Estimation of the observed F.I.M. using a 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 complex expression of the likelihood, $I(\theta)$ has no closed form expression. It is however possible to estimate it using a stochastic approximation procedure, based on Louis' formula:
\(\DDt{\log (\pmacro(\by,\bpsi;\theta))} = \esp{\DDt{\log (\pmacro(\by,\bpsi;\theta))} | \by ;\theta} + \cov{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta},
\)
where
\(\begin{eqnarray}
\cov{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta} &=&
\esp{ \left(\Dt{\log (\pmacro(\by,\bpsi;\theta))} \right)\left(\Dt{\log (\pmacro(\by,\bpsi;\theta))}\right)^{\transpose} | \by ; \theta} \\
&& - \esp{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta}\esp{\Dt{\log (\pmacro(\by,\bpsi;\theta))} | \by ; \theta}^{\transpose}
\end{eqnarray}\)
$\Dt{\log (\pmacro(\by,\bpsi;\theta))}$ 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 on-line.
At iteration $k$ of the algorithm:
- $\textbf{Stochastic approximation}$: update $D_k$, $G_k$ and $\Delta_k$ according to the following recurrent 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 $\sum_{k=1}^{\infty} \gamma_k^2 < \infty$.
- $\textbf{Estimation-step}$: 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 to compute 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)
|
his assumption means that, for any $i=1,2,\ldots N$, all the components of $\psi_i$ are random and that 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 $\log (\pmacro(\bpsi;\theta))$ for estimating the F.I.M. This can be done relatively simply in a closed form when the individual parameters are normally distributed (eventually up to a transformation $h$).
If some component of $\psi_i$ has no variability, (2) does not hold anymore but we will 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 will 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 don't have closed form expressions can be obtained by central differences.
Remarks
1. Using $\gamma_k=1/k$ for $k=1,2,\ldots$ means that is approximated each term with an empirical mean obtained from $(\bpsi^{(k)} , k=1,2,\ldots)$. For instance,
\(\begin{eqnarray}
\Delta_k &=& \Delta_{k-1} + \frac{1}{k} \left(\Dt{\log (\pmacro(\by,\bpsi^{(k)};\theta))} - \Delta_{k-1} \right)
\end{eqnarray}\)
|
(3)
|
\(\begin{eqnarray}
&=& \frac{1}{k} \sum_{j=1}^{k} \Dt{\log (\pmacro(\by,\bpsi^{(j)};\theta))}
\end{eqnarray}\)
|
(4)
|
(3) (resp. (4)) defines $\Delta_k$ using an online (resp. offline) algorithm. Writing $\Delta_k$ as in (3) instead of (4) avoids to store all the simulated sequences $(\bpsi^{(j)}, 1\leq j \leq k)$ for computing $\Delta_k$.
2. This approach is used in practice for computing the F.I.M. $I(\hat{\theta})$ where $\hat{\theta}$ is the maximum likelihood estimate of $\theta$. Then, the only difference with the Metropolis-Hastings used for SAEM is that the population parameter $\theta$ is not updated but remains fixed to $\hat{\theta}$.
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 in:
- for $i=1,2,\ldots N$, run a Metropolis-Hastings algorithm to draw a sequence $\psi_i^{(k)}$ with limiting distribution $\pmacro(\psi_i | y_i ;\hat{\theta})$,
- at iteration $k$ of the M-H algorithm, compute the first and second derivatives of $\pypsi(\by,\bpsi^{(k)};\hat{\theta})$
- 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,
\(\begin{eqnarray}
\log (\ppsii(\psi_i;\theta)) &=& -\frac{d}{2}\log(2\pi) + \sum_{\iparam=1}^d \log(h_\iparam^{\prime}(\psi_{i,\iparam}))
-\frac{1}{2} \sum_{\iparam=1}^d \log(\omega_\iparam^2)
-\sum_{\iparam=1}^d \frac{1}{2\, \omega_\iparam^2}( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2, \\
\partial \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} &=&
\frac{1}{\omega_\iparam^2} h_\iparam^{\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) ), \\
\partial \log (\ppsii(\psi_i;\theta))/\partial \omega^2_{\iparam} &=&
-\frac{1}{2\omega_\iparam^2}
+ \frac{1}{2\, \omega_\iparam^4}( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2, \\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} \partial \psi_{ {\rm pop},\jparam} &=&
\left\{
\begin{array}{ll}
\left( h_\iparam^{\prime\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )- h_\iparam^{\prime \, 2}(\psi_{ {\rm pop},\iparam}) \right)/\omega_\iparam^2, & {\rm \quad if \quad} \iparam=\jparam \\
0, & {\rm \quad otherwise,}
\end{array}
\right.
\\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \omega^2_{\iparam} \partial \omega^2_{\jparam} &=& \left\{
\begin{array}{ll}
1/(2\omega_\iparam^4) -
( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )^2/\omega_\iparam^6, & {\quad \rm if \quad} \iparam=\jparam \\
0, & {\rm \quad otherwise,}
\end{array}
\right.
\\
\partial^2 \log (\ppsii(\psi_i;\theta))/\partial \psi_{ {\rm pop},\iparam} \partial \omega^2_{\jparam} &=& \left\{
\begin{array}{ll}
-h_\iparam^{\prime}(\psi_{ {\rm pop},\iparam})( h_\iparam(\psi_{i,\iparam}) - h_\iparam(\psi_{ {\rm pop},\iparam}) )/\omega_\iparam^4 \\
0, & {\rm \quad \quad otherwise.}
\end{array}
\right.
\end{eqnarray}\)
Example 2
We consider the same model for continuous data, assuming a constant error model and assuming 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
\(\begin{equation}
\log(\pyipsii(y_i,\psi_i;\theta)) = \log(\pcyipsii(y_i | \psi_i ; a^2)) + \log(\ppsii(\psi_i;\psi_{\rm pop},\Omega)).
\end{equation}\)
where
\( \log(\pcyipsii(y_i | \psi_i ; a^2))
=-\frac{n_i}{2}\log(2\pi)- \frac{n_i}{2}\log(a^2) - frac{1}{2a^2}\sum_{j=1}^{n_i}(y_{ij} - f(t_{ij}, \psi_i))^2
\)
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
We still consider the same model for continuous data, assuming now that a subset $\xi$ of the parameters of the structural model has no variability:
\(\begin{eqnarray}
y_{ij} | \psi_i &\sim& {\cal N}(f(t_{ij}, \psi_i,\xi) \ , \ a^2) \ \ ; \ \ 1 \leq j \leq n_i \\
h(\psi_i) &\sim_{i.i.d}& {\cal N}( h(\psi_{\rm pop}) , \Omega),
\end{eqnarray}\)
$\psi$ remains the subset of individual parameters with variability. Here, $\theta_y=(\xi,a^2)$, $\theta_\psi=(\psi_{\rm pop},\Omega)$, and
\( \log(\pcyipsii(y_i | \psi_i ; \xi,a^2))
=-\frac{n_i}{2}\log(2\pi)- \frac{n_i}{2}\log(a^2) - \frac{1}{2 a^2}\sum_{j=1}^{n_i}(y_{ij} - f(t_{ij}, \psi_i,\xi))^2
\)
Derivatives of $\log(\pcyipsii(y_i |\psi_i ; \xi, a^2))$ with respect to $\xi$ require to compute the derivative of $f$ with respect to $\xi$. These derivatives are usually not available. An alternative is to numerically approximate these derivatives using finite differences.
Estimation of the F.I.M. using a linearization of the model
Consider here a model for continuous data, using the $\phi$-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 $\phi_i$ ($\hphi_i$ can for instance be the estimated mean or the estimated mode of the conditional distribution $\pmacro(\phi_i |y_i ; \hat{\theta})$).
We can therefore 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 $\teps_{ij}$'s are {\id i.i.d}, then $\Sigma_{n_i}$ is the identity matrix.
We can equivalently use the original $\psi$-parametrization using 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 order derivatives in a closed form is not straightforward. Then, finite differences can be used for approximating numerically these quantities. We can use for instance a central difference approximation of the second derivative of ${\llike}(\theta)$:
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,
\(\begin{eqnarray}
\partial_{\theta_j}{ {\llike}(\theta)} &\approx& \displaystyle{ \frac{ {\llike}(\theta+\nu^{(j)})- {\llike}(\theta-\nu^{(j)})}{2\nu} } \ , \\
\partial^2_{\theta_j,\theta_k}{ {\llike}(\theta)} &\approx& \displaystyle{ \frac{ {\llike}(\theta+\nu^{(j)}+\nu^{(k)})-{\llike}(\theta+\nu^{(j)}-\nu^{(k)})
-{\llike}(\theta-\nu^{(j)}+\nu^{(k)})+{\llike}(\theta-\nu^{(j)}-\nu^{(k)})}{4\nu^2} } .
\end{eqnarray}\)
|
(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 in:
1. for $i=1,2,\ldots N$, get some estimate $(\hpsi_i)$ of the individual parameters $(\psi_i)$ (we can average for example the last terms of the sequence $(\psi_i^{(k)})$ drawn during the last iterations of the SAEM algorithm),
2. for $i=1,2,\ldots N$, compute $\hphi_i=h(\hpsi_i)$, compute the mean and the variance of the normal distribution defined in (5), and compute ${\llike}(\theta)$ using this normal approximation,
3. use (6) to approximate the matrix of second-order derivatives of ${\llike}(\theta)$.
