On robustness of the relative belief ratio and the strength of its evidence with respect to the geometric contamination prior

Abstract

The relative belief ratio becomes a widespread tool in many hypothesis testing problems. It measures the statistical evidence that a given statement is true based on a combination of data, model and prior. Additionally, a measure of the strength is used to calibrate its value. In this paper, robustness of the relative belief ratio and its strength to the choice of the prior is studied. Specifically, the Gâteaux derivative is used to measure their sensitivity when the geometric contaminated prior is used. Examples are presented to illustrate the results.

Appendices

Appendix A Proof of Proposition 1

From (8),

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{RB_{\epsilon }(\theta |x)-RB(\theta |x)}{\epsilon }&=RB(\theta |x)\lim _{\epsilon \rightarrow 0} \frac{\frac{m(x)}{m_{\epsilon }(x)}-1}{\epsilon }\nonumber \\&=RB(\theta |x)\lim _{\epsilon \rightarrow 0} \frac{m(x)-m_{\epsilon }(x)}{\epsilon m_{\epsilon }(x)} \end{aligned}$$

(9)

Using L’Hôpital’s rule, we get

$$\begin{aligned} (9)&=RB(\theta |x)\lim _{\epsilon \rightarrow 0} \frac{-\frac{d}{d\epsilon }m_{\epsilon }(x)}{\epsilon \frac{d}{d\epsilon }m_{\epsilon }(x)+m_{\epsilon }(x)}. \end{aligned}$$

From (6),

$$\begin{aligned} \frac{d}{d\epsilon }m_{\epsilon }(x)=\int _{\Theta }\frac{d}{d\epsilon }\left\{ c(\epsilon )\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta )\right\} f(x|\theta )d\theta . \end{aligned}$$

Let $h(\epsilon )=c(\epsilon )\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta )$. Then

$$\begin{aligned} \log h(\epsilon )=\log c(\epsilon )+(1-\epsilon )\log \pi (\theta )+\epsilon \log q(\theta ). \end{aligned}$$

(10)

It follows that

$$\begin{aligned} \frac{d}{d\epsilon }h(\epsilon )&=\left[ \frac{\frac{d}{d\epsilon }c(\epsilon )}{c(\epsilon )}+ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] c(\epsilon )\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta ). \end{aligned}$$

Thus,

$$\begin{aligned} \frac{d}{d\epsilon }m_{\epsilon }(x)&=\int _{\Theta } \frac{d}{d\epsilon }c(\epsilon )\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta ) f(x|\theta )d\theta \\+ & {} \int _{\Theta } \log \left( \frac{q(\theta )}{\pi (\theta )}\right) c(\epsilon ) \pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta ) f(x|\theta )d\theta . \end{aligned}$$

As $\epsilon \rightarrow 0,$ we have

$$\begin{aligned} \frac{d}{d\epsilon }m_{\epsilon }(x)&\rightarrow \int _{\Theta } \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \pi (\theta ) f(x|\theta )d\theta \\&= m(x)\int _{\Theta } \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \frac{\pi (\theta ) f(x|\theta )}{m(x)}d\theta \\&= m(x) E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] . \end{aligned}$$

Thus,

$$\begin{aligned} (9)= RB(\theta |x)\frac{m(x) E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] }{m(x)}= RB(\theta |x)E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] . \end{aligned}$$

Appendix B Proof of Proposition 2

Let $\Pi _{\epsilon }(\cdot |x)$ be the posterior cumulative distribution function with posterior density $\pi _{\epsilon }(\cdot |x)$. By (7) and (8),

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\Pi _{\epsilon }\left( RB_{\epsilon }(\theta |x)\le RB_{\epsilon }(\theta _{0}|x)|x\right) -\Pi \left( RB(\theta |x)\le RB(\theta _{0}|x)|x\right) }{\epsilon }\right\rbrace \nonumber \\ &=\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\Pi _{\epsilon }\left( RB(\theta |x)\frac{m(x)}{m_{\epsilon }(x)}\le RB(\theta _{0}|x)\frac{m(x)}{m_{\epsilon }(x)}|x\right) -\Pi \left( RB(\theta |x)\frac{m(x)}{m_{\epsilon }(x)}\le RB(\theta _{0}|x)\frac{m(x)}{m_{\epsilon }(x)}|x\right) }{\epsilon }\right\rbrace \nonumber \\ &=\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\Pi _{\epsilon }\left( RB(\theta |x)\le RB(\theta _{0}|x)|x\right) -\Pi \left( RB(\theta |x)\le RB(\theta _{0}|x)|x\right) }{\epsilon }\right\rbrace \nonumber \\ &=\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\Pi _{\epsilon }\left( \frac{f_{\theta }(x)}{m(x)}\le \frac{f_{\theta _{0}}(x)}{m(x)}|x\right) -\Pi \left( \frac{f_{\theta }(x)}{m(x)}\le \frac{f_{\theta _{0}}(x)}{m(x)}|x\right) }{\epsilon }\right\rbrace \nonumber \\ &=\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\Pi _{\epsilon }(f_{\theta }(x)\le f_{\theta _{0}}(x)|x) -\Pi \left( f_{\theta }(x)\le f_{\theta _{0}}(x)|x\right) }{\epsilon }\right\rbrace \nonumber \\ &=I. \end{aligned}$$

(11)

Observe that,

$$\begin{aligned} \Pi _{\epsilon }\left( f_{\theta }(x)\le f_{\theta _{0}}(x)|x\right) =\int _{{\mathscr {M}}}\frac{c(\epsilon )\pi ^{(1-\epsilon )}(\theta )q^{\epsilon }(\theta )f(x|\theta )}{m_{\epsilon }(x)}\, d\theta . \end{aligned}$$

Therefore,

$$\begin{aligned} I=\lim _{\epsilon \rightarrow 0}\left\lbrace \frac{\int _{{\mathscr {M}}}\frac{c(\epsilon )\pi ^{(1-\epsilon )}(\theta )q^{\epsilon }(\theta )f(x|\theta )}{m_{\epsilon }(x)}\, d\theta -\int _{{\mathscr {M}}}\frac{\pi (\theta )f(x|\theta )}{m(x)}\, d\theta }{\epsilon }\right \rbrace . \end{aligned}$$

Using L’Hôpital’s rule,

$$\begin{aligned} I&=\lim _{\epsilon \rightarrow 0}\frac{\frac{d}{d\epsilon }\left[ \int _{{\mathscr {M}}}\frac{c(\epsilon )\pi ^{(1-\epsilon )}(\theta )q^{\epsilon }(\theta )f(x|\theta )}{m_{\epsilon }(x)}\, d\theta \right] -0}{1}\\&=\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}} \frac{d}{d\epsilon }\left[ \frac{c(\epsilon )\pi ^{(1-\epsilon )}(\theta )q^{\epsilon }(\theta )f(x|\theta )}{m_{\epsilon }(x)}\right] \, d\theta \\&=\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}} f(x|\theta )\frac{d}{d\epsilon }\left[ \frac{h(\epsilon )}{m_{\epsilon }(x)}\right] \, d\theta , \end{aligned}$$

where $h(\epsilon )=c(\epsilon )\pi ^{(1-\epsilon )}(\theta )q^{\epsilon }(\theta )$. Hence,

$$\begin{aligned} I&=\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}} f(x|\theta ) \frac{m_{\epsilon }(x)\frac{d}{d\epsilon }h(\epsilon )-h(\epsilon )\frac{d}{d\epsilon }m_{\epsilon }(x)}{(m_{\epsilon }(x))^{2}} \, d\theta \\&=\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}} f(x|\theta )\frac{\frac{d}{d\epsilon }h(\epsilon )}{m_{\epsilon }(x)}d\theta -\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}}f(x|\theta )\frac{h(\epsilon )\frac{d}{d\epsilon }m_{\epsilon }(x)}{(m_{\epsilon }(x))^{2}} d\theta . \end{aligned}$$

Applying $\frac{d}{d\epsilon }h(\epsilon )$ given by (10) in the above equation gives

$$\begin{aligned} I&=\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}}f(x|\theta ) \frac{\left[ \frac{\frac{d}{d\epsilon }c(\epsilon )}{c(\epsilon )}+ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] c(\epsilon )\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta )}{m_{\epsilon }(x)}\,d\theta \\&\quad -\lim _{\epsilon \rightarrow 0}\int _{{\mathscr {M}}}f(x|\theta ) \frac{h(\epsilon )\frac{d}{d\epsilon }m_{\epsilon }(x) }{(m_{\epsilon }(x))^{2}}\,d\theta \\&=I_1-I_2. \end{aligned}$$

After some simplification, we obtain

$$\begin{aligned}&I_1 =\lim _{\epsilon \rightarrow 0}\left\{ \int _{{\mathscr {M}}}\frac{f(x|\theta )}{m_{\epsilon }(x)}\pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta )\frac{d}{d\epsilon }c(\epsilon )\,d\theta \right. \\&\quad \left. +\int _{{\mathscr {M}}}\frac{f(x|\theta )}{m_{\epsilon }(x)} \log \left( \frac{q(\theta )}{\pi (\theta )}\right) c(\epsilon ) \pi ^{1-\epsilon }(\theta )q^{\epsilon }(\theta )\,d\theta \right\} . \end{aligned}$$

Letting $\epsilon \rightarrow 0$ makes

$$\begin{aligned} I_1\rightarrow \int _{{\mathscr {M}}}\log \left( \frac{q(\theta )}{\pi (\theta )}\right) \frac{f(x|\theta )\pi (\theta )}{m(x)}\,d\theta =\int _{{\mathscr {M}}}\log \left( \frac{q(\theta )}{\pi (\theta )}\right) \pi (\theta |x)\,d\theta . \end{aligned}$$

On the other hand, since $h(\epsilon )\rightarrow \pi (\theta )$ and $\frac{d}{d\epsilon }m_{\epsilon }(x)\rightarrow m(x) E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right]$ as $\epsilon \rightarrow 0$, we have

$$\begin{aligned} I_2&\rightarrow \int _{{\mathscr {M}}}f(x|\theta )\frac{\pi (\theta )m(x) E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] }{(m(x))^2}\, d\theta \\&=E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] \int _{{\mathscr {M}}}\frac{f(x|\theta )\pi (\theta )}{m(x)}\,d\theta \\&= E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] \Pi \left( f_{\theta }(x)\le f_{\theta _{0}}(x)|x\right) \\&= E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] \Pi \left( RB(\theta |x)\le RB(\theta _{0}|x)|x\right) . \end{aligned}$$

Thus, as $\epsilon \rightarrow 0$,

$$\begin{aligned} I&\rightarrow \int _{{\mathscr {M}}}\log \left( \frac{q(\theta )}{\pi (\theta )}\right) \pi (\theta |x)\, d\theta -E_{\pi (\theta |x)}\left[ \log \left( \frac{q(\theta )}{\pi (\theta )}\right) \right] \Pi \left( RB(\theta |x)\le RB(\theta _{0}|x)|x\right) . \end{aligned}$$