torch.distributions.Poisson

class Poisson extends Distribution

new Poisson(rate: number | Tensor, options?: DistributionOptions)

readonlyrate(Tensor): – Rate parameter (expected number of events).
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Poisson distribution: models the count of events in a fixed time interval.

Parameterized by rate λ (expected number of events). Fundamental for modeling counts and discrete phenomena. Essential for:

Modeling count data (events, arrivals, occurrences)
Point processes and event counting
Approximate binomial when n is large, p is small
Generative modeling of discrete sequences
Network analysis (degree distributions)
Rare event modeling

The probability mass function: P(X = k) = (λ^k * e^(-λ)) / k! for k = 0, 1, 2, ...

\begin{aligned} \text{PMF: } P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad k = 0,1,2,\ldots \\ \text{Mean: } \mathbb{E}[X] = \lambda \\ \text{Variance: } \text{Var}(X) = \lambda \\ \text{Mode: } \lfloor \lambda \rfloor \\ \text{Entropy: } H(X) = \lambda(1 - \ln(\lambda)) + e^{-\lambda} \sum_{k=0}^{\infty} \frac{\ln(k!)}{k!} \end{aligned}

Mean = Variance: The rate λ equals both mean and variance (special property)
Integer samples: Always produces non-negative integers 0, 1, 2, ...
Tail behavior: Probabilities decrease roughly as λ/k for large k
Binomial limit: Good approximation when n np (rare events)
Exponential duality: Inter-event times are exponential; count is Poisson
Decomposability: Sum of independent Poissons is still Poisson
Overdispersion: Real count data often has variance mean (use Negative Binomial)

Rate must be positive: rate ≤ 0 causes errors
Unbounded support: Can sample arbitrarily large integers (rare)
Model assumption: Real counts often more dispersed than Poisson (Negative Binomial)
Large rates: Computation can be inefficient for very large λ

Examples

// Average 4 events per interval
const dist = new torch.distributions.Poisson(4);
const count = dist.sample();  // typically 1-7, centered at 4
const counts = dist.sample([1000]);  // 1000 samples

// Rare event: low rate
const rare = new torch.distributions.Poisson(0.5);
const sample = rare.sample();  // mostly 0s, some 1s, rarely 2+

// Common events: high rate
const common = new torch.distributions.Poisson(10);
const sample = common.sample();  // mostly 8-12, bell-curve around 10

// Real-world example: customer arrivals per hour
const arrival_rate = 2.5;  // 2.5 customers per hour
const arrival_dist = new torch.distributions.Poisson(arrival_rate);
const customers_this_hour = arrival_dist.sample();  // count of arrivals
const customers_next_8hrs = arrival_dist.sample([8]);  // 8 hourly samples

// Network degree distribution: social network connections
const avg_connections = 5;  // average friends per person
const degree_dist = new torch.distributions.Poisson(avg_connections);
const person_connections = degree_dist.sample();  // how many friends?

// Batched distributions with different rates
const rates = torch.tensor([1, 4, 10]);  // different mean counts
const dist = new torch.distributions.Poisson(rates);
const samples = dist.sample();  // [3] shaped samples
// First usually 0-2, second around 4, third around 8-12

// Mean = Variance: distinguish from other distributions
const dist = new torch.distributions.Poisson(5);
const mean = dist.mean;  // 5
const variance = dist.variance;  // also 5 (unique property!)

// Probability of specific counts
const dist = new torch.distributions.Poisson(3);
const prob_zero = dist.log_prob(0).exp();  // P(X = 0) = e^(-3)
const prob_five = dist.log_prob(5).exp();  // P(X = 5) = 3^5 * e^(-3) / 5!