PoissonNLLLoss¶

PoissonNLLLoss computes the Negative Log-Likelihood (NLL) assuming the target values follow a Poisson distribution. It is commonly used for predicting count data, where the target represents the number of events occurring within a fixed interval.

Typical applications include event counting, object counting, medical statistics, and other regression tasks involving non-negative integer counts.

Signature¶

PoissonNLLLoss(
    log_input=True,
    full=False,
    eps=1e-8,
    reduction="mean"
)

Parameters¶

Parameter	Type	Description
`log_input`	`bool`	If `True`, the input is assumed to contain the logarithm of the predicted rate. If `False`, the input is treated as the predicted rate itself. Default is `True`.
`full`	`bool`	Whether to include Stirling's approximation for the factorial term. Default is `False`.
`eps`	`float`	Small constant added for numerical stability when `log_input=False`. Default is `1e-8`.
`reduction`	`str`	Specifies the reduction to apply: `"mean"`, `"sum"`, or `"none"`. Default is `"mean"`.

Inputs¶

Input	Description
`pred`	Predicted log-rate or predicted rate, depending on `log_input`.
`target`	Ground-truth count values.

Returns¶

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

Formula¶

When log_input=True:

Loss = exp(pred) − target × pred

When log_input=False:

Loss = pred − target × log(pred + eps)

If full=True, Stirling's approximation is added:

target × log(target)
− target
+ 0.5 × log(2π × target)

This improves the approximation for larger count values.

Example¶

import aakaar
from aakaar.losses import PoissonNLLLoss

pred = aakaar.tensor([
    0.8,
    1.3,
    0.4
])

target = aakaar.tensor([
    2.0,
    4.0,
    1.0
])

criterion = PoissonNLLLoss()

loss = criterion(
    pred,
    target
)

Using Predicted Rates Instead of Log-Rates¶

criterion = PoissonNLLLoss(
    log_input=False
)

In this mode, the input tensor should contain predicted Poisson rates rather than their logarithms.

Including Stirling's Approximation¶

criterion = PoissonNLLLoss(
    full=True
)

This produces a more complete approximation of the Poisson negative log-likelihood, especially for larger target values.

Using Different Reductions¶

Mean (default)¶

criterion = PoissonNLLLoss(
    reduction="mean"
)

Sum¶

criterion = PoissonNLLLoss(
    reduction="sum"
)

None¶

criterion = PoissonNLLLoss(
    reduction="none"
)

Returns the loss for every prediction individually.

Typical Training Loop¶

criterion = PoissonNLLLoss()

loss = criterion(
    predictions,
    targets
)

loss.backward()
optimizer.step()

Typical Applications¶

Object counting
Crowd counting
Traffic flow prediction
Medical event prediction
Insurance claim modeling
Count-based regression

Notes¶

Designed for regression tasks where the target represents count data.
Supports inputs as either log-rates (log_input=True) or rates (log_input=False).
full=True adds Stirling's approximation to better approximate the full Poisson log-likelihood.
Uses a small eps value internally for numerical stability when computing logarithms.
Supports "mean", "sum", and "none" reductions.