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API Reference

Order 3 Samplers

Implements a third-order Higher-Order Langevin Monte Carlo (HoLMC) sampler for Bayesian linear regression. The sampler uses an underdamped Langevin dynamics approach to generate posterior samples over model parameters.

Parameters

params : O3Params Parameter object containing integration coefficients and covariance settings. N : int Number of samples to draw. Required. seed : int, optional Random seed for reproducibility. show_progress : bool, default=True Whether to display a progress bar during sampling.

Source code in holmc/samplers/o3sampler.py 
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class HoLMCSamplerO3Regression:
    """
    Implements a third-order Higher-Order Langevin Monte Carlo (HoLMC) sampler
    for Bayesian linear regression. The sampler uses an underdamped Langevin
    dynamics approach to generate posterior samples over model parameters.

    Parameters
    ----------
    params : O3Params
        Parameter object containing integration coefficients and covariance
        settings.
    N : int
        Number of samples to draw. Required.
    seed : int, optional
        Random seed for reproducibility.
    show_progress : bool, default=True
        Whether to display a progress bar during sampling.
    """

    def __init__(
        self,
        params: O3Params,
        N: int = None,
        seed: int = None,
        show_progress: bool = True,
    ):
        self.p = params
        self.N = N
        self.seed = seed
        self.show_progress = show_progress
        if seed is not None:
            np.random.seed(seed)
        if N is None:
            raise ValueError("Number of samples 'N' must be provided.")

    def sample(
        self,
        X: np.ndarray,
        y: np.ndarray,
        lamb: float = None,
        theta_random: bool = True,
    ) -> np.ndarray:
        """
        Run the HoLMC sampler to draw posterior samples for Bayesian linear
        regression.

        Parameters
        ----------
        X : np.ndarray
            Design matrix of shape (n_samples, n_features).
        y : np.ndarray
            Response vector of shape (n_samples,) or (n_samples, 1).
        lamb : float
            Regularization (prior precision) parameter. Required.
        theta_random : bool, default=True
            If True, initializes theta randomly; otherwise uses the ridge
            solution.

        Returns
        -------
        samples : np.ndarray
            Array of shape (N+1, n_features) containing sampled parameter
            vectors.
        """
        if lamb is None:
            raise ValueError(
                "Regularization parameter 'lamb' must be provided."
            )
        eta = self.p.eta
        n, d = X.shape
        A = ((X.T @ X) / n) + lamb * np.eye(d)
        b = ((X.T @ y) / n).squeeze()
        L = self.p.L

        # mu parameters
        mu12, mu13 = self.p.mu12(), self.p.mu13()
        mu22, mu23 = self.p.mu22(), self.p.mu23()
        mu31, mu32 = self.p.mu31(), self.p.mu32()
        mu33 = self.p.mu33()

        # Covariance matrix
        Sigma = self.p.sigma(d)

        CL = np.linalg.cholesky(Sigma)

        # Initiate the states
        if theta_random:
            theta = np.random.randn(d)
        else:
            theta = np.linalg.solve(A, b)
        v1 = np.zeros_like(theta)
        v2 = np.zeros_like(theta)

        samples = []
        samples.append(theta.copy())

        for _ in tqdm(range(self.N), disable=not self.show_progress):
            mu = np.concatenate([theta, v1, v2])
            z = np.random.randn(3 * d)
            x = mu + CL @ z
            theta, v1, v2 = np.split(x, 3)

            # Updates in the mean vectors
            delta_f = (
                A @ (eta * theta + (np.power(eta, 2) / 2.0) * v1) - eta * b
            )

            theta = theta - (eta / (2.0 * L)) * delta_f + mu12 * v1 + mu13 * v2
            v1 = -(1 / L) * delta_f + mu22 * v1 + mu23 * v2
            v2 = (mu31 / L) * delta_f + mu32 * v1 + mu33 * v2

            samples.append(theta.copy())
        return np.array(samples)

sample(X, y, lamb=None, theta_random=True)

Run the HoLMC sampler to draw posterior samples for Bayesian linear regression.

Parameters

X : np.ndarray Design matrix of shape (n_samples, n_features). y : np.ndarray Response vector of shape (n_samples,) or (n_samples, 1). lamb : float Regularization (prior precision) parameter. Required. theta_random : bool, default=True If True, initializes theta randomly; otherwise uses the ridge solution.

Returns

samples : np.ndarray Array of shape (N+1, n_features) containing sampled parameter vectors.

Source code in holmc/samplers/o3sampler.py 
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def sample(
    self,
    X: np.ndarray,
    y: np.ndarray,
    lamb: float = None,
    theta_random: bool = True,
) -> np.ndarray:
    """
    Run the HoLMC sampler to draw posterior samples for Bayesian linear
    regression.

    Parameters
    ----------
    X : np.ndarray
        Design matrix of shape (n_samples, n_features).
    y : np.ndarray
        Response vector of shape (n_samples,) or (n_samples, 1).
    lamb : float
        Regularization (prior precision) parameter. Required.
    theta_random : bool, default=True
        If True, initializes theta randomly; otherwise uses the ridge
        solution.

    Returns
    -------
    samples : np.ndarray
        Array of shape (N+1, n_features) containing sampled parameter
        vectors.
    """
    if lamb is None:
        raise ValueError(
            "Regularization parameter 'lamb' must be provided."
        )
    eta = self.p.eta
    n, d = X.shape
    A = ((X.T @ X) / n) + lamb * np.eye(d)
    b = ((X.T @ y) / n).squeeze()
    L = self.p.L

    # mu parameters
    mu12, mu13 = self.p.mu12(), self.p.mu13()
    mu22, mu23 = self.p.mu22(), self.p.mu23()
    mu31, mu32 = self.p.mu31(), self.p.mu32()
    mu33 = self.p.mu33()

    # Covariance matrix
    Sigma = self.p.sigma(d)

    CL = np.linalg.cholesky(Sigma)

    # Initiate the states
    if theta_random:
        theta = np.random.randn(d)
    else:
        theta = np.linalg.solve(A, b)
    v1 = np.zeros_like(theta)
    v2 = np.zeros_like(theta)

    samples = []
    samples.append(theta.copy())

    for _ in tqdm(range(self.N), disable=not self.show_progress):
        mu = np.concatenate([theta, v1, v2])
        z = np.random.randn(3 * d)
        x = mu + CL @ z
        theta, v1, v2 = np.split(x, 3)

        # Updates in the mean vectors
        delta_f = (
            A @ (eta * theta + (np.power(eta, 2) / 2.0) * v1) - eta * b
        )

        theta = theta - (eta / (2.0 * L)) * delta_f + mu12 * v1 + mu13 * v2
        v1 = -(1 / L) * delta_f + mu22 * v1 + mu23 * v2
        v2 = (mu31 / L) * delta_f + mu32 * v1 + mu33 * v2

        samples.append(theta.copy())
    return np.array(samples)

Implements a third-order Higher-Order Langevin Monte Carlo (HoLMC) sampler for Bayesian logistic regression. This sampler uses higher-order approximations of the potential energy gradient to improve sampling efficiency in classification tasks.

Parameters

params : O3Params Parameter object containing integration coefficients and covariance settings. N : int Number of samples to draw. Required. seed : int Random seed for reproducibility. show_progress : bool, default=True Whether to display a progress bar during sampling.

Source code in holmc/samplers/o3sampler.py 
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class HoLMCSamplerO3Classification:
    """
    Implements a third-order Higher-Order Langevin Monte Carlo (HoLMC)
    sampler for Bayesian logistic regression. This sampler uses higher-order
    approximations of the potential energy gradient to improve sampling
    efficiency in classification tasks.

    Parameters
    ----------
    params : O3Params
        Parameter object containing integration coefficients and covariance
        settings.
    N : int
        Number of samples to draw. Required.
    seed : int
        Random seed for reproducibility.
    show_progress : bool, default=True
        Whether to display a progress bar during sampling.
    """

    def __init__(
        self,
        params: O3Params,
        N: int = None,
        seed: int = None,
        show_progress: bool = True,
    ):
        self.params = params
        self.N = N
        self.seed = seed
        self.show_progress = show_progress
        if seed is not None:
            np.random.seed(seed)
        if N is None:
            raise ValueError("Number of samples 'N' must be provided.")

    def sample(
        self, X: np.ndarray, y: np.ndarray, lamb: float = None
    ) -> np.ndarray:
        if lamb is None:
            raise ValueError(
                "Regularization parameter 'lamb' must be provided."
            )
        d = X.shape[1]
        eta = self.params.eta
        L = self.params.L

        # mu parameters
        mu12, mu13 = self.params.mu12(), self.params.mu13()
        mu22, mu23 = self.params.mu22(), self.params.mu23()
        mu31, mu32 = self.params.mu31(), self.params.mu32()
        mu33 = self.params.mu33()

        # Covariance matrix
        Sigma = self.params.sigma(d)

        CL = np.linalg.cholesky(Sigma)

        # Initiate the states
        theta = np.random.randn(d)
        # theta_star = np.random.randn(d)
        # for _ in range(self.N):
        #     g = gradient(theta_star, X, y, lamb)
        #     theta_star -= eta * g
        # theta = theta_star.copy()
        v1 = np.zeros_like(theta)
        v2 = np.zeros_like(theta)

        samples = []
        samples.append(theta.copy())

        for _ in tqdm(range(self.N), disable=not self.show_progress):
            # Sample from the multivariate normal distribution
            mu = np.concatenate([theta, v1, v2])
            z = np.random.randn(3 * d)
            x = mu + CL @ z
            theta, v1, v2 = np.split(x, 3)
            # Updates in the mean vectors
            s = sigmoid(X @ theta)
            xv1 = X @ v1
            deltaU = (
                eta * (lamb * theta + X.T @ (s - y))
                + (eta**2 / 2) * (X.T @ (s * (1 - s) * xv1) + lamb * v1)
                + (eta**3 / 6) * (X.T @ (s * (1 - s) * (1 - 2 * s) * xv1**2))
                + (eta**4 / 24)
                * (X.T @ (s * (1 - s) * (1 - 6 * s + 6 * s**2) * xv1**3))
            )
            theta = theta - (eta / (2.0 * L)) * deltaU + mu12 * v1 + mu13 * v2
            v1 = -(1 / L) * deltaU + mu22 * v1 + mu23 * v2
            v2 = (mu31 / L) * deltaU + mu32 * v1 + mu33 * v2

            samples.append(theta.copy())
        return np.array(samples)

Order 4 Samplers

Implements a fourth-order Higher-Order Langevin Monte Carlo (HoLMC) sampler for Bayesian linear regression. This sampler uses a 4-stage underdamped Langevin dynamics scheme to generate high-quality posterior samples from the Bayesian model.

Parameters

params : O4Params Parameter object containing algorithmic coefficients, step size (eta), and damping (gamma). N : int Number of MCMC samples to draw. Must be specified. seed : int Random seed for reproducibility. show_progress : bool, default=True Whether to display a progress bar during sampling.

Source code in holmc/samplers/o4sampler.py 
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class HoLMCSamplerO4Regression:
    """
    Implements a fourth-order Higher-Order Langevin Monte Carlo (HoLMC)
    sampler for Bayesian linear regression. This sampler uses a 4-stage
    underdamped Langevin dynamics scheme to generate high-quality posterior
    samples from the Bayesian model.

    Parameters
    ----------
    params : O4Params
        Parameter object containing algorithmic coefficients, step size (eta),
        and damping (gamma).
    N : int
        Number of MCMC samples to draw. Must be specified.
    seed : int
        Random seed for reproducibility.
    show_progress : bool, default=True
        Whether to display a progress bar during sampling.
    """

    def __init__(
        self,
        params: O4Params,
        N: int = None,
        seed: int = None,
        show_progress: bool = True,
    ):
        self.params = params
        self.N = N
        self.seed = seed
        self.show_progress = show_progress
        self.rg = Regression()
        if seed is not None:
            np.random.seed(seed)
        if N is None:
            raise ValueError("Number of samples 'N' must be provided.")

    def sample(
        self,
        X: np.ndarray,
        y: np.ndarray,
        lamb: float = None,
        theta_random: bool = True,
    ) -> np.ndarray:
        """
        Run the fourth-order HoLMC sampler to draw posterior samples for
        Bayesian linear regression.

        Parameters
        ----------
        X : np.ndarray
            Design matrix of shape (n_samples, n_features).
        y : np.ndarray
            Response vector of shape (n_samples,) or (n_samples, 1).
        lamb : float
            Ridge regularization parameter (controls the prior precision). Must
            be specified.
        theta_random : bool, default=True
            Whether to initialize theta randomly or from the ridge solution.

        Returns
        -------
        samples : np.ndarray
            Array of shape (N+1, n_features) containing the sequence of sampled
            parameter vectors.
        """
        if lamb is None:
            raise ValueError(
                "Regularization parameter 'lamb' must be provided."
            )
        gamma = self.params.gamma
        eta = self.params.eta
        n, d = X.shape
        A = ((X.T @ X) / n) + lamb * np.eye(d)
        b = ((X.T @ y) / n).squeeze()

        # mu parameters
        mu01, mu02 = self.params.mu01(), self.params.mu02()
        mu03, mu11 = self.params.mu03(), self.params.mu11()
        mu12, mu13 = self.params.mu12(), self.params.mu13()
        mu21, mu22 = self.params.mu21(), self.params.mu22()
        mu23, mu31 = self.params.mu23(), self.params.mu31()
        mu32, mu33 = self.params.mu32(), self.params.mu33()

        # Covariance matrix
        Sigma = self.params.sigma(d)
        CL = np.linalg.cholesky(Sigma)

        # Initiate the states
        if theta_random:
            theta = np.random.randn(d)
        else:
            theta = np.linalg.solve(A, b)
        v1 = np.zeros_like(theta)
        v2 = np.zeros_like(theta)
        v3 = np.zeros_like(theta)

        samples = []
        samples.append(theta.copy())

        for _ in tqdm(range(self.N), disable=not self.show_progress):
            mu = np.concatenate([theta, v1, v2, v3])
            z = np.random.randn(4 * d)
            x = mu + CL @ z
            theta, v1, v2, v3 = np.split(x, 4)

            # Update in the mean vectors
            theta = self.rg.update_theta(
                theta, v1, v2, v3, A, b, gamma, eta, mu01, mu02, mu03
            )
            v1 = self.rg.update_v1(
                theta, v1, v2, v3, A, b, gamma, eta, mu11, mu12, mu13
            )
            v2 = self.rg.update_v2(
                theta, v1, v2, v3, A, b, gamma, eta, mu21, mu22, mu23
            )
            v3 = self.rg.update_v3(
                theta, v1, v2, v3, A, b, gamma, eta, mu31, mu32, mu33
            )

            samples.append(theta.copy())
        return np.array(samples)

sample(X, y, lamb=None, theta_random=True)

Run the fourth-order HoLMC sampler to draw posterior samples for Bayesian linear regression.

Parameters

X : np.ndarray Design matrix of shape (n_samples, n_features). y : np.ndarray Response vector of shape (n_samples,) or (n_samples, 1). lamb : float Ridge regularization parameter (controls the prior precision). Must be specified. theta_random : bool, default=True Whether to initialize theta randomly or from the ridge solution.

Returns

samples : np.ndarray Array of shape (N+1, n_features) containing the sequence of sampled parameter vectors.

Source code in holmc/samplers/o4sampler.py 
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def sample(
    self,
    X: np.ndarray,
    y: np.ndarray,
    lamb: float = None,
    theta_random: bool = True,
) -> np.ndarray:
    """
    Run the fourth-order HoLMC sampler to draw posterior samples for
    Bayesian linear regression.

    Parameters
    ----------
    X : np.ndarray
        Design matrix of shape (n_samples, n_features).
    y : np.ndarray
        Response vector of shape (n_samples,) or (n_samples, 1).
    lamb : float
        Ridge regularization parameter (controls the prior precision). Must
        be specified.
    theta_random : bool, default=True
        Whether to initialize theta randomly or from the ridge solution.

    Returns
    -------
    samples : np.ndarray
        Array of shape (N+1, n_features) containing the sequence of sampled
        parameter vectors.
    """
    if lamb is None:
        raise ValueError(
            "Regularization parameter 'lamb' must be provided."
        )
    gamma = self.params.gamma
    eta = self.params.eta
    n, d = X.shape
    A = ((X.T @ X) / n) + lamb * np.eye(d)
    b = ((X.T @ y) / n).squeeze()

    # mu parameters
    mu01, mu02 = self.params.mu01(), self.params.mu02()
    mu03, mu11 = self.params.mu03(), self.params.mu11()
    mu12, mu13 = self.params.mu12(), self.params.mu13()
    mu21, mu22 = self.params.mu21(), self.params.mu22()
    mu23, mu31 = self.params.mu23(), self.params.mu31()
    mu32, mu33 = self.params.mu32(), self.params.mu33()

    # Covariance matrix
    Sigma = self.params.sigma(d)
    CL = np.linalg.cholesky(Sigma)

    # Initiate the states
    if theta_random:
        theta = np.random.randn(d)
    else:
        theta = np.linalg.solve(A, b)
    v1 = np.zeros_like(theta)
    v2 = np.zeros_like(theta)
    v3 = np.zeros_like(theta)

    samples = []
    samples.append(theta.copy())

    for _ in tqdm(range(self.N), disable=not self.show_progress):
        mu = np.concatenate([theta, v1, v2, v3])
        z = np.random.randn(4 * d)
        x = mu + CL @ z
        theta, v1, v2, v3 = np.split(x, 4)

        # Update in the mean vectors
        theta = self.rg.update_theta(
            theta, v1, v2, v3, A, b, gamma, eta, mu01, mu02, mu03
        )
        v1 = self.rg.update_v1(
            theta, v1, v2, v3, A, b, gamma, eta, mu11, mu12, mu13
        )
        v2 = self.rg.update_v2(
            theta, v1, v2, v3, A, b, gamma, eta, mu21, mu22, mu23
        )
        v3 = self.rg.update_v3(
            theta, v1, v2, v3, A, b, gamma, eta, mu31, mu32, mu33
        )

        samples.append(theta.copy())
    return np.array(samples)

Implements a fourth-order Higher-Order Langevin Monte Carlo (HoLMC) sampler for Bayesian logistic regression. This sampler uses a high-order underdamped Langevin dynamic with correction terms up to the third derivative of the sigmoid to approximate the gradient of the posterior.

Parameters

params : O4Params Parameter object containing algorithmic coefficients, step size (eta), and damping (gamma). N : int Number of MCMC samples to draw. Must be specified. seed : int Random seed for reproducibility. show_progress : bool, default=True Whether to display a progress bar during sampling.

Source code in holmc/samplers/o4sampler.py 
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class HoLMCSamplerO4Classification:
    """
    Implements a fourth-order Higher-Order Langevin Monte Carlo (HoLMC) sampler
    for Bayesian logistic regression. This sampler uses a high-order
    underdamped Langevin dynamic with correction terms up to the third
    derivative of the sigmoid to approximate the gradient of the posterior.

    Parameters
    ----------
    params : O4Params
        Parameter object containing algorithmic coefficients, step size (eta),
        and damping (gamma).
    N : int
        Number of MCMC samples to draw. Must be specified.
    seed : int
        Random seed for reproducibility.
    show_progress : bool, default=True
        Whether to display a progress bar during sampling.
    """

    def __init__(
        self,
        params: O4Params,
        N: int = None,
        seed: int = None,
        show_progress: bool = True,
    ):
        self.params = params
        self.N = N
        self.seed = seed
        self.show_progress = show_progress
        self.cl = Classification()
        if seed is not None:
            np.random.seed(seed)
        if N is None:
            raise ValueError("Number of samples 'N' must be provided.")

    def sample(
        self, X: np.ndarray, y: np.ndarray, lamb: float = None
    ) -> np.ndarray:
        if lamb is None:
            raise ValueError(
                "Regularization parameter 'lamb' must be provided."
            )
        d = X.shape[1]
        eta = self.params.eta
        gamma = self.params.gamma

        # mu parameters
        mu01, mu02 = self.params.mu01(), self.params.mu02()
        mu03, mu11 = self.params.mu03(), self.params.mu11()
        mu12, mu13 = self.params.mu12(), self.params.mu13()
        mu21, mu22 = self.params.mu21(), self.params.mu22()
        mu23, mu31 = self.params.mu23(), self.params.mu31()
        mu32, mu33 = self.params.mu32(), self.params.mu33()

        # Covariance matrix
        Sigma = self.params.sigma(d)
        CL = np.linalg.cholesky(Sigma)

        # Initiate the states
        theta = np.random.randn(d)
        # theta_star = np.random.randn(d)
        # for _ in range(self.N):
        #     g = gradient(theta_star, X, y, lamb)
        #     theta_star -= eta * g
        # theta = theta_star.copy()
        v1 = np.zeros_like(theta)
        v2 = np.zeros_like(theta)
        v3 = np.zeros_like(theta)

        samples = []
        samples.append(theta.copy())

        for _ in tqdm(range(self.N), disable=not self.show_progress):
            mu = np.concatenate([theta, v1, v2, v3])
            z = np.random.randn(4 * d)
            x = mu + CL @ z
            theta, v1, v2, v3 = np.split(x, 4)
            # Updates in the mean vectors
            theta = self.cl.update_theta(
                theta, v1, v2, v3, X, y, gamma, eta, lamb, mu01, mu02, mu03
            )
            v1 = self.cl.update_v1(
                theta, v1, v2, v3, X, y, gamma, eta, lamb, mu11, mu12, mu13
            )
            v2 = self.cl.update_v2(
                theta, v1, v2, v3, X, y, gamma, eta, lamb, mu21, mu22, mu23
            )
            v3 = self.cl.update_v3(
                theta, v1, v2, v3, X, y, gamma, eta, lamb, mu31, mu32, mu33
            )

            samples.append(theta.copy())
        return np.array(samples)

Parameters

Source code in holmc/utils/params.py 
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class O3Params:
    def __init__(
        self,
        gamma: float = None,
        eta: float = None,
        xi: float = None,
        L: float = 1.0
    ):
        self.gamma = gamma
        self.eta = eta
        self.xi = xi
        self.L = L

    def mu12(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (
            (1 + gamma**2 / xi**2) * eta
            - (gamma**2 / (2 * xi)) * eta**2
            - (gamma**2 / xi**3) * (1 - np.exp(-xi * eta))
        )
        return val

    def mu13(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (gamma / xi) * eta + (gamma / xi**2) * (np.exp(-xi * eta) - 1)
        return val

    def mu22(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = 1 + (gamma**2 / xi**2) * (1 - xi * eta - np.exp(-xi * eta))
        return val

    def mu23(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (gamma / xi) * (1 - np.exp(-xi * eta))
        return val

    def mu31(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (gamma / xi) - (gamma / xi**2) * ((1 - np.exp(-xi * eta)) / eta)
        return val

    def mu32(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (
            (gamma**3 / xi**2) * eta
            + (gamma**3 / xi**2) * eta * np.exp(-xi * eta)
            - (2 * gamma**3 / xi**3 + gamma / xi) * (1 - np.exp(-xi * eta))
        )
        return val

    def mu33(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        val = (
            np.exp(-xi * eta)
            + (gamma**2 / xi) * eta * np.exp(-xi * eta)
            - (gamma**2 / xi**2) * (1 - np.exp(-xi * eta))
        )
        return val

    # Sigma entries
    def sigma11(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        exp1 = np.exp(-eta * xi)
        exp2 = np.exp(-2 * eta * xi)
        val = (
            (2 * np.power(gamma, 2) * eta) / np.power(xi, 3)
            - (2 * np.power(gamma, 2) * np.power(eta, 2)) / np.power(xi, 2)
            + (2 * np.power(gamma, 2) * np.power(eta, 3)) / (3 * xi)
            - (4 * np.power(gamma, 2) * eta * exp1) / np.power(xi, 3)
            + (np.power(gamma, 2) * (1 - exp2)) / np.power(xi, 4)
        )
        return val

    def sigma12(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        L = self.L
        exp1 = np.exp(-eta * xi)
        val = (np.power(gamma, 2) * np.power((xi * eta - (1 - exp1)), 2)) / (
            np.power(xi, 3) * L
        )
        return val

    def sigma22(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        exp1 = np.exp(-eta * xi)
        exp2 = np.exp(-2 * eta * xi)
        val = (
            (2 * np.power(gamma, 2) * eta) / xi
            - (4 * np.power(gamma, 2) * (1 - exp1)) / np.power(xi, 2)
            + (np.power(gamma, 2) * (1 - exp2)) / np.power(xi, 2)
        )
        return val

    def sigma13(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        exp1 = np.exp(-eta * xi)
        exp2 = np.exp(-2 * eta * xi)
        val = (
            -(np.power(gamma, 3) * np.power(eta, 2) * (2 * exp1 + 1))
            / np.power(xi, 2)
            + eta
            * (
                (2 * np.power(gamma, 3)) / np.power(xi, 3)
                - (np.power(gamma, 3) * exp2) / np.power(xi, 3)
                - (4 * np.power(gamma, 3) * exp1) / np.power(xi, 3)
                - (2 * gamma * exp1) / xi
            )
            + (
                ((3 * np.power(gamma, 3)) / (2 * np.power(xi, 4)))
                + gamma / np.power(xi, 2)
            )
            * (1 - exp2)
        )
        return val

    def sigma23(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        exp1 = np.exp(-eta * xi)
        exp2 = np.exp(-2 * eta * xi)
        val = (
            (np.power(gamma, 3) * (exp2 - 2 * exp1 - 2) * eta)
            / np.power(xi, 2)
            + (3 * np.power(gamma, 3) * (exp2 - 4 * exp1 + 3))
            / (2 * np.power(xi, 3))
            + (gamma * np.power((1 - exp1), 2)) / xi
        )
        return val

    def sigma33(self):
        gamma = self.gamma
        eta = self.eta
        xi = self.xi
        exp1 = np.exp(-eta * xi)
        exp2 = np.exp(-2 * eta * xi)
        val = (
            -(np.power(gamma, 4) * np.power(eta, 2) * exp2) / np.power(xi, 2)
            + eta
            * (
                -(2 * np.power(gamma, 2) * exp2) / xi
                + (np.power(gamma, 4) * (-3 * exp2 + 4 * exp1 + 2))
                / np.power(xi, 3)
            )
            + (np.power(gamma, 4) * (-5 * exp2 + 16 * exp1 - 11))
            / (2 * np.power(xi, 4))
            + (
                (np.power(gamma, 2) * (-3 * exp2 + 4 * exp1 - 1))
                / np.power(xi, 2)
            )
            + (1 - exp2)
        )
        return val

    def sigma(self, d):
        s11 = self.sigma11()
        s12 = self.sigma12()
        s13 = self.sigma13()
        s22 = self.sigma22()
        s23 = self.sigma23()
        s33 = self.sigma33()
        I_d = np.eye(d)

        # Now construct the full (3*dim x 3*dim) matrix
        top = np.hstack([s11 * I_d, s12 * I_d, s13 * I_d])
        mid = np.hstack([s12 * I_d, s22 * I_d, s23 * I_d])
        bot = np.hstack([s13 * I_d, s23 * I_d, s33 * I_d])

        mat = np.vstack([top, mid, bot])
        epsilon = 0.00001
        matrix = mat + epsilon * np.eye(3 * d)

        return matrix
Source code in holmc/utils/params.py 
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class O4Params:
    def __init__(
        self,
        gamma: float = None,
        eta: float = None
    ):
        self.gamma = gamma
        self.eta = eta

    def mu01(self):
        gamma = self.gamma
        eta = self.eta
        term1 = eta
        term2 = -(gamma**2) * (eta**3) / 6
        term3 = (gamma**4) * (eta**5) / 120

        exp_term = -np.exp(-gamma * eta) / (gamma**5)
        constant_terms = (
            +1 / gamma**5
            - eta / gamma**4
            + eta**2 / (2 * gamma**3)
            - eta**3 / (6 * gamma**2)
            + eta**4 / (24 * gamma)
        )

        term4 = gamma**4 * (exp_term + constant_terms)
        val = term1 + term2 + term3 + term4
        return val

    def mu02(self):
        gamma = self.gamma
        eta = self.eta
        term1 = gamma * (eta**2) / 2
        term2 = -(gamma**3) * (eta**4) / 24

        bracket = (
            np.exp(-gamma * eta) / (gamma**4)
            - 1 / (gamma**4)
            + eta / (gamma**3)
            - eta**2 / (2 * gamma**2)
            + eta**3 / (6 * gamma)
        )

        term3 = -(gamma**3) * bracket
        val = term1 + term2 + term3

        return val

    def mu03(self):
        gamma = self.gamma
        eta = self.eta
        term1 = -(gamma**4) * (eta**5) / 120

        bracket1 = (
            -np.exp(-gamma * eta) / (gamma**3)
            + 1 / (gamma**3)
            - eta / (gamma**2)
            + eta**2 / (2 * gamma)
        )

        bracket2 = (
            -np.exp(-gamma * eta) / (gamma**5)
            + 1 / (gamma**5)
            - eta / (gamma**4)
            + eta**2 / (2 * gamma**3)
            - eta**3 / (6 * gamma**2)
            + eta**4 / (24 * gamma)
        )

        term2 = gamma**2 * bracket1
        term3 = -(gamma**4) * bracket2

        val = term1 + term2 + term3

        return val

    def mu11(self):
        gamma = self.gamma
        eta = self.eta
        term1 = 1
        term2 = -(gamma**2) * (eta**2) / 2
        term3 = (gamma**4) * (eta**4) / 24

        bracket = (
            np.exp(-gamma * eta) / (gamma**4)
            - 1 / (gamma**4)
            + eta / (gamma**3)
            - eta**2 / (2 * gamma**2)
            + eta**3 / (6 * gamma)
        )

        term4 = gamma**4 * bracket

        val = term1 + term2 + term3 + term4

        return val

    def mu12(self):
        gamma = self.gamma
        eta = self.eta
        term1 = gamma * eta
        term2 = -(gamma**3) * (eta**3) / 6

        bracket = (
            -np.exp(-gamma * eta) / (gamma**3)
            + 1 / (gamma**3)
            - eta / (gamma**2)
            + eta**2 / (2 * gamma)
        )

        term3 = -(gamma**3) * bracket

        val = term1 + term2 + term3

        return val

    def mu13(self):
        gamma = self.gamma
        eta = self.eta
        term1 = -(gamma**4) * (eta**4) / 24

        bracket1 = (
            np.exp(-gamma * eta) / (gamma**2) - 1 / (gamma**2) + eta / gamma
        )

        bracket2 = (
            np.exp(-gamma * eta) / (gamma**4)
            - 1 / (gamma**4)
            + eta / (gamma**3)
            - eta**2 / (2 * gamma**2)
            + eta**3 / (6 * gamma)
        )

        term2 = gamma**2 * bracket1
        term3 = -(gamma**4) * bracket2

        val = term1 + term2 + term3

        return val

    def mu21(self):
        gamma = self.gamma
        eta = self.eta
        # Leading terms
        term1 = -gamma * eta
        term2 = gamma**3 * eta**3 / 6
        term3 = -(gamma**5) * eta**5 / 120

        # First bracket (with gamma^5)
        bracket1 = (
            -np.exp(-gamma * eta) / gamma**5
            + 1 / gamma**5
            - eta / gamma**4
            + eta**2 / (2 * gamma**3)
            - eta**3 / (6 * gamma**2)
            + eta**4 / (24 * gamma)
        )
        term4 = -(gamma**5) * bracket1

        # Second bracket (with gamma^3)
        bracket2 = (
            -np.exp(-gamma * eta) / gamma**3
            + 1 / gamma**3
            - eta / gamma**2
            + eta**2 / (2 * gamma)
        )
        term5 = gamma**3 * bracket2

        # Third bracket (gamma^5 / 3!)
        bracket3 = (
            -6 * np.exp(-gamma * eta) / gamma**5
            + 6 / gamma**5
            - 6 * eta / gamma**4
            + 3 * eta**2 / gamma**3
            - eta**3 / gamma**2
            + eta**4 / (4 * gamma)
        )
        term6 = -(gamma**5 / 6) * bracket3

        # Fourth bracket (gamma^5)
        bracket4 = (
            4 * np.exp(-gamma * eta) / gamma**5
            - 4 / gamma**5
            + eta * np.exp(-gamma * eta) / gamma**4
            + 3 * eta / gamma**4
            - eta**2 / gamma**3
            + eta**3 / (6 * gamma**2)
        )
        term7 = -(gamma**5) * bracket4

        val = term1 + term2 + term3 + term4 + term5 + term6 + term7

        return val

    def mu22(self):
        gamma = self.gamma
        eta = self.eta
        term1 = 1
        term2 = -(gamma**2) * eta**2 / 2
        term3 = gamma**4 * eta**4 / 24

        # Bracket 1
        bracket1 = (
            np.exp(-gamma * eta) / gamma**4
            - 1 / gamma**4
            + eta / gamma**3
            - eta**2 / (2 * gamma**2)
            + eta**3 / (6 * gamma)
        )
        term4 = gamma**4 * bracket1

        # Bracket 2
        bracket2 = np.exp(-gamma * eta) / gamma**2 - 1 / gamma**2 + eta / gamma
        term5 = -(gamma**2) * bracket2

        # Bracket 3 (coefficient γ⁴ / 2!)
        bracket3 = (
            2 * np.exp(-gamma * eta) / gamma**4
            - 2 / gamma**4
            + 2 * eta / gamma**3
            - eta**2 / gamma**2
            + eta**3 / (3 * gamma)
        )
        term6 = (gamma**4 / 2) * bracket3

        # Bracket 4
        bracket4 = (
            -3 * np.exp(-gamma * eta) / gamma**4
            + 3 / gamma**4
            - eta * np.exp(-gamma * eta) / gamma**3
            - 2 * eta / gamma**3
            + eta**2 / (2 * gamma**2)
        )
        term7 = gamma**4 * bracket4

        val = term1 + term2 + term3 + term4 + term5 + term6 + term7

        return val

    def mu23(self):
        gamma = self.gamma
        eta = self.eta
        # Term 1
        term1 = gamma**5 * eta**5 / 120

        # Bracket 1 (gamma^3)
        bracket1 = (
            -np.exp(-gamma * eta) / gamma**3
            + 1 / gamma**3
            - eta / gamma**2
            + eta**2 / (2 * gamma)
        )
        term2 = -(gamma**3) * bracket1

        # Bracket 2 (gamma^5)
        bracket2 = (
            -np.exp(-gamma * eta) / gamma**5
            + 1 / gamma**5
            - eta / gamma**4
            + eta**2 / (2 * gamma**3)
            - eta**3 / (6 * gamma**2)
            + eta**4 / (24 * gamma)
        )
        term3 = gamma**5 * bracket2

        # Term 4
        term4 = 1 - np.exp(-gamma * eta)

        # Bracket 3 (gamma^5 / 3!)
        bracket3 = (
            -6 * np.exp(-gamma * eta) / gamma**5
            + 6 / gamma**5
            - 6 * eta / gamma**4
            + 3 * eta**2 / gamma**3
            - eta**3 / gamma**2
            + eta**4 / (4 * gamma)
        )
        term5 = (gamma**5 / 6) * bracket3

        # Bracket 4 (gamma^5)
        bracket4 = (
            4 * np.exp(-gamma * eta) / gamma**5
            - 4 / gamma**5
            + eta * np.exp(-gamma * eta) / gamma**4
            + 3 * eta / gamma**4
            - eta**2 / gamma**3
            + eta**3 / (6 * gamma**2)
        )
        term6 = gamma**5 * bracket4

        val = term1 + term2 + term3 + term4 + term5 + term6

        return val

    def mu31(self):
        gamma = self.gamma
        eta = self.eta
        exp = np.exp(-gamma * eta)

        # Term 1
        bracket1 = exp / gamma**2 - 1 / gamma**2 + eta / gamma
        term1 = gamma**2 * bracket1

        # Term 2
        bracket2 = (
            6 * exp / gamma**4
            - 6 / gamma**4
            + 6 * eta / gamma**3
            - 3 * eta**2 / gamma**2
            + eta**3 / gamma
        )
        term2 = -(gamma**4 / 6) * bracket2

        # Term 3
        bracket3 = (
            120 * exp / gamma**6
            - 120 / gamma**6
            + 120 * eta / gamma**5
            - 60 * eta**2 / gamma**4
            + 20 * eta**3 / gamma**3
            - 5 * eta**4 / gamma**2
            + eta**5 / gamma
        )
        term3 = (gamma**6 / 120) * bracket3

        # Term 4
        bracket4 = (
            -5 * exp / gamma**6
            + 5 / gamma**6
            - eta * exp / gamma**5
            - 4 * eta / gamma**5
            + 3 * eta**2 / (2 * gamma**4)
            - eta**3 / (3 * gamma**3)
            + eta**4 / (24 * gamma**2)
        )
        term4 = gamma**6 * bracket4

        # Term 5
        bracket5 = (
            -3 * exp / gamma**4
            + 3 / gamma**4
            - eta * exp / gamma**3
            - 2 * eta / gamma**3
            + eta**2 / (2 * gamma**2)
        )
        term5 = -(gamma**4) * bracket5

        # Term 6
        bracket6 = (
            -30 * exp / gamma**6
            + 30 / gamma**6
            - 6 * eta * exp / gamma**5
            - 24 * eta / gamma**5
            + 9 * eta**2 / gamma**4
            - 2 * eta**3 / gamma**3
            + eta**4 / (4 * gamma**2)
        )
        term6 = (gamma**6 / 6) * bracket6

        # Term 7
        bracket7 = (
            10 * exp / gamma**6
            - 10 / gamma**6
            + 4 * eta * exp / gamma**5
            + 6 * eta / gamma**5
            + eta**2 * exp / (2 * gamma**4)
            - 3 * eta**2 / (2 * gamma**4)
            + eta**3 / (6 * gamma**3)
        )
        term7 = gamma**6 * bracket7

        val = term1 + term2 + term3 + term4 + term5 + term6 + term7

        return val

    def mu32(self):
        gamma = self.gamma
        eta = self.eta
        exp = np.exp(-gamma * eta)

        # Term 1
        bracket1 = 1 / gamma - exp / gamma
        term1 = -gamma * bracket1

        # Term 2
        bracket2 = (
            -2 * exp / gamma**3
            + 2 / gamma**3
            - 2 * eta / gamma**2
            + eta**2 / gamma
        )
        term2 = (gamma**3 / 2) * bracket2

        # Term 3
        bracket3 = (
            -24 * exp / gamma**5
            + 24 / gamma**5
            - 24 * eta / gamma**4
            + 12 * eta**2 / gamma**3
            - 4 * eta**3 / gamma**2
            + eta**4 / gamma
        )
        term3 = -(gamma**5 / 24) * bracket3

        # Term 4
        bracket4 = (
            4 * exp / gamma**5
            - 4 / gamma**5
            + eta * exp / gamma**4
            + 3 * eta / gamma**4
            - eta**2 / gamma**3
            + eta**3 / (6 * gamma**2)
        )
        term4 = -(gamma**5) * bracket4

        # Term 5
        bracket5 = (
            2 * exp / gamma**3
            - 2 / gamma**3
            + eta * exp / gamma**2
            + eta / gamma**2
        )
        term5 = gamma**3 * bracket5

        # Term 6
        bracket6 = (
            8 * exp / gamma**5
            - 8 / gamma**5
            + 2 * eta * exp / gamma**4
            + 6 * eta / gamma**4
            - 2 * eta**2 / gamma**3
            + eta**3 / (3 * gamma**2)
        )
        term6 = -(gamma**5 / 2) * bracket6

        # Term 7
        bracket7 = (
            -6 * exp / gamma**5
            + 6 / gamma**5
            - 3 * eta * exp / gamma**4
            - 3 * eta / gamma**4
            - eta**2 * exp / (2 * gamma**3)
            + eta**2 / (2 * gamma**3)
        )
        term7 = -(gamma**5) * bracket7

        val = term1 + term2 + term3 + term4 + term5 + term6 + term7

        return val

    def mu33(self):
        gamma = self.gamma
        eta = self.eta
        exp = np.exp(-gamma * eta)

        # Term 1
        term1 = exp

        # Term 2
        bracket2 = (
            120 * exp / gamma**6
            - 120 / gamma**6
            + 120 * eta / gamma**5
            - 60 * eta**2 / gamma**4
            + 20 * eta**3 / gamma**3
            - 5 * eta**4 / gamma**2
            + eta**5 / gamma
        )
        term2 = -(gamma**6 / 120) * bracket2

        # Term 3
        bracket3 = (
            -3 * exp / gamma**4
            + 3 / gamma**4
            - eta * exp / gamma**3
            - 2 * eta / gamma**3
            + eta**2 / (2 * gamma**2)
        )
        term3 = gamma**4 * bracket3

        # Term 4
        bracket4 = (
            -5 * exp / gamma**6
            + 5 / gamma**6
            - eta * exp / gamma**5
            - 4 * eta / gamma**5
            + 3 * eta**2 / (2 * gamma**4)
            - eta**3 / (3 * gamma**3)
            + eta**4 / (24 * gamma**2)
        )
        term4 = -(gamma**6) * bracket4

        # Term 5
        bracket5 = -exp / gamma**2 + 1 / gamma**2 - eta * exp / gamma
        term5 = -(gamma**2) * bracket5

        # Term 6
        bracket6 = (
            -30 * exp / gamma**6
            + 30 / gamma**6
            - 6 * eta * exp / gamma**5
            - 24 * eta / gamma**5
            + 9 * eta**2 / gamma**4
            - 2 * eta**3 / gamma**3
            + eta**4 / (4 * gamma**2)
        )
        term6 = -(gamma**6 / 6) * bracket6

        # Term 7
        bracket7 = (
            3 * exp / gamma**4
            - 3 / gamma**4
            + 2 * eta * exp / gamma**3
            + eta / gamma**3
            + eta**2 * exp / (2 * gamma**2)
        )
        term7 = gamma**4 * bracket7

        # Term 8
        bracket8 = (
            10 * exp / gamma**6
            - 10 / gamma**6
            + 4 * eta * exp / gamma**5
            + 6 * eta / gamma**5
            + eta**2 * exp / (2 * gamma**4)
            - 3 * eta**2 / (2 * gamma**4)
            + eta**3 / (6 * gamma**3)
        )
        term8 = -(gamma**6) * bracket8

        val = term1 + term2 + term3 + term4 + term5 + term6 + term7 + term8

        return val

    def sigma00(self):
        gamma = self.gamma
        eta = self.eta
        val = (
            ((2 * eta) / gamma)
            - (
                (np.exp(-2 * gamma * eta) - 4 * np.exp(-gamma * eta) + 3)
                / np.power(gamma, 2)
            )
            + ((4 * np.power(eta, 3) * gamma) / 3.0)
            + 2 * np.power(eta, 2) * np.exp(-eta * gamma)
            - 2 * np.power(eta, 2)
            - ((np.power(eta, 4) * np.power(gamma, 2)) / 2)
            + ((np.power(eta, 5) * np.power(gamma, 3)) / 10)
        )
        return val

    def sigma11(self):
        gamma = self.gamma
        eta = self.eta
        val = (
            (2 * eta * gamma)
            - np.exp(-2 * eta * gamma)
            - (2 * np.power(eta, 2) * np.power(gamma, 2))
            + ((2 * np.power(eta, 3) * np.power(gamma, 3)) / 3)
            - (4 * eta * gamma * np.exp(-eta * gamma))
            + 1
        )
        return val

    def sigma22(self):
        gamma = self.gamma
        eta = self.eta
        val = (
            (4 * np.exp(-eta * gamma))
            - ((13 * np.exp(-2 * eta * gamma)) / 2)
            + (8 * eta * gamma)
            - ((4 * np.power(eta, 3) * np.power(gamma, 3)) / 3)
            + ((np.power(eta, 5) * np.power(gamma, 5)) / 10)
            - (
                10
                * (np.power(eta, 2) * np.power(gamma, 2))
                * np.exp(-eta * gamma)
            )
            - (np.power(eta, 2) * np.power(gamma, 2)
                * np.exp(-2 * eta * gamma))
            - (
                2
                * (np.power(eta, 3) * np.power(gamma, 3))
                * np.exp(-eta * gamma)
            )
            - 12 * eta * gamma * np.exp(-eta * gamma)
            - 5 * eta * gamma * np.exp(-2 * eta * gamma)
            + 2.5
        )
        return val

    def sigma33(self):
        gamma = self.gamma
        eta = self.eta
        val = (
            (283 * eta) / 4
            + 88 * np.exp(-eta * gamma)
            - (397 * np.exp(-2 * eta * gamma)) / 8
            + 32 * eta * gamma
            + 84 * eta * np.exp(-eta * gamma)
            - (101 * eta * np.exp(-2 * eta * gamma)) / 4
            + (159 * eta) / (2 * gamma)
            - 24 * eta**2 * gamma
            + 6 * eta**3 * gamma
            + 32 * eta**2 * np.exp(-eta * gamma)
            - (eta**2 * np.exp(-2 * eta * gamma)) / 2
            + (204 * np.exp(-eta * gamma)) / gamma
            - (149 * np.exp(-2 * eta * gamma)) / (4 * gamma)
            + (192 * np.exp(-eta * gamma)) / gamma**2
            - (21 * np.exp(-2 * eta * gamma)) / (2 * gamma**2)
            - (39 * eta**2) / 2
            - 667 / (4 * gamma)
            - 363 / (2 * gamma**2)
            - 8 * eta**2 * gamma**2
            + (22 * eta**3 * gamma**2) / 3
            + (2 * eta**3 * gamma**3) / 3
            - 2 * eta**4 * gamma**2
            - (4 * eta**4 * gamma**3) / 3
            + (7 * eta**5 * gamma**3) / 15
            + (eta**5 * gamma**4) / 10
            - (eta**6 * gamma**4) / 15
            + (eta**7 * gamma**5) / 210
            + (96 * eta * np.exp(-eta * gamma)) / gamma
            - (9 * eta * np.exp(-2 * eta * gamma)) / (2 * gamma)
            + 36 * eta**2 * gamma * np.exp(-eta * gamma)
            - 6 * eta**2 * gamma * np.exp(-2 * eta * gamma)
            + 4 * eta**3 * gamma * np.exp(-eta * gamma)
            - 10 * eta**2 * gamma**2 * np.exp(-eta * gamma)
            - (77 * eta**2 * gamma**2 * np.exp(-2 * eta * gamma)) / 4
            + 10 * eta**3 * gamma**2 * np.exp(-eta * gamma)
            - (eta**3 * gamma**2 * np.exp(-2 * eta * gamma)) / 2
            - 2 * eta**3 * gamma**3 * np.exp(-eta * gamma)
            - (7 * eta**3 * gamma**3 * np.exp(-2 * eta * gamma)) / 2
            + eta**4 * gamma**3 * np.exp(-eta * gamma)
            - (eta**4 * gamma**4 * np.exp(-2 * eta * gamma)) / 4
            + 8 * eta * gamma * np.exp(-eta * gamma)
            - (197 * eta * gamma * np.exp(-2 * eta * gamma)) / 4
            - 307 / 8
        )
        return val

    def sigma01(self):
        gamma = self.gamma
        eta = self.eta
        exp1 = np.exp(eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)
        inner = (
            2 * exp1 + eta**2 * gamma**2 * exp1
            - 2 * eta * gamma * exp1 - 2
        )
        val = (exp2 * inner**2) / (4 * gamma)
        return val

    def sigma02(self):
        gamma = self.gamma
        eta = self.eta

        exp1 = np.exp(-eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)

        val = (
            4 * eta
            + 2 * eta * exp1
            - eta * exp2
            - 2 * eta**2 * gamma
            + (10 * exp1) / gamma
            - (5 * exp2) / (2 * gamma)
            - (15) / (2 * gamma)
            + (eta**3 * gamma**2) / 3
            + (eta**4 * gamma**3) / 4
            - (eta**5 * gamma**4) / 10
            + 2 * eta**2 * gamma * exp1
            + eta**3 * gamma**2 * exp1
        )
        return val

    def sigma03(self):
        gamma = self.gamma
        eta = self.eta
        exp1 = np.exp(-eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)

        val = (
            (7 * eta * exp2) / 2
            - 18 * eta * exp1
            - 8 * eta
            - (21 * eta) / (2 * gamma)
            + 5 * eta**2 * gamma
            - (5 * eta**3 * gamma) / 3
            - 8 * eta**2 * exp1
            - (36 * exp1) / gamma
            + (27 * exp2) / (4 * gamma)
            - (32 * exp1) / gamma**2
            + (2 * exp2) / gamma**2
            + 3 * eta**2
            + 117 / (4 * gamma)
            + 30 / gamma**2
            - 2 * eta**3 * gamma**2
            + (2 * eta**4 * gamma**2) / 3
            + (eta**4 * gamma**3) / 4
            - (3 * eta**5 * gamma**3) / 20
            + (eta**6 * gamma**4) / 60
            - (18 * eta * exp1) / gamma
            + (eta * exp2) / (2 * gamma)
            - 12 * eta**2 * gamma * exp1
            + (eta**2 * gamma * exp2) / 2
            - eta**3 * gamma * exp1
            - 4 * eta**3 * gamma**2 * exp1
            - (eta**4 * gamma**3 * exp1) / 2
        )
        return val

    def sigma12(self):
        gamma = self.gamma
        eta = self.eta
        exp1 = np.exp(-eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)

        val = (
            (5 * exp2) / 2
            - 4 * eta * gamma
            + 2 * eta**2 * gamma**2
            + (eta**3 * gamma**3) / 3
            - (eta**4 * gamma**4) / 4
            + 3 * eta**2 * gamma**2 * exp1
            + 8 * eta * gamma * exp1
            + eta * gamma * exp2
            - 5 / 2
        )
        return val

    def sigma13(self):
        gamma = self.gamma
        eta = self.eta

        exp1 = np.exp(-eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)

        val = (
            8 * eta * gamma
            - 4 * exp1
            - (27 * exp2) / 4
            - (3 * eta) / 2
            - 12 * eta * exp1
            - (eta * exp2) / 2
            - 3 * eta**2 * gamma
            - (10 * exp1) / gamma
            - (2 * exp2) / gamma
            + 12 / gamma
            - 5 * eta**2 * gamma**2
            + (5 * eta**3 * gamma**2) / 3
            + (2 * eta**3 * gamma**3) / 3
            - (5 * eta**4 * gamma**3) / 12
            + (eta**5 * gamma**4) / 20
            - eta**2 * gamma * exp1
            - 8 * eta**2 * gamma**2 * exp1
            - (eta**2 * gamma**2 * exp2) / 2
            - eta**3 * gamma**3 * exp1
            - 22 * eta * gamma * exp1
            - (7 * eta * gamma * exp2) / 2
            + 43 / 4
        )
        return val

    def sigma23(self):
        gamma = self.gamma
        eta = self.eta
        exp1 = np.exp(-eta * gamma)
        exp2 = np.exp(-2 * eta * gamma)

        val = (
            (143 * exp2) / 8
            - 12 * exp1
            - 7 * eta
            - 16 * eta * gamma
            + 18 * eta * exp1
            + (7 * eta * exp2) / 2
            + 6 * eta**2 * gamma
            + (2 * exp1) / gamma
            + (25 * exp2) / (4 * gamma)
            - 33 / (4 * gamma)
            + 2 * eta**2 * gamma**2
            - (4 * eta**3 * gamma**2) / 3
            + (4 * eta**3 * gamma**3) / 3
            - (eta**4 * gamma**3) / 12
            - (eta**4 * gamma**4) / 4
            + (eta**5 * gamma**4) / 10
            - (eta**6 * gamma**5) / 60
            + 5 * eta**2 * gamma * exp1
            + (eta**2 * gamma * exp2) / 2
            + 20 * eta**2 * gamma**2 * exp1
            + (19 * eta**2 * gamma**2 * exp2) / 4
            + 5 * eta**3 * gamma**3 * exp1
            + (eta**3 * gamma**3 * exp2) / 2
            + (eta**4 * gamma**4 * exp1) / 2
            + 24 * eta * gamma * exp1
            + (63 * eta * gamma * exp2) / 4
            - 47 / 8
        )
        return val

    def sigma(self, d):
        s00 = self.sigma00()
        s01 = self.sigma01()
        s02 = self.sigma02()
        s03 = self.sigma03()
        s11 = self.sigma11()
        s12 = self.sigma12()
        s13 = self.sigma13()
        s22 = self.sigma22()
        s23 = self.sigma23()
        s33 = self.sigma33()

        I_d = np.eye(d)

        top_mat = np.hstack([s00 * I_d, s01 * I_d, s02 * I_d, s03 * I_d])
        mid_mat1 = np.hstack([s01 * I_d, s11 * I_d, s12 * I_d, s13 * I_d])
        mid_mat2 = np.hstack([s02 * I_d, s12 * I_d, s22 * I_d, s23 * I_d])
        bot_mat = np.hstack([s03 * I_d, s13 * I_d, s23 * I_d, s33 * I_d])

        mat = np.vstack([top_mat, mid_mat1, mid_mat2, bot_mat])
        epsilon = 0.00001
        matrix = mat + epsilon * np.eye(4 * d)

        return matrix

Base class for performing grid search over Langevin Monte Carlo hyperparameters.

Parameters

gammas : list List of gamma (friction) values to try. etas : list List of eta (step size) values to try. xis : list List of xi values (used only for third-order methods). If None, fourth-order is assumed. N : int Number of MCMC samples to draw for each parameter configuration. Required. seed : int Random seed for reproducibility. show_progress : bool, default=True Whether to show a progress bar during the grid search.

Source code in holmc/utils/gridsearch.py 
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class GridSearch:
    """
    Base class for performing grid search over Langevin Monte Carlo
    hyperparameters.

    Parameters
    ----------
    gammas : list
        List of gamma (friction) values to try.
    etas : list
        List of eta (step size) values to try.
    xis : list
        List of xi values (used only for third-order methods). If None,
        fourth-order is assumed.
    N : int
        Number of MCMC samples to draw for each parameter configuration.
        Required.
    seed : int
        Random seed for reproducibility.
    show_progress : bool, default=True
        Whether to show a progress bar during the grid search.
    """

    def __init__(
        self,
        gammas: list = None,
        etas: list = None,
        xis: list = None,
        N: int = None,
        seed: int = None,
        show_progress: bool = True,
    ):
        self.gammas = gammas
        self.etas = etas
        self.xis = xis
        self.N = N
        self.seed = seed
        self.show_progress = show_progress
        if seed is not None:
            np.random.seed(seed)
        if N is None:
            raise ValueError("N must be specified for grid search.")

Bases: GridSearch

Grid search implementation for Bayesian linear regression using HoLMC samplers.

This class runs over a grid of (eta, gamma, xi) parameters and evaluates performance using the Wasserstein-2 distance between the sampled distribution and the target posterior.

Methods

run(X, y, lamb): Executes the grid search and returns a DataFrame of results sorted by W2 distance.

Source code in holmc/utils/gridsearch.py 
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class GridSearchRegression(GridSearch):
    """
    Grid search implementation for Bayesian linear regression using HoLMC
    samplers.

    This class runs over a grid of (eta, gamma, xi) parameters and evaluates
    performance
    using the Wasserstein-2 distance between the sampled distribution and the
    target posterior.

    Methods
    -------
    run(X, y, lamb):
        Executes the grid search and returns a DataFrame of results sorted by
        W2 distance.
    """

    def run(self, X: np.ndarray, y: np.ndarray, lamb: float = None):
        """
        Run the grid search for regression.

        Parameters
        ----------
        X : np.ndarray
            Design matrix of shape (n_samples, n_features).
        y : np.ndarray
            Response vector of shape (n_samples,) or (n_samples, 1).
        lamb : float
            Regularization (prior precision) parameter. Required.

        Returns
        -------
        pd.DataFrame
            DataFrame containing the grid search results sorted by minimum
            Wasserstein-2 distance.
        """
        if lamb is None:
            raise ValueError("lamb must be specified for grid search.")

        if X is None or y is None:
            raise ValueError("X and y must be provided for grid search.")

        N = self.N

        results = []

        is_third_order = self.xis is not None

        for eta in tqdm(self.etas, disable=not self.show_progress, desc="Eta"):
            for gamma in self.gammas:
                xi_list = self.xis if is_third_order else [None]
                for xi in xi_list:
                    try:
                        if is_third_order:
                            params = O3Params(eta=eta, gamma=gamma, xi=xi)
                            sampler = HoLMCSamplerO3Regression(
                                params=params,
                                N=N,
                                seed=self.seed,
                                show_progress=False,
                            )
                        else:
                            params = O4Params(eta=eta, gamma=gamma)
                            sampler = HoLMCSamplerO4Regression(
                                params=params,
                                N=N,
                                seed=self.seed,
                                show_progress=False,
                            )
                        samples = sampler.sample(X, y, lamb)
                        metric = Wasserstein2Distance(X=X, y=y)
                        dist = metric.w2distance(samples)
                        min_dist = np.nanmin(dist)

                        result = {
                            "gamma": gamma,
                            "eta": eta,
                            "w2dist": min_dist,
                        }
                        if is_third_order:
                            result["xi"] = xi
                        results.append(result)
                    except Exception:
                        result = {"gamma": gamma, "eta": eta, "w2dist": np.nan}
                        if is_third_order:
                            result["xi"] = xi
                        results.append(result)

        self.results_df = pd.DataFrame(results).sort_values("w2dist")
        return self.results_df

run(X, y, lamb=None)

Run the grid search for regression.

Parameters

X : np.ndarray Design matrix of shape (n_samples, n_features). y : np.ndarray Response vector of shape (n_samples,) or (n_samples, 1). lamb : float Regularization (prior precision) parameter. Required.

Returns

pd.DataFrame DataFrame containing the grid search results sorted by minimum Wasserstein-2 distance.

Source code in holmc/utils/gridsearch.py 
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def run(self, X: np.ndarray, y: np.ndarray, lamb: float = None):
    """
    Run the grid search for regression.

    Parameters
    ----------
    X : np.ndarray
        Design matrix of shape (n_samples, n_features).
    y : np.ndarray
        Response vector of shape (n_samples,) or (n_samples, 1).
    lamb : float
        Regularization (prior precision) parameter. Required.

    Returns
    -------
    pd.DataFrame
        DataFrame containing the grid search results sorted by minimum
        Wasserstein-2 distance.
    """
    if lamb is None:
        raise ValueError("lamb must be specified for grid search.")

    if X is None or y is None:
        raise ValueError("X and y must be provided for grid search.")

    N = self.N

    results = []

    is_third_order = self.xis is not None

    for eta in tqdm(self.etas, disable=not self.show_progress, desc="Eta"):
        for gamma in self.gammas:
            xi_list = self.xis if is_third_order else [None]
            for xi in xi_list:
                try:
                    if is_third_order:
                        params = O3Params(eta=eta, gamma=gamma, xi=xi)
                        sampler = HoLMCSamplerO3Regression(
                            params=params,
                            N=N,
                            seed=self.seed,
                            show_progress=False,
                        )
                    else:
                        params = O4Params(eta=eta, gamma=gamma)
                        sampler = HoLMCSamplerO4Regression(
                            params=params,
                            N=N,
                            seed=self.seed,
                            show_progress=False,
                        )
                    samples = sampler.sample(X, y, lamb)
                    metric = Wasserstein2Distance(X=X, y=y)
                    dist = metric.w2distance(samples)
                    min_dist = np.nanmin(dist)

                    result = {
                        "gamma": gamma,
                        "eta": eta,
                        "w2dist": min_dist,
                    }
                    if is_third_order:
                        result["xi"] = xi
                    results.append(result)
                except Exception:
                    result = {"gamma": gamma, "eta": eta, "w2dist": np.nan}
                    if is_third_order:
                        result["xi"] = xi
                    results.append(result)

    self.results_df = pd.DataFrame(results).sort_values("w2dist")
    return self.results_df

Bases: GridSearch

Grid search implementation for Bayesian logistic regression using HoLMC samplers.

This class evaluates sampler performance over a grid of parameters using predictive accuracy based on samples averaged over time.

Methods

run(X, y, lamb): Executes the grid search and returns a DataFrame sorted by maximum accuracy. compute_accuracy(X, y, samples): Computes predictive accuracy from the sample sequence.

Source code in holmc/utils/gridsearch.py 
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class GridSearchClassification(GridSearch):
    """
    Grid search implementation for Bayesian logistic regression using HoLMC
    samplers.

    This class evaluates sampler performance over a grid of parameters using
    predictive accuracy
    based on samples averaged over time.

    Methods
    -------
    run(X, y, lamb):
        Executes the grid search and returns a DataFrame sorted by maximum
        accuracy.
    compute_accuracy(X, y, samples):
        Computes predictive accuracy from the sample sequence.
    """

    def compute_accuracy(self, X, y, samples):
        """
        Compute classification accuracy over cumulative average of sampled
        parameters.

        Parameters
        ----------
        X : np.ndarray
            Feature matrix of shape (n_samples, n_features).
        y : np.ndarray
            Binary target labels (0 or 1) of shape (n_samples,).
        samples : np.ndarray
            Array of sampled parameter vectors of shape (n_samples, n_features)

        Returns
        -------
        np.ndarray
            Array of accuracy scores as function of the sample index.
        """
        n = samples.shape[0]
        accuracies = []

        for i in range(1, n + 1):
            avg_theta = np.mean(samples[:i], axis=0)
            logits = X @ avg_theta
            probs = sigmoid(logits)
            y_pred = (probs >= 0.5).astype(int)
            acc = accuracy(y, y_pred)
            accuracies.append(acc)

        return np.array(accuracies)

    def run(self, X: np.ndarray, y: np.ndarray, lamb: float = None):
        """
        Run the grid search for classification.

        Parameters
        ----------
        X : np.ndarray
            Feature matrix of shape (n_samples, n_features).
        y : np.ndarray
            Binary target labels (0 or 1) of shape (n_samples,).
        lamb : float
            Regularization (prior precision) parameter. Required.

        Returns
        -------
        pd.DataFrame
            DataFrame containing the grid search results sorted by maximum
            accuracy.
        """
        if lamb is None:
            raise ValueError("lamb must be specified for grid search.")

        results = []
        N = self.N

        is_third_order = self.xis is not None

        for eta in tqdm(self.etas, disable=not self.show_progress, desc="Eta"):
            for gamma in self.gammas:
                xi_list = self.xis if is_third_order else [None]
                for xi in xi_list:
                    try:
                        if is_third_order:
                            params = O3Params(eta=eta, gamma=gamma, xi=xi)
                            sampler = HoLMCSamplerO3Classification(
                                params=params,
                                N=N,
                                seed=self.seed,
                                show_progress=False,
                            )
                        else:
                            params = O4Params(eta=eta, gamma=gamma)
                            sampler = HoLMCSamplerO4Classification(
                                params=params,
                                N=N,
                                seed=self.seed,
                                show_progress=False,
                            )
                        samples = sampler.sample(X, y, lamb)
                        ac = self.compute_accuracy(X, y, samples)
                        max_acc = np.nanmax(ac)

                        result = {
                            "gamma": gamma,
                            "eta": eta,
                            "MaxAcc": max_acc,
                        }
                        if is_third_order:
                            result["xi"] = xi
                        results.append(result)
                    except Exception:
                        result = {"gamma": gamma, "eta": eta, "MaxAcc": np.nan}
                        if is_third_order:
                            result["xi"] = xi
                        results.append(result)

        self.results_df = pd.DataFrame(results).sort_values(
            "MaxAcc", ascending=False
        )
        return self.results_df

compute_accuracy(X, y, samples)

Compute classification accuracy over cumulative average of sampled parameters.

Parameters

X : np.ndarray Feature matrix of shape (n_samples, n_features). y : np.ndarray Binary target labels (0 or 1) of shape (n_samples,). samples : np.ndarray Array of sampled parameter vectors of shape (n_samples, n_features)

Returns

np.ndarray Array of accuracy scores as function of the sample index.

Source code in holmc/utils/gridsearch.py 
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def compute_accuracy(self, X, y, samples):
    """
    Compute classification accuracy over cumulative average of sampled
    parameters.

    Parameters
    ----------
    X : np.ndarray
        Feature matrix of shape (n_samples, n_features).
    y : np.ndarray
        Binary target labels (0 or 1) of shape (n_samples,).
    samples : np.ndarray
        Array of sampled parameter vectors of shape (n_samples, n_features)

    Returns
    -------
    np.ndarray
        Array of accuracy scores as function of the sample index.
    """
    n = samples.shape[0]
    accuracies = []

    for i in range(1, n + 1):
        avg_theta = np.mean(samples[:i], axis=0)
        logits = X @ avg_theta
        probs = sigmoid(logits)
        y_pred = (probs >= 0.5).astype(int)
        acc = accuracy(y, y_pred)
        accuracies.append(acc)

    return np.array(accuracies)

run(X, y, lamb=None)

Run the grid search for classification.

Parameters

X : np.ndarray Feature matrix of shape (n_samples, n_features). y : np.ndarray Binary target labels (0 or 1) of shape (n_samples,). lamb : float Regularization (prior precision) parameter. Required.

Returns

pd.DataFrame DataFrame containing the grid search results sorted by maximum accuracy.

Source code in holmc/utils/gridsearch.py 
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def run(self, X: np.ndarray, y: np.ndarray, lamb: float = None):
    """
    Run the grid search for classification.

    Parameters
    ----------
    X : np.ndarray
        Feature matrix of shape (n_samples, n_features).
    y : np.ndarray
        Binary target labels (0 or 1) of shape (n_samples,).
    lamb : float
        Regularization (prior precision) parameter. Required.

    Returns
    -------
    pd.DataFrame
        DataFrame containing the grid search results sorted by maximum
        accuracy.
    """
    if lamb is None:
        raise ValueError("lamb must be specified for grid search.")

    results = []
    N = self.N

    is_third_order = self.xis is not None

    for eta in tqdm(self.etas, disable=not self.show_progress, desc="Eta"):
        for gamma in self.gammas:
            xi_list = self.xis if is_third_order else [None]
            for xi in xi_list:
                try:
                    if is_third_order:
                        params = O3Params(eta=eta, gamma=gamma, xi=xi)
                        sampler = HoLMCSamplerO3Classification(
                            params=params,
                            N=N,
                            seed=self.seed,
                            show_progress=False,
                        )
                    else:
                        params = O4Params(eta=eta, gamma=gamma)
                        sampler = HoLMCSamplerO4Classification(
                            params=params,
                            N=N,
                            seed=self.seed,
                            show_progress=False,
                        )
                    samples = sampler.sample(X, y, lamb)
                    ac = self.compute_accuracy(X, y, samples)
                    max_acc = np.nanmax(ac)

                    result = {
                        "gamma": gamma,
                        "eta": eta,
                        "MaxAcc": max_acc,
                    }
                    if is_third_order:
                        result["xi"] = xi
                    results.append(result)
                except Exception:
                    result = {"gamma": gamma, "eta": eta, "MaxAcc": np.nan}
                    if is_third_order:
                        result["xi"] = xi
                    results.append(result)

    self.results_df = pd.DataFrame(results).sort_values(
        "MaxAcc", ascending=False
    )
    return self.results_df