Negative-Channel Fisher Information

• Negative-channel Fisher information is a method that estimates the scalar Fisher information using the empirical second derivative of the log-likelihood.
• It shows improved efficiency by achieving lower sampling variance compared to the gradient-outer-product estimator in common models.
• Both asymptotic analysis and empirical simulations confirm that the negative-Hessian estimator is particularly effective for normal and signal-plus-noise models.

Negative-channel Fisher information, also known as the negative-Hessian estimator, refers to a classical approach for estimating the scalar Fisher information number (FIN) from independent observations drawn from a parametric family. The estimator leverages the empirical second derivative (Hessian) of the log-likelihood, offering an alternative to the widely used gradient-outer-product estimator. The accuracy and efficiency of this method, especially in the scalar case, have been the subject of precise asymptotic analysis, with explicit variance expansions and comparative results for finite-sample performance (Guo, 2014).

1. Definition and Formulation

Let $X_1,\dots,X_n$ be independent real-valued observations from a distribution with density $f(x;\theta)$, where $\theta$ is an unknown scalar parameter. The log-likelihood is given by

$$\ell(\theta) = \ln L(\theta) = \sum_{i=1}^n\ln f(X_i;\theta)$$

The Fisher information number (FIN) for a single observation is defined equivalently by

$I(\theta) = \E\left[-\,\frac{d^2}{d\theta^2}\ln f(X;\theta)\right] = \E\left[\left(\frac{d}{d\theta}\ln f(X;\theta)\right)^2\right]$

For $n$ independent samples, the total Fisher information is $nI(\theta)$.

When $I(\theta)$ does not have a closed-form expression, it is estimated from data using two natural sample-based estimators:

• Gradient-outer-product estimator:

$$I_G(\theta) = \frac{1}{n}\sum_{i=1}^n \left(\frac{\partial}{\partial\theta}\ln f(X_i;\theta)\right)^2$$

• Negative-channel (negative-Hessian) estimator:

$$I_H(\theta) = -\frac{1}{n} \sum_{i=1}^n \frac{\partial^2}{\partial\theta^2}\ln f(X_i;\theta)$$

Both are unbiased estimators of the per-datum FIN (Guo, 2014).

2. Asymptotic Variance and Central Limit Analysis

Under standard regularity conditions (independence, finiteness of moments, sufficient smoothness), the asymptotic distributions of these estimators follow from the Central Limit Theorem: $\sqrt{n}\big(I_G(\theta) - I(\theta)\big) \xrightarrow{d} N\big(0, \Var[g_1(\theta)^2]\big)$

$\sqrt{n}\big(I_H(\theta) - I(\theta)\big) \xrightarrow{d} N\big(0, \Var[H_1(\theta)]\big)$

where $$g_i(\theta) = \frac{\partial}{\partial\theta}\ln f(X_i;\theta)$$ and $$H_i(\theta) = \frac{\partial^2}{\partial\theta^2}\ln f(X_i;\theta)$$. The asymptotic variances $\Var[g_1^2]$ and $\Var[H_1]$ quantify the efficiency of each estimator.

3. Taylor Expansion-Based Variance Approximations

Closed-form expressions for the variances $\Var[g_1^2]$ and $\Var[H_1]$ are generally unavailable. Taylor series expansions around the population mean $\mu = \E[X]$ yield tractable approximations:

• For $g(X)^2$, the second-order expansion introduces terms involving central moments of $X$ up to order four and higher. As laid out in eq. (3.11) of (Guo, 2014), the variance expansion contains leading contributions from the variance, variance of squared deviations, and higher moments.
• For $H(X)$, an analogous expansion (see eq. (3.12)) depends on derivatives of $H$ at $\mu$, and central moments up to the third and fourth order.

When the underlying distribution is symmetric (odd central moments vanish), the difference between variances simplifies, as in eq. (3.15). In this case, a sufficient condition for the negative-Hessian estimator to be asymptotically at least as efficient as the gradient-outer-product estimator is that

$4\,g(\mu)^2\,g'(\mu)^2 \geq H'(\mu)^2$

$$\big(g'(\mu)^2 + g(\mu)g''(\mu)\big)^2 \geq H''(\mu)^2$$

along with appropriate sign conditions on derivative products.

4. Relative Efficiency and Sufficient Conditions

Setting $V_G = \Var[g(X)^2]$ and $V_H = \Var[H(X)]$, the respective estimator variances for large $n$ scale as

$\Var(I_G) \sim \frac{V_G}{n},\qquad \Var(I_H) \sim \frac{V_H}{n}$

The ratio $V_G/V_H$ thus measures their asymptotic relative efficiency. The sufficient conditions above are typically met in exponential-family models and for distributions with symmetric densities; in such scenarios, $\Var(I_G) \gg \Var(I_H)$. This suggests the negative-channel estimator $I_H$ usually achieves considerably lower sampling variance and is more accurate for the scalar case (Guo, 2014).

5. Numerical Simulations and Empirical Comparison

Empirical results presented by Guo illustrate the theoretical findings in three settings:

• For the normal distribution with unknown mean ($\theta = \mu$):

$$g(X) = \frac{X - \mu}{\sigma^2},\qquad H(X) = -\frac{1}{\sigma^2}$$

Here $H(X)$ is constant, yielding $\Var[H(X)] = 0$ and simulation confirms $\Var(I_H) \approx 0$ while $\Var(I_G) \gg 0$.

• For the normal distribution with unknown variance ($\theta = \sigma^2$), and for a signal-plus-noise model $X_i \sim N(0, \theta + q_i)$, the Taylor expansions predict, and Monte Carlo simulations confirm, that $I_H$ has substantially lower sampling variance than $I_G$. The observed variance-ratios exceed $1.6$ and average between $1.5$ and $2$ in these scenarios (Guo, 2014).

Summary of the simulation findings:

Model	$\Var(I_H)$	$\Var(I_G)$	Observed Variance Ratio $\frac{\Var(I_G)}{\Var(I_H)}$
Gaussian, $\theta = \mu$	$\approx 0$	$\gg 0$	$\gg 1$
Gaussian, $\theta = \sigma^2$	$> 0$	$> 0$	$>1.6$
Signal-plus-noise	$> 0$	$> 0$	$1.5$–$2$

These results demonstrate that the negative-channel estimator is unbiased and achieves lower or equivalent asymptotic variance in a range of common statistical models.

6. Implications and Areas for Future Research

The negative-channel Fisher information estimator is, under broad regularity conditions, asymptotically at least as efficient as its gradient-outer-product counterpart in the scalar parameter case. In finite samples, it tends to show marked improvements in sampling variance, particularly for exponential-family and symmetric models.

A plausible implication is that, when feasible to compute, the negative-Hessian estimator should be preferred for Fisher information calculation in scalar parameter settings. Limitations of the current analysis and possibilities for extension include generalization to vector parameters, relaxing independence assumptions, and addressing distributions or models where regularity conditions do not hold (Guo, 2014). Further research directions include deriving higher-order variance corrections, improving empirical variance estimation in small samples, and extending comparative studies to more complex or high-dimensional models.