Reading Parameter Correlations From Information

Why the Inverse Information Matrix Matters

A Fisher information matrix summarizes how strongly the likelihood responds to changes in the model parameters near a specified parameter value. Its diagonal entries describe information in individual parameter directions, while its off-diagonal entries reflect how the likelihood’s sensitivities to different parameters are related. To understand joint estimation uncertainty, however, it is not enough to inspect the information entries separately: the relevant local covariance approximation is obtained by inverting the matrix.

This tutorial focuses on that inverse and its interpretation. A variance from the inverse describes uncertainty in one parameter estimate; an off-diagonal covariance describes how two estimation errors vary together; and a correlation expresses that co-variation on a dimensionless scale. The matrix can also reveal combinations of parameters that are much less precisely estimated than any single diagonal entry might suggest.

Definition: Let $\theta$ be a vector of model parameters, and let $I(\theta)$ be the Fisher information matrix evaluated at $\theta$. If the likelihood is regular and $I(\theta)$ is nonsingular, the local large-sample covariance approximation for the maximum likelihood estimator is $$ \operatorname{Cov}(\hat{\theta}) \approx I(\theta)^{-1}. $$ The diagonal entries of $I(\theta)^{-1}$ approximate parameter-estimation variances; its off-diagonal entries approximate covariances between estimation errors.

Conditions for the Covariance Approximation

The inverse-information interpretation is a local, large-sample result, not an exact description of uncertainty for every data set. In the usual regular likelihood setting, observations are generated under the stated model, the true parameter is interior to the parameter space, the likelihood is sufficiently smooth, and the model is locally identifiable. Information must be finite and nonsingular at the parameter value of interest. Under these conditions, the maximum likelihood estimator is approximately multivariate normal around the true parameter, with covariance given by the inverse expected information.

In practice, the parameter value is not known. One may evaluate expected information at a nominal parameter value for a design calculation, or evaluate it at an estimate for a fitted-model approximation. Another common approximation uses the inverse of the observed information, $J(\hat{\theta})^{-1}$, where $J$ is the negative Hessian of the log-likelihood. These substitutions are useful only when the fitted point is in a regular region and the relevant matrix is numerically stable and positive definite. The approximation can be poor with small samples, boundary estimates, strong nonlinearity, or a likelihood that is far from locally quadratic.

For a covariance matrix, variances must be nonnegative and the matrix must be positive semidefinite. A nonsingular local information matrix under the regular conditions above is positive definite, so its inverse is also positive definite. If information is singular, the inverse does not exist; the preceding material on singular information and local identifiability addresses that failure. A negative eigenvalue in a symmetric information matrix signals that it is not positive definite, but does not by itself show that it is singular. A zero eigenvalue means singularity. A relatively small eigenvalue compared with the largest one, or a large condition number, can signal near-singularity and an unstable inverse. The absolute size of a determinant alone is not a reliable diagnostic, since it depends on the scale and units of the matrix.

From Covariances to Correlations

Write the approximate covariance matrix as $\Sigma=I^{-1}$. For two parameters, the correlation of their estimation errors is their covariance divided by the product of their standard errors. This is not the correlation between the underlying parameter values: it describes how estimation errors tend to move together across repeated data sets under the local approximation.

Definition: For parameters $\theta_i$ and $\theta_j$, let $\Sigma_{ij}$ be the corresponding entry of the approximate covariance matrix. Their approximate estimation-error correlation is $$ \rho_{ij} = \frac{\Sigma_{ij}}{\sqrt{\Sigma_{ii}\Sigma_{jj}}}. $$ It is dimensionless and lies between $-1$ and $1$ when the covariance matrix is valid. The diagonal entries $\Sigma_{ii}$ are variances; the standard error approximation is $\sqrt{\Sigma_{ii}}$.

For a two-parameter information matrix with entries $a$, $b$, and $d$, the matrix and its inverse are

I= \begin{pmatrix} a & b\\ b & d \end{pmatrix}, \qquad I^{-1}= \frac{1}{ad-b^2} \begin{pmatrix} d & -b\\ -b & a \end{pmatrix}, \qquad ad-b^2>0.

Consequently, the approximate correlation is $-b/\sqrt{ad}$. This follows by dividing the inverse’s off-diagonal entry, $-b/(ad-b^2)$, by the square root of the product of its diagonal entries, $\sqrt{ad}/(ad-b^2)$. The sign reversal is important: in this two-parameter case, positive off-diagonal information produces negative covariance, and negative off-diagonal information produces positive covariance.

Worked Example: Inverting a Two-Parameter Information Matrix

Suppose a hypothetical pharmacometric analysis uses two dimensionless log-scale parameters and, at its nominal parameter values, gives the information matrix

I= \begin{pmatrix} 100 & 80\\ 80 & 100 \end{pmatrix}.

The determinant is $100(100)-80^2=10{,}000-6{,}400=3{,}600$, which is positive. The eigenvalues are $180$ and $20$, so the matrix is positive definite and invertible. Applying the two-by-two inverse formula gives

I^{-1} = \frac{1}{3{,}600} \begin{pmatrix} 100 & -80\\ -80 & 100 \end{pmatrix} = \begin{pmatrix} 1/36 & -1/45\\ -1/45 & 1/36 \end{pmatrix}.

Thus each parameter has approximate variance $1/36=0.02778$ in squared log-parameter units, and the approximate standard error is $\sqrt{1/36}=1/6=0.1667$. The covariance is $-1/45=-0.02222$ in the product of the two log-parameter units. Using the exact fractions, the correlation is

\rho_{12} = \frac{-1/45}{\sqrt{(1/36)(1/36)}} = \frac{-1/45}{1/36} = -0.8000.

Under this local approximation, estimation errors in the two parameters tend to move in opposite directions: an estimate above its true value for one parameter tends to accompany an estimate below its true value for the other. This is a statement about estimation uncertainty, not evidence that the parameters have a negative biological relationship.

Weakly Estimated Parameter Combinations

The diagonal variances do not always make the most important weakness obvious. In a multivariate model, a particular linear combination can be poorly estimated even when the individual parameter variances seem moderate. For a fixed vector $c$, the approximate variance of the linear combination $c^\mathsf{T}\hat{\theta}$ is

\operatorname{Var}(c^\mathsf{T}\hat{\theta}) \approx c^\mathsf{T}I^{-1}c.

This follows by applying the variance rule for a linear transformation to the approximate covariance matrix. In particular, an eigenvector of a symmetric information matrix identifies a parameter direction: if its eigenvalue is small, changes in that direction produce relatively little information. The inverse information has a correspondingly large variance in that direction. Eigenvalues and condition numbers should be interpreted with attention to parameter scaling, because a change in parameter units can change their numerical values.

Worked Example: A Small Information Eigenvalue and Strong Correlation

A second hypothetical study gives the information matrix

I= \begin{pmatrix} 1 & 0.99\\ 0.99 & 1 \end{pmatrix}.

Its eigenvalues are $1.99$, for the direction $(1,1)^\mathsf{T}$, and $0.01$, for $(1,-1)^\mathsf{T}$. Both are positive, so the matrix is invertible, but the small eigenvalue indicates that the difference direction is much less informed than the sum direction. The condition number is $1.99/0.01=199$, indicating substantial disparity between the directions.

The determinant is $1-0.99^2=0.0199$. Therefore,

I^{-1} = \frac{1}{0.0199} \begin{pmatrix} 1 & -0.99\\ -0.99 & 1 \end{pmatrix} = \begin{pmatrix} 50.2513 & -49.7487\\ -49.7487 & 50.2513 \end{pmatrix}.

Each parameter has approximate variance $1/0.0199=50.2513$, while their covariance is $-0.99/0.0199=-49.7487$. From the exact matrix entries, the correlation is $(-0.99/0.0199)/(1/0.0199)=-0.9900$. The large marginal variances and strong negative correlation are consistent with weak information about the difference between the parameters.

For a direct check, the variance of the difference is

\operatorname{Var}(\hat{\theta}_1-\hat{\theta}_2) \approx \begin{pmatrix}1&-1\end{pmatrix} I^{-1} \begin{pmatrix}1\\-1\end{pmatrix} = \frac{1+1+0.99+0.99}{0.0199} = \frac{3.98}{0.0199} = 200.

By contrast, the variance of the sum is $(1+1-0.99-0.99)/0.0199=0.02/0.0199=1.0050$. The difference is much less precisely estimated than the sum. This is the practical meaning of the small-information direction; it is not established merely by noticing that the two diagonal entries of $I$ are equal.

Worked Example: Variances of Parameters and Contrasts

Suppose a third hypothetical analysis has three dimensionless parameters with information matrix

I= \begin{pmatrix} 2 & 1 & 0\\ 1 & 2 & 0\\ 0 & 0 & 0.25 \end{pmatrix}.

The two-by-two block has eigenvalues $3$ and $1$; the third eigenvalue is $0.25$. All are positive, so the matrix is positive definite. Inverting the block and the final diagonal entry gives

I^{-1} = \begin{pmatrix} 2/3 & -1/3 & 0\\ -1/3 & 2/3 & 0\\ 0 & 0 & 4 \end{pmatrix}.

The first two approximate variances are each $2/3=0.6667$, and their covariance is $-1/3=-0.3333$. Their correlation is $(-1/3)/\sqrt{(2/3)(2/3)}=-0.5000$. The third parameter’s variance is $4$, so its approximate standard error is $2$, larger than the first two standard errors, each $\sqrt{2/3}=0.8165$.

The first two parameters also illustrate why combinations matter. The variance of their sum is $2/3+2/3+2(-1/3)=2/3$. The variance of their difference is $2/3+2/3-2(-1/3)=2$. The sum direction has information eigenvalue $3$, whereas the difference direction has eigenvalue $1$; hence the difference is less precisely estimated. The third direction has still lower information, $0.25$, and the largest variance, $4$. These results identify uncertainty directions without treating a single parameter’s variance as a complete account of joint precision.

How to Check an Interpretation

A useful reading procedure is to verify the matrix before interpreting its entries, then separate marginal uncertainty from joint uncertainty. The following checks help prevent common numerical and conceptual errors.

Check symmetry and invertibility.
Fisher information should be symmetric. Check that the matrix is nonsingular; for a regular identifiable model at an interior point, it should also be positive definite.

Invert without premature rounding.
Keep sufficient precision in the inverse and calculate correlations from unrounded entries. Round only the final reported values.

Check variances and correlations.
Diagonal covariance entries should be nonnegative, and correlations should lie in $[-1,1]$. A result outside that range suggests a calculation error or an invalid covariance approximation.

Inspect directions and model adequacy.
Look for small relative eigenvalues or a large condition number, and consider whether the likelihood is locally quadratic and the fitted parameters are away from boundaries.

The inverse matrix is only as informative as the likelihood model and local approximation that produced it. If a model omits important variability, has a boundary estimate, or has a strongly nonquadratic likelihood, the inverse-information standard errors may not reflect the actual uncertainty well. Comparing expected-information and observed-information approximations can reveal sensitivity to the chosen local curvature. Profile likelihood or simulation-based uncertainty assessments can provide additional checks when regular large-sample approximations are doubtful; they do not make an invalid model valid.

Common Mistake and Key Takeaway

Common mistake: “The information matrix has a large positive off-diagonal entry, so the two estimated parameters must have positively correlated errors.” This reverses the relationship in the two-parameter case: the inverse matrix has off-diagonal entry $-b/(ad-b^2)$, so its sign is opposite to $b$. More generally, correlations must be computed from the inverse matrix, not read directly from the information matrix.

A related mistake is to interpret a large correlation as proof that the model is singular. A correlation near $1$ or $-1$ can signal strongly linked estimation errors and may accompany poor precision in a parameter combination, but it does not by itself establish singularity. Singularity means the information matrix has a zero eigenvalue; near-singularity is assessed relative to scale, for example through a small relative eigenvalue or a large condition number.

Key takeaway: Under regular, locally identifiable likelihood conditions, invert the Fisher information matrix to approximate the joint covariance of parameter estimates. Read its diagonal as variances, standardize its off-diagonal entries to obtain estimation-error correlations, and examine linear combinations or eigen-directions to find weakly estimated parameter combinations.

Check Your Understanding

Try each question, then check the solutions below.

A two-parameter information matrix is $\begin{pmatrix}9&3\\3&4\end{pmatrix}$. Find its inverse and the approximate correlation.
For the matrix in Question 1, what are the approximate variances and standard errors?
Why does a small eigenvalue of a positive definite information matrix matter for estimation?
Does a negative eigenvalue of a symmetric information matrix necessarily mean the matrix is singular? Explain.

Solutions. For Question 1, the determinant is $9(4)-3^2=27$, so the inverse is $\frac{1}{27}\begin{pmatrix}4&-3\\-3&9\end{pmatrix}$. The correlation is $(-3/27)/\sqrt{(4/27)(9/27)}=-3/6=-0.5000$. For Question 2, the variances are $4/27=0.1481$ and $9/27=0.3333$; the standard errors are $\sqrt{4/27}=0.3849$ and $\sqrt{9/27}=0.5774$, in the corresponding parameter units. For Question 3, the inverse has a large variance along the eigenvector associated with that eigenvalue, so the parameter combination in that direction is weakly estimated. For Question 4, no. A negative eigenvalue shows that the matrix is not positive definite, but it can still be nonsingular; singularity requires a zero eigenvalue.