Von Mises Distribution
Θ ~ VM(μ, κ)
Circular / Directional Family • Published July 22, 2026
The site's first circular/directional distribution -- the natural analogue of the already-published Normal once a straight line is replaced by a circle: an exponential-family density exp(κcos(θ-μ)) normalized by a modified Bessel function, exact uniform-circle and large-κ Normal-approximation limits confirmed numerically, closed-form MLE for the mean direction (the sample circular mean) with a root-finding MLE for concentration, and simulation via the Best-Fisher rejection algorithm verified by a Kolmogorov-Smirnov test.
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About This Reference Sheet
The site's first circular/directional distribution -- the natural analogue of the already-published Normal once a straight line is replaced by a circle: an exponential-family density exp(κcos(θ-μ)) normalized by a modified Bessel function, exact uniform-circle and large-κ Normal-approximation limits confirmed numerically, closed-form MLE for the mean direction (the sample circular mean) with a root-finding MLE for concentration, and simulation via the Best-Fisher rejection algorithm verified by a Kolmogorov-Smirnov test.
Support
θ ∈ [-π, π) -- wraps around a circle
Parameters
μ ∈ [-π, π) (mean direction), κ ≥ 0 (concentration)