inference. In order to obtain approximate posterior samples under Student-\(t\) regression models, we usually take a data augmentation approach because the joint posterior distribution of the regression coefficients and error variance does not have a standard form and is difficult to directly sample from. In particular, we use the fact that Student’s \(t\)-distribution is a scale mixture of normals and generate samples of mixing variables as well as the target parameters in order to approximate the posterior distribution.

Efficiency of Markov chains based on such algorithms has been investigated by many authors. A popular criterion is geometric ergodicity. Sufficient conditions for this property were derived by [1] in the case where the prior distribution is improper and by [2] in the proper case. Recently, [3] obtained conditions to cover the case of a Student-\(t\) error density with few degrees of freedom.

A stronger property of a Markov chain is that it is trace-class. There are several benefits related to this property (e.g., [4]). In particular, we can estimate the spectral gap of a trace-class Markov chain ([5]), which is directly related to the geometric convergence rate of the chain. Although trace-class results for MCMC algorithms for regression models were obtained by [6] and [4], their results are not applicable when we use error distributions with polynomial tails.

In this paper, we obtain trace-class results for MCMC algorithms for Student-\(t\) regression models. In Section 2, we consider the improper case where the parameters follow the invariant distributions. We show that the Markov operator associated with a standard data augmentation algorithm is not trace-class and that the Markov operator associated with another algorithm is trace-class. In Section 3, we consider the proper case where the prior distribution is normal-inverse gamma.

2 The Case of an Improper Prior↩︎

2.1 The model and conditional distributions↩︎

In this section, we consider the univariate Student-\(t\) regression model. Specifically, we consider the model given by \[\begin{align} &y_i \sim \int_{0}^{\infty } {\rm{N}} ( y_i | {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}/ u_i ) {\rm{Ga}} ( u_i | a, b) d{u_i} \text{,} \quad i = 1, \dots , n \text{,} {\nonumber} \end{align}\] where \({\text{\boldsymbol{y}}}= ( y_i )_{i = 1}^{n} \in \mathbb{R} ^n\) and \({\text{\boldsymbol{X}}}= ( {\text{\boldsymbol{x}}}_1 , \dots , {\text{\boldsymbol{x}}}_n )^{\top } \in \mathbb{R} ^{n \times p}\) are the outcome and explanatory variables, \({\text{\boldsymbol{u}}}= ( u_i )_{i = 1}^{n} \in (0, \infty )^n\) are latent variables having the mixing distribution \({\rm{Ga}} (a, b)\) for \(a, b > 0\), and \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\) and \({{{\sigma}^2}}\in (0, \infty )\) are the regression coefficients and error variance. The prior is given by \(( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}) \sim 1 / {{{\sigma}^2}}\).

In the remainder of this section, we suppress the dependence on \({\text{\boldsymbol{y}}}\). The joint posterior distribution is given by \[\begin{align} p( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}, {\text{\boldsymbol{u}}}) &\propto {1 \over {{{\sigma}^2}}} \prod_{i = 1}^{n} \Big[ {u_i}^{a - 1} e^{- b u_i} {{u_i}^{1 / 2} \over ( {{{\sigma}^2}})^{1 / 2}} \exp \Big\{ - {u_i \over 2 {{{\sigma}^2}}} ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \Big\} \Big] \text{.} {\nonumber} \end{align}\] Therefore, \[\begin{align} p( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}| {\text{\boldsymbol{u}}}) &\propto {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \exp \Big\{ - {1 \over 2 {{{\sigma}^2}}} ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) \Big\} {\nonumber}\\ &\propto {\rm{IG}} \Big( {{{\sigma}^2}}\Big| {n - p \over 2}, {{\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\over 2} \Big) {\nonumber}\\ &\quad \times {\rm{N}}_p ( {\text{\boldsymbol{\beta}}}| ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}, {{{\sigma}^2}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} ) \text{,} {\nonumber} \end{align}\] where the second line follows since \[\begin{align} ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) &= {\text{\boldsymbol{\beta}}}^{\top } {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}+ {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- 2 {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}{\nonumber}\\ &= \{ {\text{\boldsymbol{\beta}}}- ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\} ^{\top } {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}\{ {\text{\boldsymbol{\beta}}}- ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\} {\nonumber}\\ &\quad + {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\text{.} {\nonumber} \end{align}\] Meanwhile, \[\begin{align} p( {\text{\boldsymbol{u}}}, {{{\sigma}^2}}| {\text{\boldsymbol{\beta}}}) &\propto {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \prod_{i = 1}^{n} \Big( {u_i}^{1 / 2 + a - 1} \exp \Big[ - u_i \Big\{ {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big\} \Big] \Big) {\nonumber}\\ &\propto \Big[ {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \prod_{i = 1}^{n} {1 \over \{ ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 / (2 {{{\sigma}^2}}) + b \} ^{1 / 2 + a}} \Big] \prod_{i = 1}^{n} {\rm{Ga}} \Big( u_i \Big| {1 \over 2} + a, {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big) \text{.} {\nonumber} \end{align}\]

2.2 An algorithm which is not trace class↩︎

The following algorithm is a standard data augmentation algorithm for the Student-\(t\) regression model.

Although geometric ergodicity can be shown under some assumptions (see, for example, [3]), we have the following result.

Proof. By Theorem 2 of [5], the data augmentation Markov chain is trace-class if and only if \(I < \infty\), where \[\begin{align} I &= \int_{\mathbb{R} ^p \times (0, \infty ) \times (0, \infty )^n} p( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}) d( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}, {\text{\boldsymbol{u}}}) \text{.} {\nonumber} \end{align}\] The set \(\bigcup_{i = 1}^{n} \{ {\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p | | y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}}| = 0 \}\) is closed. Therefore, there exist \({\text{\boldsymbol{\beta}}}_0 \in \mathbb{R} ^p\) and \({\delta}, {\varepsilon}> 0\) such that for all \({\text{\boldsymbol{\beta}}}\in U_{{\delta}} ( {\text{\boldsymbol{\beta}}}_0 )\) and all \(i = 1, \dots , n\), we have \(| y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}}| > {\varepsilon}\). Let \(M = \sup_{{\text{\boldsymbol{\beta}}}\in U_{{\delta}} ( {\text{\boldsymbol{\beta}}}_0 )} \| {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}\| \in (0, \infty )\). Then for all \({\text{\boldsymbol{\beta}}}\in U_{{\delta}} ( {\text{\boldsymbol{\beta}}}_0 )\), we have, by Section 2.1, \[\begin{align} p( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}| {\text{\boldsymbol{u}}}) &= \frac{ \displaystyle {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \exp \Big\{ - {1 \over 2 {{{\sigma}^2}}} ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) \Big\} }{ \displaystyle \int_{(0, \infty )} {(2 \pi )^{p / 2} / | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2} \over ( {{{\sigma}^2}})^{1 + (n - p) / 2}} \exp \Big\{ - {{\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\over 2 {{{\sigma}^2}}} \Big\} d{{{\sigma}^2}}} {\nonumber}\\ &\ge \frac{ \displaystyle {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \prod_{i = 1}^{n} \exp \Big( - {M^2 \over 2 {{{\sigma}^2}}} u_i \Big) }{ \displaystyle {(2 \pi )^{p / 2} \over | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2}} {{\Gamma}((n - p) / 2) 2^{(n - p) / 2} \over \{ {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\} ^{(n - p) / 2}} } {\nonumber} \end{align}\] and \[\begin{align} p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}) &= \prod_{i = 1}^{n} \frac{ \displaystyle {u_i}^{1 / 2 + a - 1} \exp \Big[ - u_i \Big\{ {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big\} \Big] }{ \displaystyle \int_{(0, \infty )^n} {u_i}^{1 / 2 + a - 1} \exp \Big[ - u_i \Big\{ {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big\} \Big] d{u_i} } {\nonumber}\\ &\ge \prod_{i = 1}^{n} \frac{ \displaystyle {u_i}^{1 / 2 + a - 1} \exp \Big\{ - u_i \Big( {M^2 \over 2 {{{\sigma}^2}}} + b \Big) \Big\} }{ \displaystyle \int_{(0, \infty )^n} {u_i}^{1 / 2 + a - 1} \exp \Big\{ - u_i \Big( {{\varepsilon}^2 \over 2 {{{\sigma}^2}}} + b \Big) \Big\} d{u_i} } \text{.} {\nonumber} \end{align}\] Thus, \[\begin{align} I &\ge {1 \over M_1} \int_{U_{{\delta}} ( {\text{\boldsymbol{\beta}}}_0 ) \times (0, \infty ) \times (0, \infty )^n} \Big( {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2} \{ {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\} ^{(n - p) / 2} \over ( {{{\sigma}^2}})^{1 + n / 2}} {\nonumber}\\ &\quad \times \Big[ \prod_{i = 1}^{n} \frac{ \displaystyle {u_i}^{1 / 2 + a - 1} \exp \{ - u_i ( M^2 / {{{\sigma}^2}}+ b) \} }{ \displaystyle {\Gamma}(1 / 2 + a) / \{ {\varepsilon}^2 / (2 {{{\sigma}^2}}) + b \} ^{1 / 2 + a} } \Big] \Big) d( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}, {\text{\boldsymbol{u}}}) {\nonumber}\\ &= {1 \over M_1} \Big\{ \int_{U_{{\delta}} ( {\text{\boldsymbol{\beta}}}_0 )} 1 d{\text{\boldsymbol{\beta}}}\Big\} \int_{(0, \infty ) \times (0, \infty )^n} \Big( {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{V}}}{\text{\boldsymbol{X}}}|^{1 / 2} \{ {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{V}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{V}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{V}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{V}}}{\text{\boldsymbol{y}}}\} ^{(n - p) / 2} \over ( M^2 / {{{\sigma}^2}}+ b)^{p / 2 + (n - p) / 2} ( {{{\sigma}^2}})^{1 + n / 2}} {\nonumber}\\ &\quad \times \Big[ \prod_{i = 1}^{n} \frac{ \displaystyle {v_i}^{1 / 2 + a - 1} \exp (- v_i ) / {\Gamma}(1 / 2 + a) }{ \displaystyle ( M^2 / {{{\sigma}^2}}+ b)^{1 / 2 + a} / \{ {\varepsilon}^2 / (2 {{{\sigma}^2}}) + b \} ^{1 / 2 + a} } \Big] \Big) d( {{{\sigma}^2}}, {\text{\boldsymbol{v}}}) {\nonumber} \end{align}\] for some \(M_1 > 0\) and the right-hand side of the above inequality is infinite since \[\begin{align} &\int_{0}^{1} {1 \over ( M^2 / {{{\sigma}^2}}+ b)^{p / 2 + (n - p) / 2} ( {{{\sigma}^2}})^{1 + n / 2}} {1 \over [( M^2 / {{{\sigma}^2}}+ b)^{1 / 2 + a} / \{ {\varepsilon}^2 / (2 {{{\sigma}^2}}) + b \} ^{1 / 2 + a} ]^n} d{{{\sigma}^2}}= \infty \text{.} {\nonumber} \end{align}\] This completes the proof. \(\Box\)

2.3 An algorithm which is trace class↩︎

In the following collapsed Gibbs algorithm, \({\text{\boldsymbol{u}}}\) is sampled unconditionally on \({{{\sigma}^2}}\).

The conditional density of \(\log {{{\sigma}^2}}\) given \({\text{\boldsymbol{\beta}}}\) is log-concave. Therefore, we can use the method of [7].

When we use Algorithm 2, we sample \({\text{\boldsymbol{\beta}}}\sim p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}})\) and \({\text{\boldsymbol{u}}}\sim p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}})\). Therefore, this is a data augmentation algorithm based on the joint density \(p( {\text{\boldsymbol{\beta}}}, {\text{\boldsymbol{u}}})\). The transition density is given by \[\begin{align} k( {\text{\boldsymbol{\beta}}}_{\rm{new}} | {\text{\boldsymbol{\beta}}}_{\rm{old}} ) &= \int_{(0, \infty )^n} p( {\text{\boldsymbol{\beta}}}_{\rm{new}} | {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}_{\rm{old}} ) d{\text{\boldsymbol{u}}}\text{,} \quad {\text{\boldsymbol{\beta}}}_{\rm{old}} , {\text{\boldsymbol{\beta}}}_{\rm{new}} \in \mathbb{R} ^p \text{.} {\nonumber} \end{align}\]

The lower bound on \(a\) does not increase linearly with \(n\). This is in contrast for example to the case considered by [8]. In proving Proposition 2, we use ideas used in papers (e.g., [9], [10]) investigating robustness of posterior distributions in the presence of outliers.

Proof of Proposition 2. By Theorem 2 of [5], the data augmentation Markov chain is trace-class if and only if \(I < \infty\), where \[\begin{align} I &= \int_{\mathbb{R} ^p} k( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{\beta}}}) d{\text{\boldsymbol{\beta}}}= \int_{\mathbb{R} ^p \times (0, \infty )^n} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) d( {\text{\boldsymbol{\beta}}}, {\text{\boldsymbol{u}}}) \text{.} {\nonumber} \end{align}\] By Section 2.1, \[\begin{align} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) &= \frac{ \displaystyle \int_{0}^{\infty } {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \exp \Big\{ - {1 \over 2 {{{\sigma}^2}}} ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) \Big\} d{{{\sigma}^2}}}{ \displaystyle \int_{\mathbb{R} ^p \times (0, \infty )} {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \exp \Big\{ - {1 \over 2 {{{\sigma}^2}}} ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) \Big\} d( {\text{\boldsymbol{\beta}}}, {{{\sigma}^2}}) } {\nonumber}\\ &= {{\Gamma}(n / 2) / \{ \pi ^{p / 2} {\Gamma}((n - p) / 2) \} \over \{ ( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}})^{\top } {\text{\boldsymbol{U}}}( {\text{\boldsymbol{y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\beta}}}) \} ^{n / 2}} | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2} \{ {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}- {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}}\} ^{(n - p) / 2} {\nonumber} \end{align}\] and \[\begin{align} p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) &= \frac{ \displaystyle \int_{0}^{\infty } {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \prod_{i = 1}^{n} \Big( {u_i}^{1 / 2 + a - 1} \exp \Big[ - u_i \Big\{ {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big\} \Big] \Big) d{{{\sigma}^2}}}{ \displaystyle \int_{(0, \infty )^n \times (0, \infty )} {1 \over ( {{{\sigma}^2}})^{1 + n / 2}} \prod_{i = 1}^{n} \Big( {u_i}^{1 / 2 + a - 1} \exp \Big[ - u_i \Big\{ {( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \over 2 {{{\sigma}^2}}} + b \Big\} \Big] \Big) d( {\text{\boldsymbol{u}}}, {{{\sigma}^2}}) } {\nonumber}\\ &= {2^{n / 2} b^{n a + n / 2} {\Gamma}(n / 2) \over \{ {\Gamma}(1 / 2 + a) \} ^n} {\prod_{i = 1}^{n} {u_i}^{1 / 2 + a - 1} e^{- b u_i} \over \big\{ \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^{n / 2}} {\nonumber}\\ &\quad / \int_{0}^{\infty } ( {{{\sigma}^2}})^{n a - 1} \Big[ \prod_{i = 1}^{n} {1 \over \{ ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 / (2 b) + {{{\sigma}^2}}\} ^{1 / 2 + a}} \Big] d{{{\sigma}^2}}\text{.} {\nonumber} \end{align}\] Therefore, \[\begin{align} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) &\le M_1 {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2} ( {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}})^{(n - p) / 2} \over \big\{ \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^n} \Big\{ \prod_{i = 1}^{n} ( {u_i}^{1 / 2 + a - 1} e^{- b u_i} ) \Big\} {\nonumber}\\ &\quad / \int_{0}^{\infty } ( {{{\sigma}^2}})^{n a - 1} \Big[ \prod_{i = 1}^{n} {1 \over \{ ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 / (2 b) + {{{\sigma}^2}}\} ^{1 / 2 + a}} \Big] d{{{\sigma}^2}}{\nonumber} \end{align}\] for some \(M_1 > 0\). Note that \[\begin{align} | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}|^{1 / 2} ( {\text{\boldsymbol{y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{y}}})^{(n - p) / 2} \prod_{i = 1}^{n} ( {u_i}^{1 / 2 + a - 1} e^{- b u_i} ) &\le M_2 \prod_{i = 1}^{n} ( {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} ) {\nonumber} \end{align}\] for some \(M_2 > 0\) and that \[\begin{align} \int_{0}^{\infty } ( {{{\sigma}^2}})^{n a - 1} \Big[ \prod_{i = 1}^{n} {1 \over \{ ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 / (2 b) + {{{\sigma}^2}}\} ^{1 / 2 + a}} \Big] d{{{\sigma}^2}}&\ge {1 \over M_3} \int_{0}^{\infty } {( {{{\sigma}^2}})^{n a - 1} \over \prod_{i = 1}^{n} (1 + \| {\text{\boldsymbol{\beta}}}\| ^2 + {{{\sigma}^2}})^{1 / 2 + a}} d{{{\sigma}^2}}{\nonumber}\\ &= {B(n a, n / 2) \over M_3} {1 \over (1 + \| {\text{\boldsymbol{\beta}}}\| ^2 )^{n / 2}} {\nonumber} \end{align}\] for some \(M_3 > 0\). Then \[\begin{align} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) &\le M_4 {(1 + \| {\text{\boldsymbol{\beta}}}\| ^2 )^{n / 2} \over \big\{ \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^n} \prod_{i = 1}^{n} ( {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} ) {\nonumber} \end{align}\] for some \(M_4 > 0\).

Let \(0 < {\delta}< 1\) satisfy the condition of part (i) of Lemma 3. Let \({\varepsilon}> 0\) and \(R > 1 / {\varepsilon}\) satisfy the condition of part (ii) of Lemma 3. First, suppose that \(\| {\text{\boldsymbol{\beta}}}\| \le R\). Then since \[\begin{align} \Big\{ \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \Big\} ^n &\ge (n - p)^n \Big\{ \Big( {1 \over n - p} \sum_{j = 1}^{n - p} u_{i_j} {\delta}^2 \Big) ^{n - p} \Big\} ^{n / (n - p)} \ge (n - p)^n {\delta}^{2 n} \Big( \prod_{j = 1}^{n - p} u_{i_j} \Big) ^{n / (n - p)} {\nonumber} \end{align}\] for some \(1 \le i_1 < \dots < i_{n - p} \le n\), \[\begin{align} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) &\le M_5 \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - p)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] \label{ptrace95classp1} \end{align}\tag{1}\] for some \(M_5 > 0\). Next, suppose that \(\| {\text{\boldsymbol{\beta}}}\| > R\). Fix \(1 \le i_1 < \dots < i_p \le n\) and suppose that \[\begin{align} | y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}}| &\ge {\varepsilon}\| {\text{\boldsymbol{\beta}}}\| {\nonumber} \end{align}\] for all \(i \in \{ 1, \dots , n \} \setminus \{ i_1 , \dots , i_p \}\). Fix \(1 \le l \le p\) and \(1 \le j_1 < \dots < j_l \le p\) and suppose that for all \(j = 1, \dots , p\), we have \(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| < {\delta}\) if and only if \(j \in \{ j_1 , \dots , j_l \}\). Then \[\begin{align} \Big\{ \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \Big\} ^n &= (n - l)^n \Big[ \Big\{ {1 \over n - l} \sum_{i = 1}^{n} u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \Big\} ^{n - l} \Big] ^{n / (n - l)} {\nonumber}\\ &\ge (n - l)^n \Big[ \Big\{ {1 \over n - l} \sum_{i \in \{ 1, \dots , n \} \setminus \{ i_{j_1} , \dots , i_{j_l} \} } u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \Big\} ^{n - l} \Big] ^{n / (n - l)} {\nonumber}\\ &\ge (n - l)^n \Big[ \prod_{i \in \{ 1, \dots , n \} \setminus \{ i_{j_1} , \dots , i_{j_l} \} } \{ u_i ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \} \Big] ^{n / (n - l)} \text{.} {\nonumber} \end{align}\] Therefore, \[\begin{align} &p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) {\nonumber}\\ &\le M_6 {(1 + \| {\text{\boldsymbol{\beta}}}\| ^2 )^{n / 2} \over \big\{ \prod_{i \in \{ 1, \dots , n \} \setminus \{ i_1 , \dots , i_p \} } ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^{n / (n - l)}} {\nonumber}\\ &\quad \times {1 \over \big\{ \prod_{i \in \{ i_1 , \dots , i_p \} \setminus \{ i_{j_1} , \dots , i_{j_l} \} } ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^{n / (n - l)}} \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - l)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] {\nonumber}\\ &\le M_7 {(1 + \| {\text{\boldsymbol{\beta}}}\| ^2 )^{n / 2} / \| {\text{\boldsymbol{\beta}}}\| ^{2 (n - p) n / (n - l)} \over \big\{ \prod_{i \in \{ i_1 , \dots , i_p \} \setminus \{ i_{j_1} , \dots , i_{j_l} \} } ( y_i - {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\beta}}})^2 \big\} ^{n / (n - l)}} \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - l)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] {\nonumber}\\ &\le M_8 {1 \over \prod_{j \in \{ 1 , \dots , p \} \setminus \{ j_1 , \dots , j_l \} } \{ 1 + ( y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}})^2 \} ^{n / (n - l)}} \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - p)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] \label{ptrace95classp2} \end{align}\tag{2}\] for some \(M_6 , M_7 , M_8 > 0\). From (1 ) and (2 ), it follows that \[\begin{align} I &= \int_{\mathbb{R} ^p \times (0, \infty )^n} p( {\text{\boldsymbol{\beta}}}| {\text{\boldsymbol{u}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\beta}}}) d( {\text{\boldsymbol{\beta}}}, {\text{\boldsymbol{u}}}) {\nonumber}\\ &\le M_9 \int_{\mathbb{R} ^p \times (0, \infty )^n} 1( \| {\text{\boldsymbol{\beta}}}\| \le R) \Big( \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - p)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] \Big) d( {\text{\boldsymbol{\beta}}}, {\text{\boldsymbol{u}}}) {\nonumber}\\ &\quad + M_{10} \sum_{1 \le i_1 < \dots < i_p \le n} \int_{\mathbb{R} ^p \times (0, \infty )^n} \Big( \Big[ \sum_{l = 0}^{p} \sum_{1 \le j_1 < \dots < j_l \le p} {\prod_{j \in \{ j_1 , \dots , j_l \} } 1(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \le {\delta}) \over \prod_{j \in \{ 1 , \dots , p \} \setminus \{ j_1 , \dots , j_l \} } \{ 1 + ( y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}})^2 \} } \Big] {\nonumber}\\ &\quad \times \prod_{i = 1}^{n} \Big[ \Big\{ 1 + {1 \over {u_i}^{n / (n - p)}} \Big\} {u_i}^{1 / 2 + a - 1} e^{- b u_i / 2} \Big] \Big) d( {\text{\boldsymbol{\beta}}}, {\text{\boldsymbol{u}}}) {\nonumber} \end{align}\] for some \(M_9 , M_{10} > 0\), the right-hand side of which is finite. This completes the proof. \(\Box\)

Proof. Fix \(1 \le i_1 < \dots < i_{p + 1} \le n\) and let \({\delta}_{i_1 , \dots , i_{p + 1}} = \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} {\tilde{\text{\boldsymbol{y}}}}- \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \tilde{{\tilde{\text{\boldsymbol{y}}}}} \| / ( \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} \| + \| \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \| ) > 0\), where \({\tilde{\text{\boldsymbol{y}}}}= ( y_{i_1} , \dots , y_{i_p} )^{\top }\) and \(\tilde{{\tilde{\text{\boldsymbol{y}}}}} = ( y_{i_1} , \dots , y_{i_{p - 1}} , y_{i_{p + 1}} )^{\top }\) and \({\widetilde{\text{\boldsymbol{X}}}}= ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_p} )^{\top }\) and \(\widetilde{{\widetilde{\text{\boldsymbol{X}}}}} = ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_{p - 1}} , {\text{\boldsymbol{x}}}_{i_{p + 1}} )^{\top }\). Then for all \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\), \[\begin{align} {\delta}_{i_1 , \dots , i_{p + 1}} &\le ( \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} {\tilde{\text{\boldsymbol{y}}}}- {\text{\boldsymbol{\beta}}}\| + \| \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \tilde{{\tilde{\text{\boldsymbol{y}}}}} - {\text{\boldsymbol{\beta}}}\| ) / ( \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} \| + \| \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \| ) {\nonumber}\\ &\le ( \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} \| \| {\tilde{\text{\boldsymbol{y}}}}- {\widetilde{\text{\boldsymbol{X}}}}{\text{\boldsymbol{\beta}}}\| + \| \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \| \| \tilde{{\tilde{\text{\boldsymbol{y}}}}} - \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} {\text{\boldsymbol{\beta}}}\| ) / ( \| {\widetilde{\text{\boldsymbol{X}}}}^{- 1} \| + \| \widetilde{{\widetilde{\text{\boldsymbol{X}}}}} ^{- 1} \| ) {\nonumber}\\ &\le \| ( y_{i_1} , \dots , y_{i_{p + 1}} )^{\top } - ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_{p + 1}} )^{\top } {\text{\boldsymbol{\beta}}}\| \text{,} {\nonumber} \end{align}\] which implies that \(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {\delta}_{i_1 , \dots , i_{p + 1}} / \sqrt{p + 1}\) for some \(j = 1, \dots , p + 1\).

Let \({\delta}= ( \min_{1 \le i_1 < \dots < i_{p + 1} \le n} {\delta}_{i_1 , \dots , i_{p + 1}} ) / \sqrt{p + 1} > 0\). Fix \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\). Then for all \(1 \le i_1 < \dots < i_{p + 1} \le n\), there exists \(j = 1, \dots , p + 1\) such that \(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {\delta}\). Therefore, there exist \(1 \le i_1 < \dots < i_{n - p} \le n\) such that for all \(j = 1, \dots , n - p\), we have \(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {\delta}\). \(\Box\)

Proof. Fix \(1 \le i_1 < \dots < i_p \le n\) and let \[\begin{align} {\varepsilon}_{i_1 , \dots , i_p} &= \inf_{{\text{\boldsymbol{\beta}}}\in \{ {\widetilde{\text{\boldsymbol{\beta}}}}\in \mathbb{R} ^p | \| {\widetilde{\text{\boldsymbol{\beta}}}}\| = 1 \} } \| ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_p} )^{\top } {\text{\boldsymbol{\beta}}}\| \text{.} {\nonumber} \end{align}\] Then, since \(\| ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_p} )' {\text{\boldsymbol{\beta}}}\| > 0\) for all \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p \setminus \{ \boldsymbol{0} ^{(p)} \}\), by continuity \(0 < {\varepsilon}_{i_1 , \dots , i_p} < \infty\). For all \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p \setminus \{ \boldsymbol{0} ^{(p)} \}\), there exists \(j = 1, \dots , p\) such that \[\begin{align} {| {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \over \| {\text{\boldsymbol{\beta}}}\| } &\ge {\| ( {\text{\boldsymbol{x}}}_{i_1} , \dots , {\text{\boldsymbol{x}}}_{i_p} )^{\top } {\text{\boldsymbol{\beta}}}\| \over \sqrt{p} \| {\text{\boldsymbol{\beta}}}\| } \ge {{\varepsilon}_{i_1 , \dots , i_p} \over \sqrt{p}} \text{.} {\nonumber} \end{align}\]

Let \[\begin{align} {\varepsilon}&= \min_{1 \le i_1 < \dots < i_p \le n} {{\varepsilon}_{i_1 , \dots , i_p} \over \sqrt{p}} > 0 \text{.} {\nonumber} \end{align}\] Fix \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\). Then for all \(1 \le i_1 < \dots < i_p \le n\), there exists \(j = 1, \dots , p\) such that \(| {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {\varepsilon}\| {\text{\boldsymbol{\beta}}}\|\). Therefore, there exist \(1 \le i_1 < \dots < i_{n - p + 1} \le n\) such that for all \(j = 1, \dots , n - p + 1\), we have \(| {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {\varepsilon}\| {\text{\boldsymbol{\beta}}}\|\). \(\Box\)

Proof. First, part (i) follows from Lemma 1. Next, by Lemma 2, there exists \({{\varepsilon}}' > 0\) such that for all \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\), there are \(1 \le i_1 < \dots < i_{n - p + 1} \le n\) satisfying \(\min_{1 \le j \le n - p + 1} | {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge {{\varepsilon}}' \| {\text{\boldsymbol{\beta}}}\|\). Let \({\varepsilon}= {{\varepsilon}}' / 2 > 0\) and \(R = 2 \| {\text{\boldsymbol{y}}}\| / {{\varepsilon}}' > 0\). Then for all \({\text{\boldsymbol{\beta}}}\in \mathbb{R} ^p\) satisfyig \(\| {\text{\boldsymbol{\beta}}}\| \ge R\), there exist \(1 \le i_1 < \dots < i_{n - p + 1} \le n\) such that for all \(j = 1, \dots , n - p + 1\), we have \(| y_{i_j} - {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| \ge | {{\text{\boldsymbol{x}}}_{i_j}}^{\top } {\text{\boldsymbol{\beta}}}| - | y_{i_j} | \ge {{\varepsilon}}' \| {\text{\boldsymbol{\beta}}}\| / 2 + {{\varepsilon}}' R / 2 - | y_{i_j} | \ge {\varepsilon}\| {\text{\boldsymbol{\beta}}}\|\). This proves part (ii). \(\Box\)

3 The Case of a Proper Prior↩︎

In this section, we consider the multivariate linear regression model of [2]. We follow the notation of [3]. Let \(h \colon (0, \infty ) \to [0, \infty )\) be a normalized mixing density and suppose that \[\begin{align} &{\text{\boldsymbol{y}}}_i \sim {1 \over | {\text{\boldsymbol{{\Sigma}}}}|^{1 / 2}} f_h ( {\text{\boldsymbol{{\Sigma}}}}^{- 1 / 2} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )) {\nonumber} \end{align}\] for \(i = 1, \dots , n\), where \({\text{\boldsymbol{Y}}}= ( {\text{\boldsymbol{y}}}_1 , \dots , {\text{\boldsymbol{y}}}_n )^{\top } \in \mathbb{R} ^{n \times d}\) and \({\text{\boldsymbol{X}}}= ( {\text{\boldsymbol{x}}}_1 , \dots , {\text{\boldsymbol{x}}}_n )^{\top } \in \mathbb{R} ^{n \times p}\) are outcome and explanatory variables, \({\text{\boldsymbol{\mathcal{B}}}}\in \mathbb{R} ^{p \times d}\) and \({\text{\boldsymbol{{\Sigma}}}}> {\text{\boldsymbol{O}}}^{(d)}\) are the matrix of regression coefficients and the covariance matrix, and where \[\begin{align} f_h ( {\text{\boldsymbol{\varepsilon}}}) &= \int_{0}^{\infty } {u^{d / 2} \over (2 \pi )^{d / 2}} \exp \Big( - {u \over 2} \| {\text{\boldsymbol{\varepsilon}}}\| ^2 \Big) h(u) du \text{,} \quad {\text{\boldsymbol{\varepsilon}}}\in \mathbb{R} ^d \text{,} {\nonumber} \end{align}\] is the error density. The prior distribution is given by \[\begin{align} p( {\text{\boldsymbol{\mathcal{B}}}}| {\text{\boldsymbol{{\Sigma}}}}) p( {\text{\boldsymbol{{\Sigma}}}}) &= {\rm{N}}_{p, d} ( {\text{\boldsymbol{\mathcal{B}}}}| {\text{\boldsymbol{B}}}, {\text{\boldsymbol{A}}}, {\text{\boldsymbol{{\Sigma}}}}) {\rm{IW}}_d ( {\text{\boldsymbol{{\Sigma}}}}| \nu , {\text{\boldsymbol{C}}}) \text{,} {\nonumber} \end{align}\] where \({\text{\boldsymbol{B}}}\in \mathbb{R} ^{p \times d}\), \({\text{\boldsymbol{A}}}> {\text{\boldsymbol{O}}}^{(p)}\), \(\nu > d - 1\), and \({\text{\boldsymbol{C}}}> {\text{\boldsymbol{O}}}^{(d)}\) are arbitrary. The following data augmentation algorithm is from [2].

The following result is proved in the Supplemetary Material.

Acknowledgments↩︎

Research of the author was supported in part by JSPS KAKENHI Grant Number JP25K21163 from Japan Society for the Promotion of Science.

4 Proof of Proposition 3↩︎

Here, we prove Proposition 3.

Proof of Proposition 3. Let \(I = \int p( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}| {\text{\boldsymbol{u}}}, {\text{\boldsymbol{y}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{y}}}) d( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}})\). We want to show that \(I < \infty\). By Algorithm 3, \[\begin{align} p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{y}}}) &= \prod_{i = 1}^{n} {{u_i}^{d / 2 + a - 1} \exp [- u_i \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ] \over {\Gamma}(d / 2 + a) / \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a}} {\nonumber}\\ &= C_1 \Big[ \prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} \Big] \Big[ \prod_{i = 1}^{n} \{{u_i}^{d / 2 + a - 1} \exp (- b u_i ) \} \Big] {\nonumber}\\ &\quad \times \exp \Big[ - {1 \over 2} {\rm tr\,}\{ {\text{\boldsymbol{U}}}( {\text{\boldsymbol{Y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{Y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}})^{\top } \} \Big] {\nonumber} \end{align}\] for some \(C_1 > 0\), where \[\begin{align} {\rm tr\,}\{ {\text{\boldsymbol{U}}}( {\text{\boldsymbol{Y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{Y}}}- {\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}})^{\top } \} &= {\rm tr\,}\{ {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- 2 {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}) \} \text{.} {\nonumber} \end{align}\] Meanwhile, \[\begin{align} p( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}| {\text{\boldsymbol{u}}}, {\text{\boldsymbol{y}}}) &= C_2 {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big( - {1 \over 2} {\text{\boldsymbol{\Psi}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} \Big) {1 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {1 \over | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Omega}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}})^{\top } \Big\} {\nonumber}\\ &= {C_2 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {1 \over | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big[ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{\Psi}}}+ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}})^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}}) \} \Big] {\nonumber} \end{align}\] for some \(C_2 > 0\), where \[\begin{align} &{\text{\boldsymbol{\Psi}}}+ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}})^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{{\Gamma}}}}) = {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}- 2 {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}\text{.} {\nonumber} \end{align}\] Therefore, \[\begin{align} &p( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}| {\text{\boldsymbol{u}}}, {\text{\boldsymbol{y}}}) p( {\text{\boldsymbol{u}}}| {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{y}}}) {\nonumber}\\ &= {C_2 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {1 \over | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}- 2 {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}) \Big\} {\nonumber}\\ &\quad \times C_1 \Big[ \prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} \Big] \Big[ \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i ) \} \Big] {\nonumber}\\ &\quad \times {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- 2 {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}{\text{\boldsymbol{\mathcal{B}}}}) \Big\} {\nonumber}\\ &= {C_2 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {1 \over | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \Big\} {\nonumber}\\ &\quad \times C_1 \Big[ \prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} \Big] \Big[ \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i ) \} \Big] {\nonumber}\\ &\quad \times {\rm etr\,}\Big[ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{\mathcal{B}}}}^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} ) {\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- 2 ( {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} ) {\text{\boldsymbol{\mathcal{B}}}}\} \Big] \text{,} {\nonumber} \end{align}\] where \[\begin{align} &{\text{\boldsymbol{\mathcal{B}}}}^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} ) {\text{\boldsymbol{\mathcal{B}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- 2 ( {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} ) {\text{\boldsymbol{\mathcal{B}}}}{\nonumber}\\ &= ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } {\text{\boldsymbol{W}}}( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) + {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- {\text{\boldsymbol{V}}}^{\top } {\text{\boldsymbol{W}}}{\text{\boldsymbol{V}}}{\nonumber} \end{align}\] for \({\text{\boldsymbol{W}}}= {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1}\) and \({\text{\boldsymbol{V}}}= ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}})\). Thus, \[\begin{align} I &= C_1 C_2 \int \Big( {1 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \Big\} {1 \over | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} |^{d / 2}} {\nonumber}\\ &\quad \times {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- {\text{\boldsymbol{V}}}^{\top } {\text{\boldsymbol{W}}}{\text{\boldsymbol{V}}}) \Big\} {\nonumber}\\ &\quad \times \Big[ \prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} \Big] \Big[ \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i ) \} \Big] {\nonumber}\\ &\quad \times (2 \pi )^{p d / 2} {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} |^{d / 2} \over (2 \pi )^{p d / 2} | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } {\text{\boldsymbol{W}}}( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) \Big\} \Big) d( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}}) \text{.} \label{pproperp1} \end{align}\tag{3}\]

We have \[\begin{align} &\prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} {\nonumber}\\ &\le \Big\{ \prod_{i = 1}^{n} ( {{\text{\boldsymbol{y}}}_i}^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{y}}}_i + {{\text{\boldsymbol{x}}}_i}^{\top } {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i + b) \Big\} ^{d / 2 + a} {\nonumber}\\ &= \Big[ \prod_{i = 1}^{n} \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{y}}}_i {{\text{\boldsymbol{y}}}_i}^{\top } ) + {\rm tr\,}( {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i {{\text{\boldsymbol{x}}}_i}^{\top } ) + b \} \Big] ^{d / 2 + a} {\nonumber}\\ &\le M_3 \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) + {\rm tr\,}( {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } ) + 1 \} ^{n (d / 2 + a)} {\nonumber}\\ &\le M_4 [ \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} + \{ {\rm tr\,}( {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } ) \} ^{n (d / 2 + a)} + 1] {\nonumber} \end{align}\] for some \(M_3 , M_4 > 0\), where \[\begin{align} \{ {\rm tr\,}( {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } ) \} ^{n (d / 2 + a)} &\le [ 2 {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} + 2 {\rm tr\,}( {\text{\boldsymbol{V}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } )]^{n (d / 2 + a)} {\nonumber}\\ &\le M_5 [ 2 {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} ]^{n (d / 2 + a)} + M_5 \{ 2 {\rm tr\,}( {\text{\boldsymbol{V}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } ) \} ^{n (d / 2 + a)} {\nonumber} \end{align}\] for some \(M_5 > 0\). Since \({\text{\boldsymbol{I}}}\le M_6 {\text{\boldsymbol{A}}}^{- 1} {\nonumber}\le M_6 {\text{\boldsymbol{{\Omega}}}}^{- 1} {\nonumber}\le M_6 {\text{\boldsymbol{W}}}\) for some \(M_6 > 0\), \[\begin{align} {\rm tr\,}( {\text{\boldsymbol{V}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } ) &= {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } {\text{\boldsymbol{V}}}) \le M_6 {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } {\text{\boldsymbol{W}}}{\text{\boldsymbol{V}}}) {\nonumber}\\ &= M_6 {\rm tr\,}\{ {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}})^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \} \text{.} {\nonumber} \end{align}\] Therefore, \[\begin{align} {\rm tr\,}( {\text{\boldsymbol{V}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{V}}}^{\top } ) &\le 2 M_6 {\rm tr\,}[ {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}})^{- 1} {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Omega}}}}{\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}\} ] {\nonumber}\\ &\le 2 M_6 {\rm tr\,}[ {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}})^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{A}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}) \} ] {\nonumber}\\ &\le 2 M_6 {\rm tr\,}\{ {\text{\boldsymbol{{\Sigma}}}}^{- 1} (3 {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ 2 {\text{\boldsymbol{B}}}^{\top } {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}) \} {\nonumber}\\ &\le M_7 [ \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} {\rm tr\,}( {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}) + {\rm tr\,}{\text{\boldsymbol{{\Sigma}}}}^{- 1} ] \le M_8 \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} \{ ( u_1 + \dots + u_n ) + 1 \} {\nonumber} \end{align}\] for some \(M_7 , M_8 > 0\). Thus, \[\begin{align} &\{ {\rm tr\,}( {\text{\boldsymbol{\mathcal{B}}}}{\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{\mathcal{B}}}}^{\top } ) \} ^{n (d / 2 + a)} {\nonumber}\\ &\le M_5 [ 2 {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} ]^{n (d / 2 + a)} + M_9 \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} \{ ( u_1 + \dots + u_n ) + 1 \} ^{n (d / 2 + a)} {\nonumber} \end{align}\] for some \(M_9 > 0\). Hence, \[\begin{align} &\prod_{i = 1}^{n} \{ ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i )^{\top } {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{y}}}_i - {\text{\boldsymbol{\mathcal{B}}}}^{\top } {\text{\boldsymbol{x}}}_i ) / 2 + b \} ^{d / 2 + a} {\nonumber}\\ &\le M_{10} ([ {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} ]^{n (d / 2 + a)} + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} \{ 1 + ( u_1 + \dots + u_n )^{n (d / 2 + a)} \} + 1) {\nonumber}\\ &\le M_{10} (1 + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} + [ {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} ]^{n (d / 2 + a)} ) \{ 1 + ( u_1 + \dots + u_n )^{n (d / 2 + a)} \} \label{pproperp2} \end{align}\tag{4}\] for some \(M_{10} > 0\).

By (3 ) and (4 ), \[\begin{align} I &\le M_{11} \int \Big\{ {1 \over | {\text{\boldsymbol{{\Omega}}}}|^{d / 2}} {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \Big\} {1 \over | {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} |^{d / 2}} {\nonumber}\\ &\quad \times {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- {\text{\boldsymbol{V}}}^{\top } {\text{\boldsymbol{W}}}{\text{\boldsymbol{V}}}) \Big\} (1 + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} + [ {\rm tr\,}\{ ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } \} ]^{n (d / 2 + a)} ) {\nonumber}\\ &\quad \times \Big[ \{ 1 + ( u_1 + \dots + u_n )^{n (d / 2 + a)} \} \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i ) \} \Big] {\nonumber}\\ &\quad \times {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} |^{d / 2} \over (2 \pi )^{p d / 2} | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } {\text{\boldsymbol{W}}}( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) \Big\} \Big\} d( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}}) {\nonumber}\\ &\le M_{12} \int \Big( {| {\text{\boldsymbol{\Psi}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big[ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}{\nonumber}\\ &\quad - ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}})^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \} \Big] {\nonumber}\\ &\quad \times [1 + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} ]^{n (d / 2 + a)} \Big[ \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i / 2) \} \Big] {\nonumber}\\ &\quad \times {| {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} |^{d / 2} \over (2 \pi )^{p d / 2} | {\text{\boldsymbol{{\Sigma}}}}|^{p / 2}} {\rm etr\,}\Big\{ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}})^{\top } ( {\text{\boldsymbol{W}}}/ 2) ( {\text{\boldsymbol{\mathcal{B}}}}- {\text{\boldsymbol{V}}}) \Big\} \Big) d( {\text{\boldsymbol{\mathcal{B}}}}, {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}}) {\nonumber}\\ &\le M_{13} \int \Big( {| {\text{\boldsymbol{C}}}^{- 1} + {\text{\boldsymbol{B}}}^{\top } {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}|^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big[ - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} \{ {\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}{\nonumber}\\ &\quad - ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}})^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) \} \Big] {\nonumber}\\ &\quad \times [1 + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} ]^{n (d / 2 + a)} \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i / 2) \} \Big) d( {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}}) {\nonumber} \end{align}\] for some \(M_{11} , M_{12} , M_{13} > 0\). Note that \[\begin{align} &{\text{\boldsymbol{\Psi}}}+ {\text{\boldsymbol{{\Gamma}}}}^{\top } {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}})^{\top } ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} )^{- 1} ( {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{{\Omega}}}}^{- 1} {\text{\boldsymbol{{\Gamma}}}}) {\nonumber}\\ & = {\text{\boldsymbol{C}}}^{- 1} + {\text{\boldsymbol{B}}}^{\top } {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}- (2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}})^{\top } (2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{A}}}^{- 1} )^{- 1} (2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}) \ge {\text{\boldsymbol{C}}}{\nonumber} \end{align}\] since \[\begin{align} &\begin{pmatrix} {\text{\boldsymbol{B}}}^{\top } {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}+ 2 {\text{\boldsymbol{Y}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}& (2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}})^{\top } \\ 2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{Y}}}+ {\text{\boldsymbol{A}}}^{- 1} {\text{\boldsymbol{B}}}& 2 {\text{\boldsymbol{X}}}^{\top } {\text{\boldsymbol{U}}}{\text{\boldsymbol{X}}}+ {\text{\boldsymbol{A}}}^{- 1} \end{pmatrix} {\nonumber}\\ &= \begin{pmatrix} {\text{\boldsymbol{B}}}^{\top } & {\text{\boldsymbol{O}}}^{(d, p)} \\ {\text{\boldsymbol{O}}}^{(p)} & {\text{\boldsymbol{I}}}^{(p)} \end{pmatrix} \begin{pmatrix} {\text{\boldsymbol{A}}}^{- 1} & {\text{\boldsymbol{A}}}^{- 1} \\ {\text{\boldsymbol{A}}}^{- 1} & {\text{\boldsymbol{A}}}^{- 1} \end{pmatrix} \begin{pmatrix} {\text{\boldsymbol{B}}}& {\text{\boldsymbol{O}}}^{(p)} \\ {\text{\boldsymbol{O}}}^{(p, d)} & {\text{\boldsymbol{I}}}^{(p)} \end{pmatrix} + 2 \begin{pmatrix} {\text{\boldsymbol{Y}}}^{\top } & {\text{\boldsymbol{O}}}^{(d, n)} \\ {\text{\boldsymbol{O}}}^{(p, n)} & {\text{\boldsymbol{X}}}^{\top } \end{pmatrix} \begin{pmatrix} {\text{\boldsymbol{U}}}& {\text{\boldsymbol{U}}}\\ {\text{\boldsymbol{U}}}& {\text{\boldsymbol{U}}}\end{pmatrix} \begin{pmatrix} {\text{\boldsymbol{Y}}}& {\text{\boldsymbol{O}}}^{(n, p)} \\ {\text{\boldsymbol{O}}}^{(n, d)} & {\text{\boldsymbol{X}}}\end{pmatrix} \ge {\text{\boldsymbol{O}}}^{(p + d)} \text{.} {\nonumber} \end{align}\] Then \[\begin{align} I &\le M_{14} \int \Big( {|(1 + {\rm tr\,}{\text{\boldsymbol{U}}}) {\text{\boldsymbol{I}}}^{(d)} |^{(n + \nu ) / 2} \over | {\text{\boldsymbol{{\Sigma}}}}|^{(n + \nu + d + 1) / 2}} {\rm etr\,}\Big( - {1 \over 2} {\text{\boldsymbol{{\Sigma}}}}^{- 1} {\text{\boldsymbol{C}}}\Big) [1 + \{ {\rm tr\,}( {\text{\boldsymbol{{\Sigma}}}}^{- 1} ) \} ^{n (d / 2 + a)} ]^{n (d / 2 + a)} {\nonumber}\\ &\quad \times \prod_{i = 1}^{n} \{ {u_i}^{d / 2 + a - 1} \exp (- b u_i / 2) \} \Big) d( {\text{\boldsymbol{{\Sigma}}}}, {\text{\boldsymbol{u}}}) {\nonumber} \end{align}\] for some \(M_{14} > 0\). Since the right-hand side of the above inequality is finite, the result follows. \(\Box\)

References↩︎