psiphy.toy_models

class psiphy.toy_models.gaussian.GaussianSignal(N=20, mean_range=(-5, 5), sigma_range=(0.1, 5))

Simple 1-D Gaussian signal model with known analytic likelihood.

Parameters:

• N (int) – Number of data points per simulation.

• mean_range (tuple) – Prior range for the mean parameter.

• sigma_range (tuple) – Prior range for the sigma parameter (must be > 0).

Notes

Parameters: theta = [mean, sigma]. Analytic log-likelihood available via log_prob().

param_names = ['mean', 'sigma']

sample_prior()

simulate(theta)

log_prob(theta, x)

analytic_posterior(x, mean_prior_sigma=10.0)

Return (posterior_mean, posterior_sigma) for a flat-ish Gaussian prior on mean.

Assumes sigma is known to equal the sample std (plug-in).

psiphy.toy_models.gaussian.gaussian_signal

alias of GaussianSignal

class psiphy.toy_models.line_fitting.NoisyLine(Nx=50, x_range=(0, 10), noise_sigma=0.5, slope_range=(-3, 3), intercept_range=(0, 10))

Noisy linear model: y = slope * x + intercept + noise.

Parameters:

• Nx (int) – Number of x points.

• x_range (tuple) – Range of x values.

• noise_sigma (float) – Fixed Gaussian noise std (used for log_prob).

• slope_range (tuple) – Prior range for slope.

• intercept_range (tuple) – Prior range for intercept.

Notes

Parameters: theta = [slope, intercept]. Analytic log-likelihood available via log_prob() (assumes fixed noise_sigma).

param_names = ['slope', 'intercept']

sample_prior()

simulate(theta)

log_prob(theta, x)

psiphy.toy_models.line_fitting.noisy_line

alias of NoisyLine

class psiphy.toy_models.ma2.MA2Model(n_obs=100)

Moving Average(2) process — standard LFI benchmark.

Generative model: x_t = w_t + theta_1 * w_{t-1} + theta_2 * w_{t-2}, where w_t ~ N(0, 1) i.i.d.

Parameters:

n_obs (int) – Length of the time series.

Notes

Parameters: theta = [theta_1, theta_2]. The prior is uniform on the MA(2) invertibility region: |theta_2| < 1, theta_2 + theta_1 > -1, theta_2 - theta_1 > -1.

The likelihood is intractable; log_prob returns None. Summary statistics returned by simulate: [mean, variance, lag-1 autocov, lag-2 autocov].

param_names = ['theta_1', 'theta_2']

sample_prior()

simulate_raw(theta)

Return the raw time series (length n_obs).

simulate(theta)

Return 4-element summary-statistic vector: [mean, var, lag-1 autocov, lag-2 autocov].

log_prob(theta, x)

class psiphy.toy_models.cosmological.PowerLawSpectrum(ells=None, noise_sigma=0.05, ell_pivot=100.0, A_s_range=(0.5, 2.0), n_s_range=(0.85, 1.15))

Simple power-law angular power spectrum toy model.

Simulates noisy band-power measurements of a CMB-like spectrum:

C_ell = A_s * (ell / ell_pivot) ** (n_s - 1)

with independent Gaussian noise on each band power.

Parameters:

• ells (array-like or None) – Multipole values at which the spectrum is evaluated. Defaults to 20 log-spaced values from 10 to 2000.

• noise_sigma (float) – Noise std per band power (in same units as C_ell).

• ell_pivot (float) – Pivot multipole for the tilt (default 100).

• A_s_range (tuple) – Prior range for the amplitude A_s.

• n_s_range (tuple) – Prior range for the spectral index n_s.

Notes

Parameters: theta = [A_s, n_s]. Analytic Gaussian log-likelihood available via log_prob().

param_names = ['A_s', 'n_s']

sample_prior()

simulate(theta)

log_prob(theta, x)

class psiphy.toy_models.lotka_volterra.LotkaVolterra(t_obs=None, x0=10.0, y0=5.0, noise_sigma=0.5, alpha_range=(0.5, 2.0), beta_range=(0.02, 0.2), delta_range=(0.02, 0.2), gamma_range=(0.5, 2.0))

Lotka-Volterra predator-prey model.

Stochastic observations of the ODE system:

dx/dt =  alpha * x - beta  * x * y   (prey)
dy/dt =  delta * x * y - gamma * y   (predator)

with independent Gaussian noise added to each species at each observation time, making the likelihood tractable.

Parameters:

• t_obs (array-like or None) – Observation times. Defaults to 20 equally-spaced points on [0, 15].

• x0 (float) – Initial prey population.

• y0 (float) – Initial predator population.

• noise_sigma (float) – Gaussian noise std added to each observation.

• alpha_range (tuple) – Prior range for prey growth rate alpha.

• beta_range (tuple) – Prior range for predation rate beta.

• delta_range (tuple) – Prior range for predator growth rate delta.

• gamma_range (tuple) – Prior range for predator death rate gamma.

Notes

Parameters: theta = [alpha, beta, delta, gamma].

Typical values: alpha ~ 1.0, beta ~ 0.1, delta ~ 0.075, gamma ~ 1.5.

simulate() returns a 1-D array of length 2 * len(t_obs): prey observations followed by predator observations. log_prob() returns the analytic Gaussian log-likelihood.

param_names = ['alpha', 'beta', 'delta', 'gamma']

sample_prior()

simulate(theta)

Integrate the ODE and add Gaussian noise.

Returns:

Prey observations followed by predator observations. Returns NaN vector if the ODE solver fails.

Return type:

np.ndarray, shape (2 * n_obs,)

noiseless(theta)

Return noiseless ODE solution, shape (2, n_obs): [prey, predator].

log_prob(theta, x)

Gaussian log-likelihood of observations x given parameters theta.

Parameters:

• theta (array-like, shape (4,))

• x (array-like, shape (2 * n_obs,))

Return type:

float or -inf if ODE diverges.