Distribution  Parameter(s)  Natural parameter(s)  Inverse parameter mapping  Base measure  Sufficient statistic  Logpartition  Logpartition 

Bernoulli distribution  p 

 
binomial distribution with known number of trials n  p  
Poisson distribution  λ  
negative binomial distribution with known number of failures r  p  
exponential distribution  λ  
Pareto distribution with known minimum value x_{m}  α  
Weibull distribution with known shape k  λ  
Laplace distribution with known mean μ  b  
chisquared distribution  ν  
normal distribution known variance  μ  
normal distribution  μ,σ^{2}  
lognormal distribution  μ,σ^{2}  
inverse Gaussian distribution  μ,λ  
gamma distribution  α,β  
k, θ  
inverse gamma distribution  α,β  
scaled inverse chisquared distribution  ν,σ^{2}  
beta distribution  α,β  
multivariate normal distribution  μ,Σ  
categorical distribution (variant 1)  p_{1},...,p_{k} where  where 
 
categorical distribution (variant 2)  p_{1},...,p_{k} where 
where 
 
categorical distribution (variant 3)  p_{1},...,p_{k} where 


 
multinomial distribution (variant 1) with known number of trials n  p_{1},...,p_{k} where  where  
multinomial distribution (variant 2) with known number of trials n  p_{1},...,p_{k} where 
where  
multinomial distribution (variant 3) with known number of trials n  p_{1},...,p_{k} where  
 
Dirichlet distribution  α_{1},...,α_{k}  
Wishart distribution  V,n 
 
NOTE: Uses the fact that i.e. the trace of a matrix product is much like a dot product. The matrix parameters are assumed to be vectorized (laid out in a vector) when inserted into the exponential form. Also, V and X are symmetric, so e.g.  
inverse Wishart distribution  Ψ,m   
normalgamma distribution  α,β,μ,λ 
