Find the least-biased distribution consistent with known constraints
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MaxEnt principle (Jaynes 1957): Among all distributions satisfying constraints, choose the one maximizing Shannon entropy H = −∫p(x)ln p(x)dx.
Using Lagrange multipliers: p*(x) = exp(λ₀ + λ₁x + λ₂x²).
• No constraints → Uniform (max entropy on bounded support).
• Fixed mean → Exponential p(x)=λe^(−λx), x≥0.
• Fixed mean + variance → Gaussian p(x)=N(μ,σ²).
Current entropy: — nats. The green curve is the MaxEnt distribution; blue shows the entropy functional value.