# Information Geometry
## Overview
Information Geometry studies the geometric structure of probability distributions and statistical models. It provides a rigorous mathematical framework for understanding statistical inference, machine learning, and active inference through the lens of differential geometry.
## Mathematical Foundation
### 1. Statistical Manifold
```math
\mathcal{M} = \{p_θ : θ ∈ Θ\}
```
where:
- M is the manifold of probability distributions
- θ are the parameters
- Θ is the parameter space
### 2. Fisher Information Metric
```math
g_{ij}(θ) = \mathbb{E}_{p_θ}\left[\frac{∂\log p_θ}{∂θ^i}\frac{∂\log p_θ}{∂θ^j}\right]
```
### 3. α-Connections
```math
Γ_{ijk}^{(α)} = \mathbb{E}_{p_θ}\left[\frac{∂^2\log p_θ}{∂θ^j∂θ^k}\frac{∂\log p_θ}{∂θ^i} + \frac{1-α}{2}\frac{∂\log p_θ}{∂θ^i}\frac{∂\log p_θ}{∂θ^j}\frac{∂\log p_θ}{∂θ^k}\right]
```
## Core Components
### 1. [[statistical_manifolds|Statistical Manifolds]]
```julia
struct StatisticalManifold{T<:Distribution}
# Dimension of parameter space
dim::Int
# Parameter space
Θ::AbstractVector{Float64}
# Distribution family
family::Type{T}
# Metric tensor
g::Function
# Connection coefficients
Γ::Function
end
function compute_metric(manifold::StatisticalManifold,
θ::Vector{Float64})
n = manifold.dim
g = zeros(n, n)
for i in 1:n, j in 1:n
g[i,j] = expectation(manifold.family(θ)) do x
∂i = ∂log_likelihood(x, θ, i)
∂j = ∂log_likelihood(x, θ, j)
∂i * ∂j
end
end
return g
end
```
### 2. [[natural_gradient|Natural Gradient]]
```julia
struct NaturalGradient
# Manifold
manifold::StatisticalManifold
# Learning rate
η::Float64
end
function update!(grad::NaturalGradient,
θ::Vector{Float64},
∇L::Vector{Float64})
# Compute Fisher information matrix
G = compute_metric(grad.manifold, θ)
# Compute natural gradient
∇̃L = G \ ∇L
# Update parameters
return θ - grad.η * ∇̃L
end
```
### 3. [[geodesics|Geodesics]]
```julia
function compute_geodesic(manifold::StatisticalManifold,
θ₀::Vector{Float64},
θ̇₀::Vector{Float64},
T::Float64,
dt::Float64)
# Initialize trajectory
ts = 0:dt:T
θs = Vector{Vector{Float64}}(undef, length(ts))
θs[1] = θ₀
# Integrate geodesic equation
for i in 1:length(ts)-1
# Current state
θ = θs[i]
# Compute Christoffel symbols
Γ = manifold.Γ(θ)
# Update velocity
θ̇ = θ̇₀ - 0.5 * sum(Γ .* θ̇₀ .* θ̇₀)
# Update position
θs[i+1] = θ + dt * θ̇
end
return ts, θs
end
```
## Applications
### 1. [[variational_inference|Variational Inference]]
```julia
function natural_variational_inference(manifold::StatisticalManifold,
target::Distribution,
q_init::Distribution)
# Initialize variational parameters
θ = parameters(q_init)
# Natural gradient descent
for iter in 1:max_iters
# Compute ELBO gradient
∇ELBO = compute_elbo_gradient(q_init, target)
# Natural gradient update
θ = update!(NaturalGradient(manifold, 0.01), θ, ∇ELBO)
# Update variational distribution
q_init = manifold.family(θ)
end
return q_init
end
```
### 2. [[active_inference|Active Inference]]
```julia
function information_geometric_policy_selection(
manifold::StatisticalManifold,
agent::ActiveInferenceAgent)
# Generate policies
policies = generate_policies(agent)
# Compute geodesic distances to preferred states
distances = Float64[]
for π in policies
# Predicted distribution
p_pred = predict_distribution(agent, π)
# Compute geodesic distance
d = geodesic_distance(manifold,
p_pred,
agent.preferences)
push!(distances, d)
end
return policies[argmin(distances)]
end
```
### 3. [[exponential_families|Exponential Families]]
```julia
struct ExponentialFamily
# Sufficient statistics
T::Vector{Function}
# Log-partition function
A::Function
# Base measure
h::Function
function natural_parameters(η::Vector{Float64})
# Compute moment parameters
μ = ∇A(η)
# Compute Fisher information
G = ∇²A(η)
return μ, G
end
end
function compute_divergence(ef::ExponentialFamily,
p::Distribution,
q::Distribution)
# Get natural parameters
η_p = natural_parameters(p)
η_q = natural_parameters(q)
# Compute Bregman divergence
return ef.A(η_q) - ef.A(η_p) - dot(∇A(η_p), η_q - η_p)
end
```
## Theoretical Results
### 1. [[dually_flat|Dually Flat Structure]]
```julia
struct DuallyFlatManifold <: StatisticalManifold
# Potential function
ψ::Function
# Dual potential
φ::Function
# Legendre transform
∇ψ::Function
∇φ::Function
function divergence(self, p::Distribution, q::Distribution)
# Compute Bregman divergence
η_p = natural_parameters(p)
η_q = natural_parameters(q)
return self.ψ(η_q) - self.ψ(η_p) -
dot(self.∇ψ(η_p), η_q - η_p)
end
end
```
### 2. [[information_projection|Information Projection]]
```julia
function e_projection(manifold::StatisticalManifold,
p::Distribution,
constraint::Function)
# Initialize parameters
θ = parameters(p)
# Minimize KL divergence subject to constraint
for iter in 1:max_iters
# Compute gradient
∇KL = compute_kl_gradient(p, manifold.family(θ))
# Project gradient onto constraint surface
∇proj = project_gradient(∇KL, constraint)
# Update parameters
θ = update!(NaturalGradient(manifold, 0.01), θ, ∇proj)
end
return manifold.family(θ)
end
```
### 3. [[cramer_rao|Cramér-Rao Bounds]]
```julia
function cramer_rao_bound(manifold::StatisticalManifold,
estimator::Function)
# Compute Fisher information
G = compute_metric(manifold, θ)
# Compute estimator covariance
Σ = cov(estimator)
# Check Cramér-Rao inequality
return is_positive_definite(Σ - inv(G))
end
```
## Best Practices
### 1. Implementation
- Use stable numerical methods
- Implement efficient tensor operations
- Cache geometric quantities
- Handle singularities
### 2. Optimization
- Monitor metric regularity
- Adapt learning rates
- Check geodesic stability
- Validate projections
### 3. Validation
- Test with known geometries
- Verify invariance properties
- Check bound satisfaction
- Monitor convergence
## References
1. Amari, S. I. (2016). Information Geometry and Its Applications
2. Ay, N., et al. (2017). Information Geometry
3. Nielsen, F. (2020). An Elementary Introduction to Information Geometry
4. Cencov, N. N. (1982). Statistical Decision Rules and Optimal Inference
5. Lebanon, G. (2005). Information Geometry, the Embedding Principle, and Document Classification