Estimation using stochastic approximation
The observed Fisher information matrix (F.I.M.) is a function of $\theta$ defined as
\(\begin{eqnarray}
I(\theta) &=& -\DDt{\log ({\like}(\theta;\by))} \\
&=& -\DDt{\log (\py(\by;\theta))} .
\end{eqnarray}\)
|
(1)
|
Due to the likelihood being quite complex, $I(\theta)$ usually has no closed form expression. It is however possible to estimate it using a stochastic approximation procedure based on Louis' formula:
\(\DDt{\log (\pmacro(\by;\theta))} = \esp{\DDt{\log (\pmacro(\by,\bpsi;\theta))} | \by ;\theta} + \cov{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta},
\)
where
\(\begin{eqnarray}
\cov{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta} &=&
\esp{ \left(\Dt{\log (\pmacro(\by,\bpsi;\theta))} \right)\left(\Dt{\log (\pmacro(\by,\bpsi;\theta))}\right)^{\transpose} | \by ; \theta} \\
&& - \esp{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta}\esp{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta}^{\transpose} .
\end{eqnarray}\)
Thus, $\DDt{\log (\pmacro(\by;\theta))}$ is defined as a combination of conditional expectations. Each of these conditional expectations can be estimated by Monte Carlo, or equivalently approximated using a stochastic approximation algorithm.
We can then draw a sequence $(\psi_i^{(k)})$ using a Metropolis-Hasting algorithm and estimate the observed F.I.M. online. At iteration $k$ of the algorithm:
- Simulation step: for $i=1,2,\ldots,N$, draw $\psi_i^{(k)}$ from $m$ iterations of the Metropolis-Hastings algorithm described in The Metropolis-Hastings algorithm section with $\pmacro(\psi_i |y_i ;{\theta})$ as the limit distribution.
- Stochastic approximation: update $D_k$, $G_k$ and $\Delta_k$ according to the following recurrence relations:
\(\begin{eqnarray}
\Delta_k & = & \Delta_{k-1} + \gamma_k \left(\Dt{\log (\pmacro(\by,\bpsi^{(k)};{\theta}))} - \Delta_{k-1} \right) \\
D_k & = & D_{k-1} + \gamma_k \left(\DDt{\log (\pmacro(\by,\bpsi^{(k)};{\theta}))} - D_{k-1} \right)\\
G_k & = & G_{k-1} + \gamma_k \left((\Dt{\log (\pmacro(\by,\bpsi^{(k)};{\theta}))})(\Dt{\log (\pmacro(\by,\bpsi^{(k)};{\theta}))})^\transpose -G_{k-1} \right),
\end{eqnarray}\)
- where $(\gamma_k)$ is a decreasing sequence of positive numbers such that $\gamma_1=1$, $ \sum_{k=1}^{\infty} \gamma_k = \infty$, and $\sum_{k=1}^{\infty} \gamma_k^2 < \infty$.
- Estimation step: update the estimate $H_k$ of the F.I.M. according to
\(H_k = D_k + G_k - \Delta_k \Delta_k^{\transpose}. \)
Implementing this algorithm therefore requires computation of the first and second derivatives of
\(\log (\pmacro(\by,\bpsi;\theta))=\sum_{i=1}^{N} \log (\pmacro(y_i,\psi_i;\theta)).\)
Assume first that the joint distribution of $\by$ and $\bpsi$ decomposes as
\(
\pypsi(\by,\bpsi;\theta) = \pcypsi(\by | \bpsi)\ppsi(\bpsi;\theta).
\)
|
(2)
|
This assumption means that for any $i=1,2,\ldots,N$, all of the components of $\psi_i$ are random and there exists a sufficient statistic ${\cal S}(\bpsi)$ for the estimation of $\theta$. It is then sufficient to compute the first and second derivatives of $\log (\pmacro(\bpsi;\theta))$ in order to estimate the F.I.M. This can be done relatively simply in closed form when the individual parameters are normally distributed (or a transformation $h$ of them is).
If some component of $\psi_i$ has no variability, (2) no longer holds, but we can decompose $\theta$ into $(\theta_y,\theta_\psi)$ such that
\(
\pyipsii(y_i,\psi_i;\theta) = \pcyipsii(y_i | \psi_i ; \theta_y)\ppsii(\psi_i;\theta_\psi).
\)
We then need to compute the first and second derivatives of $\log(\pcyipsii(y_i |\psi_i ; \theta_y))$ and $\log(\ppsii(\psi_i;\theta_\psi))$. Derivatives of $\log(\pcyipsii(y_i |\psi_i ; \theta_y))$ that do not have a closed form expression can be obtained using central differences.
Remarks
1. Using $\gamma_k=1/k$ for $k \geq 1$ means that each term is approximated with an empirical mean obtained from $(\bpsi^{(k)}, k \geq 1)$. For instance,
\(\begin{eqnarray}
\Delta_k
&=& \Delta_{k-1} + \displaystyle{ \frac{1}{k} } \left(\Dt{\log (\pmacro(\by,\bpsi^{(k)};\theta))} - \Delta_{k-1} \right)
\end{eqnarray}\)
|
(3)
|
\(\begin{eqnarray}
&=& \displaystyle{ \frac{1}{k} }\sum_{j=1}^{k} \Dt{\log (\pmacro(\by,\bpsi^{(j)};\theta))} .
\end{eqnarray}\)
|
(4)
|
(3) (resp. (4)) defines $\Delta_k$ using an online (resp. offline) algorithm. Writing $\Delta_k$ as in (3) instead of (4) avoids having to store all simulated sequences $(\bpsi^{(j)}, 1\leq j \leq k)$ when computing $\Delta_k$.
2. This approach is used for computing the F.I.M. $I(\hat{\theta})$ in practice, where $\hat{\theta}$ is the maximum likelihood estimate of $\theta$. The only difference with the
Metropolis-Hastings used for SAEM is that the population parameter $\theta$ is not updated and remains fixed at $\hat{\theta}$.
In summary, for a given estimate $\hat{\theta}$ of the population parameter $\theta$, a stochastic approximation algorithm for estimating the observed Fisher Information Matrix $I(\hat{\theta)}$ consists of:
1. For $i=1,2,\ldots,N$, run a Metropolis-Hastings algorithm to draw a sequence $\psi_i^{(k)}$ with limit distribution $\pmacro(\psi_i |y_i ;\hat{\theta})$.
2. At iteration $k$ of the Metropolis-Hastings algorithm, compute the first and second derivatives of $\pypsi(\by,\bpsi^{(k)};\hat{\theta})$.
3.Update $\Delta_k$, $G_k$, $D_k$ and compute an estimate $H_k$ of the F.I.M.
Example 1
Consider the model
\(\begin{eqnarray}
y_i | \psi_i &\sim& \pcyipsii(y_i | \psi_i) \\
h(\psi_i) &\sim_{i.i.d}& {\cal N}( h(\psi_{\rm pop}) , \Omega),
\end{eqnarray}\)
where $\Omega = {\rm diag}(\omega_1^2,\omega_2^2,\ldots,\omega_d^2)$ is a diagonal matrix and $h(\psi_i)=(h_1(\psi_{i,1}), h_2(\psi_{i,2}), \ldots , h_d(\psi_{i,d}) )^{\transpose}$.
The vector of population parameters is $\theta = (\psi_{\rm pop} , \Omega)=(\psi_{ {\rm pop},1},\ldots,\psi_{ {\rm pop},d},\omega_1^2,\ldots,\omega_d^2)$.
Here,
\(
\log (\pyipsii(y_i,\psi_i;\theta)) = \log (\pcyipsii(y_i | \psi_i)) + \log (\ppsii(\psi_i;\theta)).
\)
Then,
\(\begin{eqnarray}
\Dt{\log (\pyipsii(y_i,\psi_i;\theta))} &=& \Dt{\log (\ppsii(\psi_i;\theta))} \\
\DDt{\log (\pyipsii(y_i,\psi_i;\theta))} &=& \DDt{\log (\ppsii(\psi_i;\theta))} .
\end{eqnarray}\)
More precisely,
\(\begin{eqnarray}
\log (\ppsii(\psi_i;\theta)) &=& -\displaystyle{\frac{d}{2} }\log(2\pi) + \sum_{\iparam=1}^d \log(h_\iparam^{\prime}(\psi_{i,\iparam}))
-\displaystyle{ \frac{1}{2} } \sum_{\iparam=1}^d \log(\omega_\iparam^2)
-\sum_{\iparam=1}^d \displaystyle{ \frac{1}{2\, \omega_\iparam^2} }( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2 \\
\partial \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} &=&
\displaystyle{\frac{1}{\omega_\iparam^2} }h_\iparam^{\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) ) \\
\partial \log (\ppsii(\psi_i;\theta))/\partial \omega^2_{\iparam} &=&
-\displaystyle{ \frac{1}{2\omega_\iparam^2} }
+\displaystyle{\frac{1}{2\, \omega_\iparam^4} }( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2 \\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} \partial \psi_{ {\rm pop},\jparam} &=&
\left\{
\begin{array}{ll}
\left( h_\iparam^{\prime\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )- h_\iparam^{\prime \, 2}(\psi_{ {\rm pop},\iparam}) \right)/\omega_\iparam^2 & {\rm if \quad } \iparam=\jparam \\
0 & {\rm otherwise}
\end{array}
\right.
\\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \omega^2_{\iparam} \partial \omega^2_{\jparam} &=& \left\{
\begin{array}{ll}
1/(2\omega_\iparam^4) -
( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2/\omega_\iparam^6 & {\rm if \quad} \iparam=\jparam \\
0 & {\rm otherwise}
\end{array}
\right.
\\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} \partial \omega^2_{\jparam} &=& \left\{
\begin{array}{ll}
-h_\iparam^{\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )/\omega_\iparam^4 & {\rm if \quad} \iparam=\jparam \\
0 & {\rm otherwise.}
\end{array}
\right.
\end{eqnarray}\)
Example 2
We consider the same model for continuous data, assuming a constant error model and that the variance $a^2$ of the residual error has no variability:
\(\begin{eqnarray}
y_{ij} | \psi_i &\sim& {\cal N}(f(t_{ij}, \psi_i) \ , \ a^2), \ \ 1 \leq j \leq n_i \\
h(\psi_i) &\sim_{i.i.d}& {\cal N}( h(\psi_{\rm pop}) , \Omega).
\end{eqnarray}\)
Here, $\theta_y=a^2$, $\theta_\psi=(\psi_{\rm pop},\Omega)$ and
\(
\log(\pyipsii(y_i,\psi_i;\theta)) = \log(\pcyipsii(y_i | \psi_i ; a^2)) + \log(\ppsii(\psi_i;\psi_{\rm pop},\Omega)),
\)
where
\(
\log(\pcyipsii(y_i | \psi_i ; a^2))
=-\displaystyle{\frac{n_i}{2} }\log(2\pi)- \displaystyle{\frac{n_i}{2} }\log(a^2) - \displaystyle{\frac{1}{2a^2} }\sum_{j=1}^{n_i}(y_{ij} - f(t_{ij}, \psi_i))^2 .
\)
Derivatives of $\log(\pcyipsii(y_i | \psi_i ; a^2))$ with respect to $a^2$ are straightforward to compute. Derivatives of $\log(\ppsii(\psi_i;\psi_{\rm pop},\Omega))$ with respect to $\psi_{\rm pop}$ and $\Omega$ remain unchanged.
Example 3
Consider again the same model for continuous data, assuming now that a subset $\xi$ of the parameters of the structural model has no variability:
\(\begin{eqnarray}
y_{ij} | \psi_i &\sim& {\cal N}(f(t_{ij}, \psi_i,\xi) \ , \ a^2), \ \ 1 \leq j \leq n_i \\
h(\psi_i) &\sim_{i.i.d}& {\cal N}( h(\psi_{\rm pop}) , \Omega).
\end{eqnarray}\)
Let $\psi$ remain as the subset of individual parameters with variability. Here, $\theta_y=(\xi,a^2)$, $\theta_\psi=(\psi_{\rm pop},\Omega)$, and
\(
\log(\pcyipsii(y_i | \psi_i ; \xi,a^2))
=-\displaystyle{\frac{n_i}{2} }\log(2\pi)- \displaystyle{\frac{n_i}{2} }\log(a^2) - \displaystyle{\frac{1}{2 a^2} }\sum_{j=1}^{n_i}(y_{ij} - f(t_{ij}, \psi_i,\xi))^2 .
\)
Derivatives of $\log(\pcyipsii(y_i | \psi_i ; \xi, a^2))$ with respect to $\xi$ require computation of the derivative of $f$ with respect to $\xi$. These derivatives are usually not calculable. One possibility is to numerically approximate them using finite differences.
Estimation using linearization of the model
Consider here a model for continuous data that uses a $\phi$-parametrization for the individual parameters:
\(\begin{eqnarray}
y_{ij} &= & f(t_{ij} , \phi_i) + g(t_{ij} , \phi_i)\teps_{ij} \\
\phi_i &=& \phi_{\rm pop} + \eta_i .
\end{eqnarray}\)
Let $\hphi_i$ be some predicted value of $\phi_i$, such as for instance the estimated mean or estimated mode of the conditional distribution $\pmacro(\phi_i |y_i ; \hat{\theta})$.
We can then choose to linearize the model for the observations $(y_{ij}, 1\leq j \leq n_i)$ of individual $i$ around the vector of predicted individual parameters. Let $\Dphi{f(t , \phi)}$ be the row vector of derivatives of $f(t , \phi)$ with respect to $\phi$. Then,
\(\begin{eqnarray}
y_{ij} &\simeq& f(t_{ij} , \hphi_i) + \Dphi{f(t_{ij} , \hphi_i)} \, (\phi_i - \hphi_i) + g(t_{ij} , \hphi_i)\teps_{ij} \\
&\simeq& f(t_{ij} , \hphi_i) + \Dphi{f(t_{ij} , \hphi_i)} \, (\phi_{\rm pop} - \hphi_i)
+ \Dphi{f(t_{ij} , \hphi_i)} \, \eta_i + g(t_{ij} , \hphi_i)\teps_{ij} .
\end{eqnarray}\)
Then, we can approximate the marginal distribution of the vector $y_i$ as a normal distribution:
\(
y_{i} \approx {\cal N}\left(f(t_{i} , \hphi_i) + \Dphi{f(t_{i} , \hphi_i)} \, (\phi_{\rm pop} - \hphi_i) ,
\Dphi{f(t_{i} , \hphi_i)} \Omega \Dphi{f(t_{i} , \hphi_i)}^{\transpose} + g(t_{i} , \hphi_i)\Sigma_{n_i} g(t_{ij} , \hphi_i)^{\transpose} \right),
\)
|
(5)
|
where $\Sigma_{n_i}$ is the variance-covariance matrix of $\teps_{i,1},\ldots,\teps_{i,n_i}$. If the $\teps_{ij}$ are i.i.d., then
$\Sigma_{n_i}$ is the identity matrix.
We can equivalently use the original $\psi$-parametrization and the fact that $\phi_i=h(\psi_i)$. Then,
\(
\Dphi{f(t_{i} , \hphi_i)} = \Dpsi{f(t_{i} , \hpsi_i)} J_h(\hpsi_i)^{\transpose} , \)
where $J_h$ is the Jacobian of $h$.
We then can approximate the observed log-likelihood ${\llike}(\theta) = \log(\like(\theta;\by))=\sum_{i=1}^N \log(\pyi(y_i;\theta))$ using this normal approximation. We can also derive the F.I.M. by computing the matrix of second-order partial derivatives of ${\llike}(\theta)$.
Except for very simple models, computing these second-order partial derivatives in closed form is not straightforward. In such cases, finite differences can be used for numerically approximating them. We can use for instance a central difference approximation of the second derivative of $\llike(\theta)$. To this end, let $\nu>0$. For $j=1,2,\ldots, m$, let $\nu^{(j)}=(\nu^{(j)}_{k}, 1\leq k \leq m)$ be the $m$-vector such that
\(
\nu^{(j)}_{k} = \left\{
\begin{array}{ll}
\nu & {\rm if \quad j= k} \\
0 & {\rm otherwise.}
\end{array}
\right.
\)
Then, for $\nu$ small enough,
\(\begin{eqnarray}
\partial_{\theta_j}{ {\llike}(\theta)} &\approx& \displaystyle{ \frac{ {\llike}(\theta+\nu^{(j)})- {\llike}(\theta-\nu^{(j)})}{2\nu} } \\
\end{eqnarray}\)
\(\begin{eqnarray}
\partial^2_{\theta_j,\theta_k}{ {\llike}(\theta)} &\approx& \displaystyle{\frac{ {\llike}(\theta+\nu^{(j)}+\nu^{(k)})- {\llike}(\theta+\nu^{(j)}-\nu^{(k)})
-{\llike}(\theta-\nu^{(j)}+\nu^{(k)})+{\llike}(\theta-\nu^{(j)}-\nu^{(k)})}{4\nu^2} } .
\end{eqnarray}\)
|
(6)
|
In summary, for a given estimate $\hat{\theta}$ of the population parameter $\theta$, the algorithm for approximating the Fisher Information Matrix $I(\hat{\theta)}$ using a linear approximation of the model consists of:
1. For $i=1,2,\ldots,N$, obtain some estimate $(\hpsi_i)$ of the individual parameters $(\psi_i)$ (we can average for example the final terms of the sequence $(\psi_i^{(k)})$ drawn during the final iterations of the SAEM algorithm).
2. For $i=1,2,\ldots,N$, compute $\hphi_i=h(\hpsi_i)$, the mean and the variance of the normal distribution defined in (5), and ${\llike}(\theta)$ using this normal approximation.
3. Use (6) to approximate the matrix of second-order derivatives of ${\llike}(\theta)$.
