dnorm(μ, τ)

Returns an instance of Normal with mean $μ$ and standard deviation $\frac{1}{√τ}$.

\[p(x|μ,τ) = \sqrt{\frac{τ}{2π}} e^{-τ \frac{(x-μ)^2}{2}}\]

dlogis(μ, τ)

Return an instance of Logistic with location parameter $μ$ and scale parameter $\frac{1}{√τ}$.

\[p(x|μ,τ) = \frac{\sqrt{τ} e^{-\sqrt{τ}(x-μ)}}{(1+e^{-\sqrt{τ}(x-μ)})^2}\]

dt(μ, τ, ν)

If $μ = 0$ and $σ = 1$, the function returns an instance of TDist with $ν$ degrees of freedom, location $μ$, and scale $σ = \frac{1}{\sqrt{τ}}$. Otherwise, it returns an instance of TDistShiftedScaled.

\[p(x|ν,μ,σ) = \frac{Γ((ν+1)/2)}{Γ(ν/2) \sqrt{νπσ}} \left(1+\frac{1}{ν}\left(\frac{x-μ}{σ}\right)^2\right)^{-\frac{ν+1}{2}}\]

TDistShiftedScaled(ν, μ, σ)

Student's t-distribution with $ν$ degrees of freedom, location $μ$, and scale $σ$.

This struct allows for a shift (determined by $μ$) and a scale (determined by $σ$) of the standard Student's t-distribution provided by the Distributions.jl package.

pdf, logpdf, rand, minimum, and maximum are implemented for this distribution; the last three are what a model needs to initialize a parameter with this prior and to transform it to unconstrained space.

See Also

TDist

ddexp(μ, τ)

Return an instance of Laplace (Double Exponential) with location $μ$ and scale $\frac{1}{\sqrt{τ}}$.

\[p(x|μ,τ) = \frac{\sqrt{τ}}{2} e^{-\sqrt{τ} |x-μ|}\]

dflat()

Returns an instance of Flat or TruncatedFlat if truncated.

Flat represents a flat (uniform) prior over the real line, which is an improper distribution. And TruncatedFlat represents a truncated version of the Flat distribution.

Only pdf, logpdf, minimum, and maximum are implemented for these Distributions.

When use in a model, the parameters always need to be initialized.

Flat

The flat distribution mimicking the behavior of the dflat distribution in the BUGS family of softwares.

TruncatedFlat

Truncated version of the Flat distribution.

dexp(λ)

Returns an instance of Exponential with rate $\frac{1}{λ}$.

\[p(x|λ) = λ e^{-λ x}\]

dchisqr(k)

Returns an instance of Chi-squared with $k$ degrees of freedom.

\[p(x|k) = \frac{1}{2^{k/2} Γ(k/2)} x^{k/2 - 1} e^{-x/2}\]

dweib(a, b)

Returns an instance of Weibull distribution object with shape parameter $a$ and scale parameter $b^{-1/a}$.

BUGS parameterizes the Weibull by shape $a$ and "rate-like" $b$ (the density below), whereas Distributions.Weibull(α, θ) uses shape $α$ and scale $θ$ with density $\frac{α}{θ} (x/θ)^{α-1} e^{-(x/θ)^α}$. Matching the two gives $θ = b^{-1/a}$; the two agree with $θ = 1/b$ only when $a = 1$.

The Weibull distribution is a common model for event times. The hazard or instantaneous risk of the event is $abx^{a-1}$. For $a < 1$ the hazard decreases with $x$; for $a > 1$ it increases. $a = 1$ results in the exponential distribution with constant hazard.

\[p(x|a,b) = abx^{a-1}e^{-b x^a}\]

dlnorm(μ, τ)

Returns an instance of LogNormal with location $μ$ and scale $\frac{1}{\sqrt{τ}}$.

\[p(x|μ,τ) = \frac{\sqrt{τ}}{x\sqrt{2π}} e^{-τ/2 (\log(x) - μ)^2}\]

dgamma(a, b)

Returns an instance of Gamma with shape $a$ and scale $\frac{1}{b}$.

\[p(x|a,b) = \frac{b^a}{Γ(a)} x^{a-1} e^{-bx}\]

dpar(a, b)

Returns an instance of Pareto with scale parameter $b$ and shape parameter $a$.

\[p(x|a,b) = \frac{a b^a}{x^{a+1}}\]

dgev(μ, σ, η)

Returns an instance of GeneralizedExtremeValue with location $μ$, scale $σ$, and shape $η$.

\[p(x|μ,σ,η) = \frac{1}{σ} \left(1 + η \frac{x - μ}{σ}\right)^{-\frac{1}{η} - 1} e^{-\left(1 + η \frac{x - μ}{σ}\right)^{-\frac{1}{η}}}\]

where $\frac{η(x - μ)}{σ} > -1$.

dgpar(μ, σ, η)

Returns an instance of GeneralizedPareto with location $μ$, scale $σ$, and shape $η$.

\[p(x|μ,σ,η) = \frac{1}{σ} (1 + η ((x - μ)/σ))^{-1/η - 1}\]

df(n, m, μ=0, τ=1)

Returns an instance of F-distribution object with $n$ and $m$ degrees of freedom, location $μ$, and scale $τ$. This function is only valid when $μ = 0$ and $τ = 1$,

\[p(x|n, m, μ, τ) = \frac{\Gamma\left(\frac{n+m}{2}\right)}{\Gamma\left(\frac{n}{2}\right) \Gamma\left(\frac{m}{2}\right)} \left(\frac{n}{m}\right)^{\frac{n}{2}} \sqrt{τ} \left(\sqrt{τ}(x - μ)\right)^{\frac{n}{2}-1} \left(1 + \frac{n \sqrt{τ}(x-μ)}{m}\right)^{-\frac{n+m}{2}}\]

where $\frac{n \sqrt{τ} (x - μ)}{m} > -1$.

dunif(a, b)

Returns an instance of Uniform with lower bound $a$ and upper bound $b$.

\[p(x|a,b) = \frac{1}{b - a}\]

dbeta(a, b)

Returns an instance of Beta with shape parameters $a$ and $b$.

\[p(x|a,b) = \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)} x^{a-1} (1 - x)^{b-1}\]

dmnorm(μ::AbstractVector, T::AbstractMatrix)

Returns an instance of Multivariate Normal with mean vector μ and covariance matrix $T^{-1}$.

\[p(x|μ,T) = (2π)^{-k/2} |T|^{1/2} e^{-1/2 (x-μ)' T (x-μ)}\]

where $k$ is the dimension of x.

dmt(μ::AbstractVector, T::AbstractMatrix, k)

Returns an instance of Multivariate T with mean vector $μ$, scale matrix $T^{-1}$, and $k$ degrees of freedom.

\[p(x|k,μ,Σ) = \frac{\Gamma((k+d)/2)}{\Gamma(k/2) (k\pi)^{p/2} |Σ|^{1/2}} \left(1 + \frac{1}{k} (x-μ)^T Σ^{-1} (x-μ)\right)^{-\frac{k+p}{2}}\]

where $p$ is the dimension of $x$.

dwish(R::AbstractMatrix, k)

Returns an instance of Wishart with $k$ degrees of freedom and the scale matrix $T^{-1}$.

\[p(X|R,k) = |R|^{k/2} |X|^{(k-p-1)/2} e^{-(1/2) tr(RX)} / (2^{kp/2} Γ_p(k/2))\]

where $p$ is the dimension of $X$, and it should be less than or equal to $k$.

ddirich(θ::AbstractVector)

Return an instance of Dirichlet with parameters $θ_i$.

\[p(x|θ) = \frac{Γ(\sum θ)}{∏ Γ(θ)} ∏ x_i^{θ_i - 1}\]

where $\theta_i > 0, x_i \in [0, 1], \sum_i x_i = 1$

dbern(p)

Return an instance of Bernoulli with success probability p.

\[p(x|p) = p^x (1 - p)^{1-x}\]

dbin(p, n)

Returns an instance of Binomial with number of trials n and success probability p.

\[p(x|n,p) = \binom{n}{x} p^x (1 - p)^{n-x}\]

end

where $\theta \in [0, 1], n \in \mathbb{Z}^+,$ and $x = 0, \ldots, n$.

dcat(p)

Returns an instance of Categorical with probabilities p.

\[p(x|p) = p[x]\]

dpois(θ)

Returns an instance of Poisson with mean (and variance) θ.

\[p(x|θ) = e^{-θ} θ^x / x!\]

dgeom(θ)

Returns an instance of Geometric with success probability θ.

\[p(x|θ) = (1 - θ)^{x-1} θ\]

dnegbin(p, r)

Returns an instance of Negative Binomial with number of failures r and success probability p.

\[P(x|r,p) = \binom{x + r - 1}{x} (1 - p)^x p^r\]

where $x \in \mathbb{Z}^+$.

dbetabin(a, b, n)

Returns an instance of Beta Binomial with number of trials n and shape parameters a and b.

\[P(x|a, b, n) = \frac{\binom{n}{x} \binom{a + b - 1}{a + x - 1}}{\binom{a + b + n - 1}{n}}\]

dhyper(n₁, n₂, m₁, ψ=1)

Returns an instance of Hypergeometric. This distribution is used when sampling without replacement from a population consisting of $n₁$ successes and $n₂$ failures, with $m₁$ being the number of trials or the sample size. The function currently only allows for $ψ = 1$.

\[p(x | n₁, n₂, m₁, \psi) = \frac{\binom{n₁}{x} \binom{n₂}{m₁ - x} \psi^x}{\sum_{i=u_0}^{u_1} \binom{n1}{i} \binom{n2}{m₁ - i} \psi^i}\]

where $u_0 = \max(0, m₁-n₂), u_1 = \min(n₁,m₁),$ and $u_0 \leq x \leq u_1$

dmulti(θ::AbstractVector, n)

Returns an instance Multinomial with number of trials n and success probabilities θ.

\[P(x|n,θ) = \frac{n!}{∏_{r} x_{r}!} ∏_{r} θ_{r}^{x_{r}}\]