class

RelaxedBernoulli

extendsDistribution

RelaxedBernoulli(temperature: Tensor | float, probs: Tensor | float | None = None, logits: Tensor | float | None = None, validate_args: bool | None = None)

source edit

Concrete (Gumbel-sigmoid) relaxation of the Bernoulli distribution.

RelaxedBernoulli(temperature=τ, probs=p) defines a continuous distribution over $(0, 1)$ whose samples are differentiable surrogates for Bernoulli samples. As $\tau \to 0$ the distribution concentrates on $\{0, 1\}$ , recovering the discrete Bernoulli. As $\tau \to \infty$ the distribution approaches $\operatorname{Uniform}(0, 1)$ .

The Concrete distribution (Maddison et al. 2017) / Gumbel-softmax (Jang et al. 2017) trick enables gradient-based optimisation through discrete latent variables in variational autoencoders and related models.

Parameters

temperatureTensor | float

Temperature parameter

\tau > 0

. Controls the sharpness of the relaxation. Small values give near-discrete samples; large values give near-uniform samples.

probsTensor | float | None= None

Bernoulli success probability

p \in (0, 1)

. Mutually exclusive with logits.

logitsTensor | float | None= None

Log-odds

l = \log(p/(1-p)) \in \mathbb{R}

. Mutually exclusive with probs.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Attributes

temperatureTensor

Temperature

\tau

probsTensor

Success probability — stored when given, otherwise derived from logits on access.

logitsTensor

Log-odds — stored when given, otherwise derived from probs on access.

Notes

Reparameterised sampling (Gumbel-sigmoid trick):

y = \sigma\!\left(\frac{l + g_1 - g_2}{\tau}\right), \quad g_1, g_2 \overset{\text{iid}}{\sim} \operatorname{Gumbel}(0, 1)

where $\sigma(\cdot)$ is the sigmoid function. Gradients propagate through both $l$ and $\tau$ .

Log-PDF (Maddison et al. 2017, Eq. 2):

\log p(y; l, \tau) = \log\tau + l - (\tau+1)\log\!\left(e^{\tau\,\text{logit}(y) - l} + 1\right) - \tau\,\text{logit}(y)

where $\text{logit}(y) = \log(y/(1-y))$ .

Examples

>>> import lucid
>>> from lucid.distributions import RelaxedBernoulli
>>> lucid.manual_seed(0)
>>> dist = RelaxedBernoulli(temperature=0.5, probs=0.7)
>>> samples = dist.rsample((100,))
>>> # Samples are in (0, 1)
>>> bool(((samples > 0) & (samples < 1)).all())
True

Used by 1

lucid.distributions

Constructors

dunder

init

→None

__init__(temperature: Tensor | float, probs: Tensor | float | None = None, logits: Tensor | float | None = None, validate_args: bool | None = None)

source edit

Construct a RelaxedBernoulli (Binary Concrete) distribution.

Parameters

temperatureTensor | float

Temperature

\lambda > 0

controlling the relaxation tightness. As

\lambda \to 0

, samples concentrate on

\{0, 1\}

recovering a hard Bernoulli; as

\lambda \to \infty

, samples concentrate around

1/2

probsTensor | float | None= None

Probability

p \in (0, 1)

parameter. Mutually exclusive with logits.

logitsTensor | float | None= None

Log-odds

\log\frac{p}{1-p}

parameter. Mutually exclusive with probs.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Raises

ValueError

If both or neither of probs and logits are provided.

Instance methods

log_prob

→Tensor

log_prob(value: Tensor)

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Log-probability density of the Concrete/RelaxedBernoulli distribution.

Parameters

valueTensor

Point(s)

y \in (0, 1)

at which to evaluate the density.

Returns

Tensor

Log-density values of the same shape as value.

logits

→Tensor

logits()

source edit

Log-odds $\log(p/(1-p))$ — as given, or derived on access.

Derived from probs clamped one epsilon inside $[0, 1]$ , as the reference framework derives it.

probs

→Tensor

probs()

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Success probability — as given, or sigmoid(logits) on access.

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

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Draw a reparameterised sample via the Gumbel-sigmoid trick.

Parameters

sample_shapetuple[int, ...]= ()

Leading shape of the output sample batch.

Returns

Tensor

Samples $y \in (0, 1)$ of shape (*sample_shape, *batch_shape), with gradients flowing through both logits and temperature.

>>> import lucid >>> from lucid.distributions import RelaxedBernoulli >>> lucid.manual_seed(0) >>> dist = RelaxedBernoulli(temperature=0.5, probs=0.7) >>> samples = dist.rsample((100,)) >>> # Samples are in (0, 1) >>> bool(((samples > 0) & (samples < 1)).all()) True

dunder

init

→None

__init__(temperature: Tensor | float, probs: Tensor | float | None = None, logits: Tensor | float | None = None, validate_args: bool | None = None)

source edit

Construct a RelaxedBernoulli (Binary Concrete) distribution.

Parameters

temperatureTensor | float

Temperature

\lambda > 0

controlling the relaxation tightness. As

\lambda \to 0

, samples concentrate on

\{0, 1\}

recovering a hard Bernoulli; as

\lambda \to \infty

, samples concentrate around

1/2

probsTensor | float | None= None

Probability

p \in (0, 1)

parameter. Mutually exclusive with logits.

logitsTensor | float | None= None

Log-odds

\log\frac{p}{1-p}

parameter. Mutually exclusive with probs.

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Raises

ValueError

If both or neither of probs and logits are provided.