FISHER INFORMATION METRIC
Fisher Metric
The Fisher information metric is the unique Riemannian metric on statistical manifolds invariant under sufficient statistics (Chentsov's theorem). For Gaussian returns, the metric is the precision matrix (inverse covariance).
Fisher Metric Computed:
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THEORY
Information Geometry
The Fisher metric measures statistical distinguishability between nearby parameter values. Two states are "close" if their distributions are hard to tell apart.
> For returns r ~ N(mu, Sigma), the Fisher information matrix is:
g_ij = (Sigma^{-1})_ij (the precision matrix)
> Economic distance = statistical distinguishability between states
> Curvature from eigenvalue structure captures concentration of risk
> Chentsov's theorem: Fisher metric is the UNIQUE invariant Riemannian metric on statistical manifolds
COMPUTE FISHER CURVATURE
Parameters
Window size controls how many observations are used to estimate the covariance matrix at each point. Larger windows are more stable but less responsive to changes.
INSTRUCTIONS
- First, fetch price data from the Data page
- Select window size and step size above
- Click "Compute Fisher Curvature"
- Results will show curvature time series and comparison with Ricci curvature
> Fisher metric operates directly on return data (no network construction needed)
NAVIGATION