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\section{ConstantAndMultiplicativeGaussianErrorModel}
    An error model which assumes that the model error is a mixture between a
    Gaussian base-level noise and a Gaussian heteroscedastic noise.

    A mixture between a Gaussian base-level noise and a Gaussian
    heteroscedastic noise assumes that the observable biomarker $X$
    is related to the :class:`MechanisticModel` output by

    $$
        X(t, \psi , \sigma _{\text{base}}, \sigma _{\text{rel}}) =
        x^{\text{m}} + \left( \sigma _{\text{base}} + \sigma _{\text{rel}}
        x^{\text{m}}\right) \, \epsilon ,
    $$

    where $x^{\text{m}} := x^{\text{m}}(t, \psi )$ is the mechanistic
    model output with parameters $\psi$, and $\epsilon$ is a
    i.i.d. standard Gaussian random variable

    $$
        \epsilon \sim \mathcal{N}(0, 1).
    $$

    As a result, this model assumes that the observed biomarker values
    $x^{\text{obs}}$ are realisations of the random variable
    $X$.

    At each time point $t$ the distribution of the observable biomarkers
    can be expressed in terms of a Gaussian distribution

    $$
        p(x | \psi , \sigma _{\text{base}}, \sigma _{\text{rel}}) =
        \frac{1}{\sqrt{2\pi} \sigma _{\text{tot}}}
        \exp{\left(-\frac{\left(x-x^{\text{m}}\right) ^2}
        {2\sigma^2 _{\text{tot}}} \right)},
    $$

    where $\sigma _{\text{tot}} = \sigma _{\text{base}} +
    \sigma _{\text{rel}}x^{\text{m}}$.

    Extends :class:`ErrorModel`.

\section{GaussianErrorModel}
    An error model which assumes that the model error follows a Gaussian
    distribution.

    A Gaussian error model assumes that the observable
    biomarker $X$ is related to the :class:`MechanisticModel`
    output by

    $$
        X(t, \psi , \sigma) = x^{\text{m}} + \sigma \epsilon ,
    $$

    where $x^{\text{m}} := x^{\text{m}}(t, \psi )$ is the mechanistic
    model output with parameters $\psi$, and $\epsilon$ is a
    i.i.d. standard Gaussian random variable

    $$
        \epsilon \sim \mathcal{N}(0, 1).
    $$

    As a result, this model assumes that the observed biomarker values
    $x^{\text{obs}}$ are realisations of the random variable
    $X$.

    At each time point $t$ the distribution of the observable biomarkers
    can be expressed in terms of a Gaussian distribution

    $$
        p(x | \psi , \sigma ) =
        \frac{1}{\sqrt{2\pi} \sigma }
        \exp{\left(-\frac{\left(x-x^{\text{m}}\right) ^2}
        {2\sigma^2 } \right)}.
    $$

    Extends :class:`ErrorModel`.

\section{LogNormalErrorModel}
    An error model which assumes that the model error follows a Log-normal
    distribution.

    A log-normal error model assumes that the observable
    biomarker $X$ is related to the :class:`MechanisticModel`
    output by

    $$
        X(t, \psi , \sigma _{\mathrm{log}}) =
        y \, \mathrm{e}^{\mu + \sigma _{\mathrm{log}} \varepsilon },
    $$

    where $y := y(t, \psi )$ is the mechanistic
    model output with parameters $\psi$, and $\varepsilon$ is a
    i.i.d. standard Gaussian random variable

    $$
        \varepsilon \sim \mathcal{N}(0, 1).
    $$

    Here, $\sigma _{\mathrm{log}}$ is the standard deviation of
    $\log X$ and
    $\mu := -\sigma _{\mathrm{log}} ^2 / 2$ is chosen such that

    $$
        \mathbb{E}[X] = y.
    $$

    As a result, this model assumes that the observed biomarker values
    $x^{\text{obs}}$ are realisations of the random variable
    $X$.

    At each time point $t$ the distribution of the observable biomarkers
    can be expressed in terms of a log-normal distribution

    $$
        p(x | \psi , \sigma _{\mathrm{log}} ) =
        \frac{1}{\sqrt{2\pi} \sigma _{\mathrm{log}} x}
        \exp{\left(-\frac{
            \left(\log x - \log y + \sigma _{\mathrm{log}}^2/2\right) ^2}
        {2\sigma _{\mathrm{log}}^2 } \right)}.
    $$

    Extends :class:`ErrorModel`.

\section{MultiplicativeGaussianErrorModel}
    An error model which assumes that the model error is a Gaussian
    heteroscedastic noise.

    A Gaussian heteroscedastic noise model assumes that the observable
    biomarker $X$ is related to the :class:`MechanisticModel`
    output by

    $$
        X(t, \psi , \sigma _{\text{rel}}) =
        x^{\text{m}} + \sigma _{\text{rel}} x^{\text{m}} \, \epsilon ,
    $$

    where $x^{\text{m}} := x^{\text{m}}(t, \psi )$ is the mechanistic
    model output with parameters $\psi$, and $\epsilon$ is a
    i.i.d. standard Gaussian random variable

    $$
        \epsilon \sim \mathcal{N}(0, 1).
    $$

    As a result, this model assumes that the observed biomarker values
    $x^{\text{obs}}$ are realisations of the random variable
    $X$.

    At each time point $t$ the distribution of the observable biomarkers
    can be expressed in terms of a Gaussian distribution

    $$
        p(x | \psi , \sigma _{\text{base}}, \sigma _{\text{rel}}) =
        \frac{1}{\sqrt{2\pi} \sigma _{\text{tot}}}
        \exp{\left(-\frac{\left(x-x^{\text{m}}\right) ^2}
        {2\sigma^2 _{\text{tot}}} \right)},
    $$

    where $\sigma _{\text{tot}} = \sigma _{\text{rel}}x^{\text{m}}$.

    Extends :class:`ErrorModel`.

\end{document}