Estimation of the observed Fisher information matrix

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 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:

where

Thus, $\DDt{\log (\pmacro(\by,\bpsi;\theta))}$ 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. 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:

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

Implementing this algorithm therefore requires computation of the first and second derivatives of

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

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.

Estimation using linearization of the model

Consider here a model for continuous data that uses a $\phi$-parametrization for the individual parameters:

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,

Then, we can approximate the marginal distribution of the vector $y_i$ as a normal distribution:

1. REDIRECTION Modèle:EquationWithRef

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,

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

Then, for $\nu$ small enough,

\(\begin{eqnarray} \partial^2_{\theta_j,\theta_k}{ {\llike}(\theta)} &\approx& \displaystyle{\frac{ {\llike}(\theta+\nu^{(j)}+\nu^{(k)})- {\llike}(\theta+\nu^{(j)}-\nu^{(k)}) -{\llike}(\theta-\nu^{(j)}+\nu^{(k)})+{\llike}(\theta-\nu^{(j)}-\nu^{(k)})}{4\nu^2} } . \end{eqnarray}\)

(6)

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:

where

$\Dt{\log (\pmacro(\by,\bpsi;\theta))}$ 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:

• $\textbf{Simulation-step}$: for $i=1,2,\ldots N$, draw $\psi_i^{(k)}$ from $m$ iterations of the Metropolis-Hastings algorithm described in The Metropolis-Hastings algorithm for simulating the individual parameters with $\pmacro(\psi_i |y_i ;{\theta})$ as limiting distribution.

• $\textbf{Stochastic approximation}$: update $D_k$, $G_k$ and $\Delta_k$ according to the following recurrent relations:

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

Implementing this algorithm therefore requires to compute the first and second derivatives of

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

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.

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:

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,

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 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,

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

Then, for $\nu$ small enough,

\(\begin{eqnarray} \partial_{\theta_j}{ {\llike}(\theta)} &\approx& \displaystyle{ \frac{ {\llike}(\theta+\nu^{(j)})- {\llike}(\theta-\nu^{(j)})}{2\nu} } \ , \end{eqnarray}\)

(6)

\(\begin{eqnarray} \partial^2_{\theta_j,\theta_k}{ {\llike}(\theta)} &\approx& \displaystyle{ \frac{ {\llike}(\theta+\nu^{(j)}+\nu^{(k)})-{\llike}(\theta+\nu^{(j)}-\nu^{(k)}) -{\llike}(\theta-\nu^{(j)}+\nu^{(k)})+{\llike}(\theta-\nu^{(j)}-\nu^{(k)})}{4\nu^2} } . \end{eqnarray}\)

(7)

The Metropolis-Hastings algorithm for simulating the individual parameters	Estimation of the log-likelihood via importance sampling