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GaussianNoiseLikelihood

Wraps a forward model and observed data into a log-likelihood for Gaussian measurement noise.

Supports three modes:

Mode noise_std marginalise_noise Noise in theta?
Fixed σ float or array — No
Estimated σ None False Yes (log_σ appended)
Marginalised σ None True No

Per-channel noise is supported via groups.

import mcmckit as mc

# Fixed scalar noise
ll = mc.GaussianNoiseLikelihood(forward_model, y_obs, noise_std=0.05)

# Estimated noise (last element of theta = log_sigma)
ll = mc.GaussianNoiseLikelihood(forward_model, y_obs)

# Marginalised noise — no noise parameter in theta
ll = mc.GaussianNoiseLikelihood(
    forward_model, y_obs,
    marginalise_noise=True,
    inv_gamma_alpha=2.0,
    inv_gamma_beta=0.05**2,
)

# Per-channel (different sigma per measurement type)
ll = mc.GaussianNoiseLikelihood(
    forward_model, y_obs,
    groups=[[0,1,2], [3,4,5]],
    marginalise_noise=True,
    inv_gamma_alpha=[2.0, 2.0],
    inv_gamma_beta=[0.05**2, 0.10**2],
)

GaussianNoiseLikelihood

Log-likelihood for a forward model with Gaussian observation noise.

Supports scalar or per-channel noise, where a "channel" is a group of observations sharing one noise parameter (e.g. all measurements of the 1st natural frequency form one channel, all measurements of the 2nd natural frequency form another).

Three noise modes

Fixed σ — noise_std is a float or array:

log p(y|θ) = -0.5 · Σᵢ ||rᵢ||² / σᵢ² - nᵢ·log(σᵢ)

Estimated σ — noise_std=None: the last n_channels elements of theta are [log σ₁, …, log σₖ] and are sampled jointly with the model parameters.

Marginalised σ — noise_std=None, marginalise_noise=True: each σᵢ² is given an Inverse-Gamma(αᵢ, βᵢ) prior and integrated out analytically. Works for any nonlinear forward model:

log p(y|θ) = Σᵢ  -(αᵢ + nᵢ/2) · log(βᵢ + 0.5·||rᵢ||²)

Parameters:

Name Type Description Default
forward_model callable

f(theta_model) -> array-like, shape (n_obs,).

 required
y_obs (array - like, shape(n_obs))

Observed data (flat vector).

 required
noise_std float, array-like, or None

Fixed noise standard deviation(s). Options:

  • float — same σ for every observation.
  • array of length n_obs — per-observation σ.
  • array of length n_channels — one σ per channel group; requires groups to be set.
  • None — σ is free (estimated or marginalised).
 None
groups list of array-like of int, or None

Index groups defining channels. E.g.::

groups=[[0,1,2,3], [4,5,6,7]]   # 2 channels of 4 obs each

None (default) treats all observations as one channel. When noise_std is a per-channel array or None, groups is required.

 None
marginalise_noise bool

If True (and noise_std=None), analytically marginalise each σᵢ² using independent Inverse-Gamma priors. Default False.

 False
inv_gamma_alpha float or array - like

Shape parameter(s) α of the Inverse-Gamma prior on σᵢ². Scalar → same for all channels. Default 1.0.

 1.0
inv_gamma_beta float or array - like

Scale parameter(s) β of the Inverse-Gamma prior on σᵢ². Scalar → same for all channels. Default 1e-4.

 0.0001

Examples:

Scalar fixed noise::

ll = GaussianNoiseLikelihood(forward_model=f, y_obs=y, noise_std=0.05)

Per-channel fixed noise (2 frequencies, 4 repetitions each)::

ll = GaussianNoiseLikelihood(
    forward_model=f, y_obs=y,
    noise_std=[0.05, 0.10],
    groups=[[0,1,2,3], [4,5,6,7]],
)

Per-channel estimated noise (2 free log_σ params appended to theta)::

ll = GaussianNoiseLikelihood(
    forward_model=f, y_obs=y,
    groups=[[0,1,2,3], [4,5,6,7]],
)
# theta = [*model_params, log_sigma_1, log_sigma_2]

Per-channel marginalised noise::

ll = GaussianNoiseLikelihood(
    forward_model=f, y_obs=y,
    groups=[[0,1,2,3], [4,5,6,7]],
    marginalise_noise=True,
    inv_gamma_alpha=[2.0, 2.0],
    inv_gamma_beta=[0.05**2, 0.10**2],
)
Source code in mcmckit/core/noise.py 
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class GaussianNoiseLikelihood:
    """Log-likelihood for a forward model with Gaussian observation noise.

    Supports **scalar** or **per-channel** noise, where a "channel" is a
    group of observations sharing one noise parameter (e.g. all measurements
    of the 1st natural frequency form one channel, all measurements of the
    2nd natural frequency form another).

    Three noise modes
    -----------------
    **Fixed σ** — ``noise_std`` is a float or array:

        log p(y|θ) = -0.5 · Σᵢ ||rᵢ||² / σᵢ² - nᵢ·log(σᵢ)

    **Estimated σ** — ``noise_std=None``: the last ``n_channels`` elements of
    ``theta`` are ``[log σ₁, …, log σₖ]`` and are sampled jointly with the
    model parameters.

    **Marginalised σ** — ``noise_std=None, marginalise_noise=True``:
    each σᵢ² is given an Inverse-Gamma(αᵢ, βᵢ) prior and integrated out
    analytically.  Works for *any* nonlinear forward model:

        log p(y|θ) = Σᵢ  -(αᵢ + nᵢ/2) · log(βᵢ + 0.5·||rᵢ||²)

    Parameters
    ----------
    forward_model : callable
        ``f(theta_model) -> array-like, shape (n_obs,)``.
    y_obs : array-like, shape (n_obs,)
        Observed data (flat vector).
    noise_std : float, array-like, or None
        Fixed noise standard deviation(s).  Options:

        - ``float`` — same σ for every observation.
        - ``array of length n_obs`` — per-observation σ.
        - ``array of length n_channels`` — one σ per channel group;
          requires ``groups`` to be set.
        - ``None`` — σ is free (estimated or marginalised).

    groups : list of array-like of int, or None
        Index groups defining channels.  E.g.::

            groups=[[0,1,2,3], [4,5,6,7]]   # 2 channels of 4 obs each

        ``None`` (default) treats all observations as one channel.
        When ``noise_std`` is a per-channel array or ``None``, ``groups``
        is required.

    marginalise_noise : bool
        If ``True`` (and ``noise_std=None``), analytically marginalise each
        σᵢ² using independent Inverse-Gamma priors.  Default ``False``.

    inv_gamma_alpha : float or array-like
        Shape parameter(s) α of the Inverse-Gamma prior on σᵢ².
        Scalar → same for all channels.  Default 1.0.

    inv_gamma_beta : float or array-like
        Scale parameter(s) β of the Inverse-Gamma prior on σᵢ².
        Scalar → same for all channels.  Default 1e-4.

    Examples
    --------
    Scalar fixed noise::

        ll = GaussianNoiseLikelihood(forward_model=f, y_obs=y, noise_std=0.05)

    Per-channel fixed noise (2 frequencies, 4 repetitions each)::

        ll = GaussianNoiseLikelihood(
            forward_model=f, y_obs=y,
            noise_std=[0.05, 0.10],
            groups=[[0,1,2,3], [4,5,6,7]],
        )

    Per-channel estimated noise (2 free log_σ params appended to theta)::

        ll = GaussianNoiseLikelihood(
            forward_model=f, y_obs=y,
            groups=[[0,1,2,3], [4,5,6,7]],
        )
        # theta = [*model_params, log_sigma_1, log_sigma_2]

    Per-channel marginalised noise::

        ll = GaussianNoiseLikelihood(
            forward_model=f, y_obs=y,
            groups=[[0,1,2,3], [4,5,6,7]],
            marginalise_noise=True,
            inv_gamma_alpha=[2.0, 2.0],
            inv_gamma_beta=[0.05**2, 0.10**2],
        )
    """

    def __init__(self, forward_model, y_obs,
                 noise_std: Union[float, list, np.ndarray, None] = None,
                 groups=None,
                 marginalise_noise: bool = False,
                 inv_gamma_alpha: Union[float, list, np.ndarray] = 1.0,
                 inv_gamma_beta: Union[float, list, np.ndarray] = 1e-4):

        self._forward = forward_model
        self._y_obs = np.asarray(y_obs, dtype=float).ravel()
        n_obs = self._y_obs.size

        # --- resolve groups --------------------------------------------------
        if groups is None:
            self._groups = [np.arange(n_obs)]
        else:
            self._groups = [np.asarray(g, dtype=int) for g in groups]
            # validate
            all_idx = np.concatenate(self._groups)
            if len(all_idx) != n_obs or set(all_idx.tolist()) != set(range(n_obs)):
                raise ValueError(
                    "groups must partition range(n_obs) exactly once."
                )
        n_channels = len(self._groups)

        # --- resolve noise_std -----------------------------------------------
        if marginalise_noise and noise_std is not None:
            raise ValueError(
                "marginalise_noise=True is incompatible with a fixed noise_std."
            )
        self._marginalise = marginalise_noise

        if noise_std is None:
            self._sigmas = None          # free or marginalised
        else:
            arr = np.asarray(noise_std, dtype=float).ravel()
            if arr.size == 1:
                # broadcast scalar to all obs
                self._sigmas = np.full(n_obs, float(arr[0]))
            elif arr.size == n_channels:
                # one sigma per channel → broadcast to per-obs
                self._sigmas = np.empty(n_obs)
                for sigma_i, idx in zip(arr, self._groups):
                    self._sigmas[idx] = sigma_i
            elif arr.size == n_obs:
                self._sigmas = arr
            else:
                raise ValueError(
                    f"noise_std has {arr.size} elements; expected 1, "
                    f"{n_channels} (n_channels), or {n_obs} (n_obs)."
                )
            if np.any(self._sigmas <= 0):
                raise ValueError("All noise_std values must be positive.")

        self._n_channels = n_channels

        # --- InvGamma hyperparameters ----------------------------------------
        alpha_arr = np.asarray(inv_gamma_alpha, dtype=float).ravel()
        beta_arr  = np.asarray(inv_gamma_beta,  dtype=float).ravel()
        self._alpha = np.broadcast_to(alpha_arr, (n_channels,)).copy()
        self._beta  = np.broadcast_to(beta_arr,  (n_channels,)).copy()

    # -------------------------------------------------------------------------

    @property
    def estimate_noise(self):
        """True if log σᵢ are free parameters (last n_channels elements of theta)."""
        return self._sigmas is None and not self._marginalise

    @property
    def marginalise_noise(self):
        """True if σᵢ² are analytically marginalised out."""
        return self._marginalise

    @property
    def n_noise_params(self):
        """Number of free noise parameters appended to theta (0 if fixed/marginalised)."""
        return self._n_channels if self.estimate_noise else 0

    def __call__(self, theta):
        """Evaluate log p(y | theta).

        Parameters
        ----------
        theta : array-like
            - Fixed / marginalised: model parameters only.
            - Estimated: model parameters followed by ``n_noise_params``
              values of ``log σᵢ``.

        Returns
        -------
        float
        """
        theta = np.asarray(theta, dtype=float)

        if self.estimate_noise:
            theta_model = theta[:-self._n_channels]
            log_sigmas   = theta[-self._n_channels:]
            sigmas = np.exp(log_sigmas)
            # broadcast to per-obs
            sigma_arr = np.empty(self._y_obs.size)
            for s, idx in zip(sigmas, self._groups):
                sigma_arr[idx] = s
        elif self._sigmas is not None:
            theta_model = theta
            sigma_arr   = self._sigmas
        else:
            theta_model = theta
            sigma_arr   = None   # marginalised

        y_pred = np.asarray(self._forward(theta_model), dtype=float).ravel()
        residuals = self._y_obs - y_pred

        if self._marginalise:
            ll = 0.0
            for k, idx in enumerate(self._groups):
                r = residuals[idx]
                rss = float(np.dot(r, r))
                n_k = len(idx)
                a = self._alpha[k] + 0.5 * n_k
                b = self._beta[k]  + 0.5 * rss
                ll -= a * np.log(b)
            return ll

        # fixed or estimated
        # Both residual factors must be scaled: the exponent is
        # sum(r_i^2 / sigma_i^2), not sum(r_i^2 / sigma_i).
        scaled = residuals / sigma_arr
        return float(-0.5 * np.dot(scaled, scaled) - np.sum(np.log(sigma_arr)))

    def posterior_sigma(self, theta):
        """Posterior mean of each σᵢ given model parameters (marginalised mode).

        Parameters
        ----------
        theta : array-like
            Model parameters only.

        Returns
        -------
        np.ndarray, shape (n_channels,)
            ``sqrt(E[σᵢ² | theta, y])`` for each channel.
        """
        if not self._marginalise:
            raise RuntimeError(
                "posterior_sigma is only available in marginalised mode."
            )
        theta = np.asarray(theta, dtype=float)
        y_pred = np.asarray(self._forward(theta), dtype=float).ravel()
        residuals = self._y_obs - y_pred

        result = np.empty(self._n_channels)
        for k, idx in enumerate(self._groups):
            r = residuals[idx]
            rss = float(np.dot(r, r))
            n_k = len(idx)
            a = self._alpha[k] + 0.5 * n_k
            b = self._beta[k]  + 0.5 * rss
            if a <= 1:
                raise RuntimeError(
                    f"alpha[{k}] + n_k/2 must be > 1 for a finite posterior mean."
                )
            result[k] = np.sqrt(b / (a - 1))
        return result

Attributes

estimate_noise property 

estimate_noise

True if log σᵢ are free parameters (last n_channels elements of theta).

marginalise_noise property 

marginalise_noise

True if σᵢ² are analytically marginalised out.

n_noise_params property 

n_noise_params

Number of free noise parameters appended to theta (0 if fixed/marginalised).

Methods:

__call__ 

__call__(theta)

Evaluate log p(y | theta).

Parameters:

Name Type Description Default
theta array - like
  • Fixed / marginalised: model parameters only.
  • Estimated: model parameters followed by n_noise_params values of log σᵢ.
 required

Returns:

Type Description
float
Source code in mcmckit/core/noise.py 
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def __call__(self, theta):
    """Evaluate log p(y | theta).

    Parameters
    ----------
    theta : array-like
        - Fixed / marginalised: model parameters only.
        - Estimated: model parameters followed by ``n_noise_params``
          values of ``log σᵢ``.

    Returns
    -------
    float
    """
    theta = np.asarray(theta, dtype=float)

    if self.estimate_noise:
        theta_model = theta[:-self._n_channels]
        log_sigmas   = theta[-self._n_channels:]
        sigmas = np.exp(log_sigmas)
        # broadcast to per-obs
        sigma_arr = np.empty(self._y_obs.size)
        for s, idx in zip(sigmas, self._groups):
            sigma_arr[idx] = s
    elif self._sigmas is not None:
        theta_model = theta
        sigma_arr   = self._sigmas
    else:
        theta_model = theta
        sigma_arr   = None   # marginalised

    y_pred = np.asarray(self._forward(theta_model), dtype=float).ravel()
    residuals = self._y_obs - y_pred

    if self._marginalise:
        ll = 0.0
        for k, idx in enumerate(self._groups):
            r = residuals[idx]
            rss = float(np.dot(r, r))
            n_k = len(idx)
            a = self._alpha[k] + 0.5 * n_k
            b = self._beta[k]  + 0.5 * rss
            ll -= a * np.log(b)
        return ll

    # fixed or estimated
    # Both residual factors must be scaled: the exponent is
    # sum(r_i^2 / sigma_i^2), not sum(r_i^2 / sigma_i).
    scaled = residuals / sigma_arr
    return float(-0.5 * np.dot(scaled, scaled) - np.sum(np.log(sigma_arr)))

posterior_sigma 

posterior_sigma(theta)

Posterior mean of each σᵢ given model parameters (marginalised mode).

Parameters:

Name Type Description Default
theta array - like

Model parameters only.

 required

Returns:

Type Description
(ndarray, shape(n_channels))

sqrt(E[σᵢ² | theta, y]) for each channel.
Source code in mcmckit/core/noise.py 
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def posterior_sigma(self, theta):
    """Posterior mean of each σᵢ given model parameters (marginalised mode).

    Parameters
    ----------
    theta : array-like
        Model parameters only.

    Returns
    -------
    np.ndarray, shape (n_channels,)
        ``sqrt(E[σᵢ² | theta, y])`` for each channel.
    """
    if not self._marginalise:
        raise RuntimeError(
            "posterior_sigma is only available in marginalised mode."
        )
    theta = np.asarray(theta, dtype=float)
    y_pred = np.asarray(self._forward(theta), dtype=float).ravel()
    residuals = self._y_obs - y_pred

    result = np.empty(self._n_channels)
    for k, idx in enumerate(self._groups):
        r = residuals[idx]
        rss = float(np.dot(r, r))
        n_k = len(idx)
        a = self._alpha[k] + 0.5 * n_k
        b = self._beta[k]  + 0.5 * rss
        if a <= 1:
            raise RuntimeError(
                f"alpha[{k}] + n_k/2 must be > 1 for a finite posterior mean."
            )
        result[k] = np.sqrt(b / (a - 1))
    return result