GaussianNLLLoss¶

GaussianNLLLoss computes the Negative Log-Likelihood (NLL) assuming the target values follow a Gaussian (normal) distribution. Unlike losses such as MSELoss, it allows the model to predict both the mean and the uncertainty (variance) of each prediction.

This loss is commonly used in probabilistic regression, uncertainty estimation, and Bayesian deep learning.

Signature¶

GaussianNLLLoss(
    full=False,
    eps=1e-6,
    reduction="mean"
)

Parameters¶

Parameter	Type	Description
`full`	`bool`	Whether to include the constant term \( \frac{1}{2}\log(2\pi) \) in the loss. Default is `False`.
`eps`	`float`	Small value added to the predicted variance for numerical stability. Default is `1e-6`.
`reduction`	`str`	Specifies the reduction to apply: `"mean"`, `"sum"`, or `"none"`. Default is `"mean"`.

Inputs¶

Input	Description
`pred`	Predicted mean values.
`target`	Ground-truth values.
`var`	Predicted variance for each element. Must be positive.

Returns¶

A scalar loss when using "mean" or "sum", or a tensor containing the per-element losses when using "none".

Formula¶

For each prediction:

Loss = 0.5 × (log(var) + (pred - target)² / var)

When full=True, the following constant term is also added:

0.5 × log(2π)

Internally, Aakaar computes:

var_clamped = var + eps

to improve numerical stability.

Example¶

import aakaar
from aakaar.losses import GaussianNLLLoss

pred = aakaar.tensor([2.5, 1.8, 4.1])

target = aakaar.tensor([2.8, 2.0, 3.9])

variance = aakaar.tensor([0.4, 0.2, 0.6])

criterion = GaussianNLLLoss()

loss = criterion(
    pred,
    target,
    variance
)

Including the Constant Term¶

criterion = GaussianNLLLoss(
    full=True
)

This produces the complete Gaussian negative log-likelihood.

Using Different Reductions¶

Mean (default)¶

criterion = GaussianNLLLoss(
    reduction="mean"
)

Sum¶

criterion = GaussianNLLLoss(
    reduction="sum"
)

None¶

criterion = GaussianNLLLoss(
    reduction="none"
)

Returns the loss for every prediction individually.

Typical Training Loop¶

criterion = GaussianNLLLoss()

loss = criterion(
    predicted_mean,
    targets,
    predicted_variance
)

loss.backward()
optimizer.step()

Typical Applications¶

Probabilistic regression
Uncertainty estimation
Bayesian neural networks
Time-series forecasting
Scientific prediction
Autonomous systems

Notes¶

Requires three inputs: predicted mean, target values, and predicted variance.
The predicted variance should always be positive.
A small eps value is added internally to the variance for numerical stability.
Supports "mean", "sum", and "none" reductions.
Setting full=True includes the constant Gaussian normalization term.
Particularly useful when predicting both values and their associated uncertainty.