projnormal.formulas.projected_normal_iso

Formulas for the projected normal distribution with isotropic covariance matrix of \(x\).

empirical_moments(mean_x, var_x, n_samples)

Compute the mean, covariance and second moment of the variable \(y = x/\sqrt{x^T x}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\), by sampling from the distribution.

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.Tensor) – Variance of x. Shape is ().

• n_samples (int) – Number of samples.

Returns:

Dictionary with the keys mean, covariance, and second_moment, containing the empirical moments of the projected normal distribution.

Return type:

dict

log_pdf(mean_x, var_x, y)

Compute the log-pdf at points y for the distribution of the variable \(y = x/\sqrt{x^T x}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\) (isotropic covariance matrix).

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.tensor) – variance of x. shape is ().

• y (torch.Tensor) – Points where to evaluate the PDF. Shape is (n_points, n_dim).

Returns:

Log-PDF evaluated at each y. Shape is (n_points,).

Return type:

torch.Tensor

mean(mean_x, var_x)

Compute the mean of \(y = x/\sqrt{x^T x}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\). This is done using the exact analytic formulas.

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.tensor) – variance of x. shape is ().

Returns:

Expected value for the projected normal. Shape is (n_dim,).

Return type:

torch.Tensor

pdf(mean_x, var_x, y)

Compute the pdf at points y for the distribution of the variable \(y = x/\sqrt{x^T x}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\) (isotropic covariance matrix).

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.tensor) – variance of x. shape is ().

• y (torch.Tensor) – Points where to evaluate the PDF. Shape is (n_points, n_dim).

Returns:

PDF evaluated at each y. Shape is (n_points,).

Return type:

torch.Tensor

sample(mean_x, var_x, n_samples)

Sample the variable \(y = x/\sqrt{x^T x}\) where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\).

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.tensor) – variance of x. shape is ().

• n_samples (int) – Number of samples.

Returns:

Samples from the projected normal. Shape is (n_samples, n_dim).

Return type:

torch.Tensor

second_moment(mean_x, var_x)

Compute the second moment matrix of \(y = x/\sqrt{x^T x}\), where \(x \sim \mathcal{N}(\mu_x, \Sigma_x)\) and \(\Sigma_x = \sigma^2 I\). This is done using the exact analytic formulas.

Parameters:

• mean_x (torch.Tensor) – Mean of x. Shape is (n_dim,).

• var_x (torch.tensor) – variance of x. shape is ().

Returns:

Second moment matrix of \(y\). Shape is (n_dim, n_dim).

Return type:

torch.Tensor

Modules

moments	Exact moments of the projected normal distribution with isotropic covariance.
probability	Probability density function (PDF) for the projected normal distribution with isotropic covariance of the unprojected Gaussian.
sampling	Sampling functions for the isotropic projected normal distribution.