# Likelihoods

GaussianProcesses.BernLikType
BernLik <: Likelihood

Bernoulli likelihood

$p(y = k | f) = θᵏ (1 - θ)^{1-k}$

for $k ∈ \{0,1\}$, where $θ = Φ(f)$ and $f$ is the latent Gaussian process.

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GaussianProcesses.BinLikType
BinLik <: Likelihood

Binomial likelihood

$p(y = k | f) = k!/(n!(n-k)!) θᵏ(1 - θ)^{n-k}$

for number of successes $k ∈ \{0, 1, …, n\}$ out of $n$ Bernoulli trials, where $θ = \exp(f)/(1 + \exp(f))$ and $f$ is the latent Gaussian process.

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GaussianProcesses.GaussLikType
GaussLik <: Likelihood

Gaussian, a.k.a. Normal, likelihood

$p(y | f, σ) = 1 / √(2πσ²) \exp(-(y - f)²/(2σ²)),$

where standard deviation $σ$ is a non-fixed hyperparameter and $f$ is the latent Gaussian process.

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GaussianProcesses.PoisLikType
PoisLik <: Likelihood

Poisson likelihood

$p(yᵢ = k | fᵢ) = θᵏ\exp(-θ)/k!$

for $k ∈ N₀$, where $θ = \exp(f)$ and $f$ is the latent Gaussian process.

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GaussianProcesses.StuTLikType
StuTLik <: Likelihood

Student-t likelihood (a.k.a. non-standardized Student's t-distribution)

$p(y | f, σ) = Γ((ν + 1)/2)/[Γ(ν/2)√(πν)σ](1 + 1/ν((y - f)/σ)²)^{-(ν + 1)/2}$

with degrees of freedom $ν ∈ N₀$, where scale $σ$ is a non-fixed hyperparameter and $f$ is the latent Gaussian process.

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