Last updated: 2017-12-21
Code version: 6e42447
We are considering the normal means problem with heteroskedastic noise. Furthermore, The noise are not independent, with unknown pairwise correlation \(\rho_{ij}\).
\[ \begin{array}{rcl} \hat\beta_j |\beta_j, \hat s_j &=& \beta_j + \hat s_j z_j \ ;\\ z_j &\sim& N(0, 1) \text{ marginally} \ ;\\ z_j &:& \text{correlated} \ . \end{array} \]
Under ash framework, a prior is put on exchangeable \(\beta_j\):
\[ g(\beta_j) = \sum_k \pi_k g_k(\beta_j)\ . \]
According to previous exploration, correlated standard normal \(z\) scores can be seen as iid from a density composed of Gaussian derivatives.
\[ f_0(z_j) = \sum_{l=0}^\infty w_l \varphi^{(l)}(z_j) \ . \] By change of variables, the likelihood
\[ p(\hat\beta_j | \beta_j, \hat s_j) = \frac{1}{\hat s_j}f_0\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) = \sum_l w_l \frac{1}{\hat s_j}\varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) \ . \]
Thus for each pair of observation \((\hat\beta_j, \hat s_j)\), the likelihood becomes
\[
\begin{array}{rcl}
f(\hat\beta_j|\hat s_j) &=& \displaystyle\int p(\hat\beta_j | \beta_j, \hat s_j)g(\beta_j)d\beta_j\\
&=&\displaystyle\int \sum_l w_l \frac{1}{\hat s_j}\varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) \sum_k \pi_k g_k(\beta_j)d\beta_j\\
&=& \displaystyle \sum_k \sum_l \pi_k w_l
\int\frac{1}{\hat s_j}
\varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)
g_k(\beta_j)d\beta_j\\
&:=& \displaystyle \sum_k \sum_l \pi_k w_l
f_{jkl} \ .
\end{array}
\] Hence, it’s all boiled down to calculate \(f_{jkl}\), which is the convolution of a Gaussian derivative \(\varphi^{(l)}\) and a prior component \(g_k\), with some change of variables manipulation. Note that in usual ASH settings, \(g_k\) is either uniform or normal, both of which can be handled without conceptual difficulty.
If \(g_k\) is a uniform, the convolution of a Gaussian derivative and a uniform is just another Gaussian derivative in a lower order, such as
\[ \varphi^{(l)} * \text{Unif} \leadsto \varphi^{(l-1)} \ . \] On the other hand, if \(g_k\) is a normal, we can use a fact about convolution that
\[ \begin{array}{rcl} && \frac{d}{dx}(f*g) = \frac{df}{dx} * g = f * \frac{dg}{dx}\\ &\Rightarrow& \varphi^{(l)} * N(\mu, \sigma^2) = \left(\varphi * N(\mu, \sigma^2)\right)^{(l)} \leadsto \tilde\varphi^{(l)} \ . \end{array} \] Because the convolution of \(\varphi\), a Gaussian, and another Gaussian is still a Gaussian, the convolution of a Gaussian derivative and a normal gives another Gaussian derivative.
With \(f_{jkl}\) computed, the goal is then to maximize the joint likeliood of the observation \(\left\{\left(\hat\beta_1, \hat s_1\right), \cdots,\left(\hat\beta_n, \hat s_n\right) \right\}\) which is \[ \max\limits_{\pi, w}\prod_j f(\hat\beta_j|\hat s_j) = \prod_j \left(\displaystyle \sum_k \sum_l \pi_k w_l f_{jkl}\right) \ , \] subject to reasonable, well designed constraints on \(\pi_k\) and especially \(w_l\).
The exact form of \(f_{jkl}\) can be derived analytically. Below is a method using characteristic functions or Fourier transforms.
Let \(\zeta_X(t) = \zeta_F(t) =\zeta_f(t) := E[e^{itX}]\) be the characteristic function of the random variable \(X\sim dF = f\). Either the random variable, its distribution, or its density function can be put in the subscript, depending on the circumstances.
On the other hand, the characteristic function of a random variable is also closely related to the Fourier transform of its density. In particular,
\[ \zeta_f(t) = E[e^{itX}] = \int e^{itx}dF(x) = \int e^{itx}f(x)dx := \mathscr{F}_f(t) \ . \] Note that in usual definition, the Fourier transform is given by \[ \hat f(\xi) := \int f(x)e^{-2\pi ix\xi}dx \ , \] with a normalizing factor \(2\pi\) and a negative sign. However, even defined in different ways, \(\mathscr{F}_f(t)\) carries over many nice properties of \(\hat f(\xi)\), for example, as we’ll show,
\[ \mathscr{F}_{f^{(m)}}(t) = (-it)^m\mathscr{F}_f(t) \ , \]
where \(f^{(m)}\) is the \(m^\text{th}\) derivative of \(f\).
Also, \[ \mathscr{F}_{f*g}(t) = \mathscr{F}_f(t)\mathscr{F}_g(t) \ , \] where \(*\) stands for convolution.
With these properties, the Fourier transform tool could be very useful when dealing with Gaussian derivatives and their convolution with other (density) functions.
Under this definition, the inversion formula for the characteristic functions, also known as the inverse Fourier transform, is
\[ f(x) = \frac1{2\pi} \int e^{-itx}\zeta_f(t)dt = \frac{1}{2\pi}\int e^{-itx}\mathscr{F}_f(t)dt \ . \]
Then the characteristic function of \(\hat\beta_j | \hat s_j\)
\[ \zeta_{\hat\beta_j|\hat s_j}(t) = E\left[e^{it\hat\beta_j}\mid\hat s_j\right] = E\left[e^{it(\beta_j + \hat s_j z_j)}\mid\hat s_j\right] = E\left[e^{it\beta_j}\right]E\left[e^{it\hat s_jz_j}\mid\hat s_j\right] = \zeta_{\beta}(t)\zeta_{f_0}\left(\hat s_jt\right) \ . \]
Let’s take care of \(\zeta_{\beta}(t)\) and \(\zeta_{f_0}\left(\hat s_jt\right)\) one by one. Note that
\[ \begin{array}{rrcl} &\beta_j &\sim& \sum_k\pi_kg_k \\ \Rightarrow & \zeta_{\beta}(t) &=& \displaystyle\int e^{it\beta_j}p(\beta_j)d\beta_j = \int e^{it\beta_j}\sum_k\pi_kg_k(\beta_j)d\beta_j =\sum_k\pi_k\int e^{it\beta_j}g_k(\beta_j)d\beta_j \\ &&=& \sum_k\pi_k\zeta_{g_k}(t) \ . \end{array} \]
Meanwhile, using the fact that Gaussian derivatives should be absolutely integrable,
\[ \begin{array}{rrcl} &f_0(z_j) &=& \sum_l w_l \varphi^{(l)}(z_j) \\ \Rightarrow & \zeta_{f_0}(t) &=& \displaystyle \int e^{itz_j}f_0(z_j)dz_j = \int e^{itz_j}\sum_lw_l\varphi^{(l)}(z_j)dz_j = \sum_lw_l\int e^{itz_j}\varphi^{(l)}(z_j)dz_j\\ &&=& \sum_l w_l \mathscr{F}_{\varphi^{(l)}}(t) \ . \end{array} \] Each Fourier transform of gaussian derivatives
\[ \begin{array}{rcl} \mathscr{F}_{\varphi^{(l)}}(t) &=& \displaystyle\int e^{itz_j}\varphi^{(l)}(z_j)dz_j\\ &=& \displaystyle e^{itz_j}\varphi^{(l-1)}(z_j)|_{-\infty}^\infty - \int it e^{itz_j}\varphi^{(l-1)}(z_j)dz_j\\ &=&\displaystyle 0-it\int e^{itz_j}\varphi^{(l-1)}(z_j)dz_j\\ &=&\displaystyle -it\int e^{itz_j}\varphi^{(l-1)}(z_j)dz_j\\ &=&\displaystyle \cdots\\ &=&\displaystyle (-it)^l \int e^{itz_j}\varphi(z_j)dz_j\\ &=&(-it)^l \mathscr{F}_{\varphi}(t)\\ &=&(-it)^l \zeta_\varphi(t)\\ &=&\displaystyle (-it)^l e^{-\frac12t^2} \ . \end{array} \] Thus,
\[ \begin{array}{rrcl} &\zeta_{\hat\beta_j|\hat s_j}(t) &=&\zeta_{\beta}(t)\zeta_{f_0}(\hat s_jt)\\ &&=&\left(\sum_k\pi_k\zeta_{g_k}(t)\right) \left(\sum_lw_l(-i\hat s_jt)^l \zeta_\varphi(\hat s_jt)\right)\\ &&=&\sum_k\sum_l\pi_kw_l(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)\\ &&=&\sum_k\sum_l\pi_kw_l(-i\hat s_jt)^l \zeta_{g_k}(t) e^{-\frac12\hat s_j^2t^2}\\ \Rightarrow & f(\hat\beta_j|\hat s_j) &=&\displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}\zeta_{\hat\beta_j|\hat s_j}(t)dt\\ &&=&\displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}\left(\sum_k\sum_l\pi_kw_l(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)\right)dt\\ &&=&\displaystyle\sum_k\sum_l\pi_kw_l \frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)dt\\ &&:=& \sum_k\sum_l\pi_kw_lf_{jkl} \ . \end{array} \] It is essentially the equivalent expression of \(f_{jkl}\) as
\[ \begin{array}{rcl} f_{jkl} &=& \displaystyle \int\frac{1}{\hat s_j} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) g_k(\beta_j)d\beta_j\\ &=& \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)dt \ . \end{array} \]
If the prior of \(\beta\) is a mixture of uniforms, \(\beta \sim g = \sum_k\pi_kg_k\) where \(g_k = \text{Unif}\left[a_k, b_k\right]\),
\[ \begin{array}{rrcl} &\zeta_{g_k}(t) &=& \displaystyle\frac{e^{itb_k} - e^{ita_k}}{it(b_k - a_k)}\\ \Rightarrow &\zeta_{g_k}(t)\zeta_\varphi(\hat s_jt) &=& \displaystyle\frac{e^{itb_k} - e^{ita_k}}{it(b_k - a_k)} e^{-\frac12\hat s_j^2t^2}\\ &&=& \displaystyle\frac{e^{itb_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)} -\displaystyle\frac{e^{ita_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}\\ \Rightarrow & f_{jkl} &=& \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)dt\\ &&=& \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l\displaystyle\frac{e^{itb_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}dt - \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l\displaystyle\frac{e^{ita_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}dt \ . \end{array} \]
It looks pretty complicated, but if we take advantage of usual Fourier transform tricks things get clearer.
Let \(\varphi_{\mu, \sigma^2}\) denote the density function of \(N(\mu, \sigma^2)\),
\[ \begin{array}{rl} &\varphi_{\mu, \sigma^2}(z) = \frac1\sigma\varphi\left(\frac{z - \mu}{\sigma}\right)\\ \Rightarrow & \varphi_{\mu, \sigma^2}^{(m)}(z) = \frac1{\sigma^{m+1}}\varphi^{(m)}\left(\frac{z - \mu}{\sigma}\right) \\ \Rightarrow & \zeta_{\varphi_{\mu, \sigma^2}}(t) = e^{it\mu - \frac12\sigma^2t^2} \ . \end{array} \]
Take the first part of \(f_{jkl}\), \[ \begin{array}{rrcl} & (-i\hat s_jt)^l\displaystyle\frac{e^{itb_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)} &=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k}(-it)^{l-1}\zeta_{\varphi_{b_k, \hat s_j^2}}(t)\\ &&=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k}(-it)^{l-1}\mathscr{F}_{\varphi_{b_k, \hat s_j^2}}(t)\\ &&=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k}\mathscr{F}_{\varphi_{b_k, \hat s_j^2}^{(l-1)}}(t)\\ \Rightarrow & \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l\displaystyle\frac{e^{itb_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}dt &=& -\displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j} \frac{\hat s_j^l}{b_k - a_k}\mathscr{F}_{\varphi_{b_k, \hat s_j^2}^{(l-1)}}(t)dt \\ &&=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k} \frac1{2\pi} \int e^{-it\hat\beta_j} \mathscr{F}_{\varphi_{b_k, \hat s_j^2}^{(l-1)}}(t) dt\\ &&=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k} \varphi_{b_k, \hat s_j^2}^{(l-1)}\left(\hat\beta_j\right)\\ &&=&\displaystyle -\frac{\hat s_j^l}{b_k - a_k} \frac1{\hat s_j^{l}}\varphi^{(l-1)}\left(\frac{\hat\beta_j- b_k}{\hat s_j}\right)\\ &&=&\displaystyle - \frac{\varphi^{(l-1)}\left(\frac{\hat\beta_j- b_k}{\hat s_j}\right)}{b_k-a_k} \ . \end{array} \] Therefore,
\[ \begin{array}{rcl} f_{jkl} &=& \displaystyle \frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l\displaystyle\frac{e^{itb_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}dt - \frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l\displaystyle\frac{e^{ita_k-\frac12\hat s_j^2t^2}}{it(b_k - a_k)}dt\\ &=& \displaystyle \left(- \frac{\varphi^{(l-1)}\left(\frac{\hat\beta_j- b_k}{\hat s_j}\right)}{b_k-a_k}\right) - \left(- \frac{\varphi^{(l-1)}\left(\frac{\hat\beta_j- a_k}{\hat s_j}\right)}{b_k-a_k}\right)\\ &=&\displaystyle \frac{\varphi^{(l-1)}\left(\frac{\hat\beta_j- a_k}{\hat s_j}\right) - \varphi^{(l-1)}\left(\frac{\hat\beta_j- b_k}{\hat s_j}\right)}{b_k-a_k} \end{array} \]
Likewise, if the prior of \(\beta\) is a mixture of normals, \(\beta \sim g = \sum_k\pi_kg_k\) where \(g_k = N(\mu_k, \sigma_k^2)\),
\[ \begin{array}{rrcl} &\zeta_{g_k}(t) &=& \displaystyle e^{it\mu_k - \frac12\sigma_k^2t^2}\\ \Rightarrow &\zeta_{g_k}(t)\zeta_\varphi(\hat s_jt) &=& e^{it\mu_k - \frac12\sigma_k^2t^2} e^{-\frac12\hat s_j^2t^2}\\ &&=& e^{it\mu_k - \frac12\left(\sigma_k^2+\hat s_j^2\right)t^2} \\ &&=& \zeta_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}}(t)\\ &&=& \mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}}(t) \\ \Rightarrow & (-i\hat s_jt)^l\zeta_{g_k}(t)\zeta_\varphi(\hat s_jt) &=& (-i\hat s_jt)^l\mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}}(t)\\ &&=& \hat s_j^l(-it)^l\mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}}(t) \\ &&=& \hat s_j^l \mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}^{(l)}}(t) \\ \Rightarrow & f_{jkl} &=& \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j}(-i\hat s_jt)^l \zeta_{g_k}(t) \zeta_\varphi(\hat s_jt)dt\\ &&=& \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j} \hat s_j^l \mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}^{(l)}}(t)dt\\ &&=& \hat s_j^l \displaystyle\frac1{2\pi}\int e^{-it\hat\beta_j} \mathscr{F}_{\varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}^{(l)}}(t)dt\\ &&=& \hat s_j^l \varphi_{\mu_k, \sigma_k^2 + \hat s_j^2}^{(l)}\left(\hat\beta_j\right)\\ &&=& \displaystyle \hat s_j^l \frac{1}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right)\\ &&=& \displaystyle \frac{\hat s_j^l}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) \ . \end{array} \] In common applications, \(\mu_k\equiv\mu\equiv0\).
The problem is now boiled down to maximize the joint likelihood
\[ \begin{array}{rcl} \max\limits_{\pi, w}\prod\limits_j f(\hat\beta_j|\hat s_j) &=& \max\limits_{\pi, w}\prod\limits_j \left(\sum_k\sum_l\pi_k w_l f_{jkl}\right)\\ &\Leftrightarrow& \max\limits_{\pi, w}\sum_j\log\left(\sum_k\sum_l\pi_k w_l f_{jkl}\right) \ , \end{array} \]
subject to appropriate constraints on \(\pi_k\) and especially \(w_l\), where the specific form of \(f_{jkl}\) depends on the mixture component of the prior \(g\) of \(\beta_j\). Here we consider two cases.
\[ \begin{array}{rrcl} &\beta_j &\sim & \sum_k \pi_k \text{ Unif }[a_k, b_k]\\ \Rightarrow & f_{jkl} &= & \displaystyle\frac{\varphi^{(l-1)}\left(\frac{\hat\beta_j-a_k}{\hat s_j}\right) - \varphi^{(l-1)}\left(\frac{\hat\beta_j-b_k}{\hat s_j}\right)}{b_k - a_k} \ . \end{array} \]
\[ \begin{array}{rrcl} &\beta_j &\sim & \sum_k \pi_k N\left(\mu_k, \sigma_k^2\right) \\ \Rightarrow & f_{jkl} &= & \displaystyle\frac{\hat s_j^l}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}} \varphi^{(l)}\left(\frac{ \hat\beta_j - \mu_k }{ \sqrt{\sigma_k^2 + \hat s_j^2} }\right) \ . \end{array} \]
Our goal is then to estimate \(\hat\pi\) by solving the following constrained biconvex optimization problem
\[ \begin{array}{rl} \max\limits_{\pi,w} & \sum_j\log\left(\sum_k\sum_l\pi_k w_l f_{jkl}\right)\\ \text{subject to} & \sum_k\pi_k = 1\\ & w_0 = 1\\ & \sum_l w_l \varphi^{l}(z) \geq 0, \forall z\in \mathbb{R}\\ & w_l \text{ decay reasonably fast.} \end{array} \]
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