To induce sparseness into draws from the Dirichlet distribution we need the parameters of the Dirichlet distribution to be:

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The Dirichlet distribution is parameterized by a vector of concentration parameters α = (α1, α2, ..., αK). For sparseness in the draws, we typically want small concentration parameters, specifically αi in the range (0, 1). This causes the distribution to place more mass on vectors where only a few components are large and the rest are near zero, creating sparse probability allocations.

Option 1: non-negative. While αi being non-negative (i.e., αi ≥ 0) is a necessary relaxation in some contexts, the Dirichlet parameters must actually be strictly positive (αi > 0). If any αi = 0, the distribution is not well-defined or degenerates, so this is not the precise requirement for inducing sparseness.

Option 2: between 0 and 1. This correctly captures the common regime used to promote sparsity in Dirichlet draws, since αi ∈ (0, 1) biases mass toward extreme coordinates, yielding sparse outcomes.

Option 3: sum to 1. The constraint that the components of a Dirichlet-distributed vector sum to 1 is a property of the draw Y ~ Dirichlet(α), not a property of the α parameters themselves. The α parameters do not need to sum to 1; instead, they can be any positive values, and the resulting draw will automatically have components summing to 1. This choice describes the output space, not the parameter values that control sparseness.

In summary, to encourage sparsity in samples from Dirichlet(α), we typically choose αi in the (0, 1) interval, which fosters concentration on few components while keeping the overall draw valid.

SP2025.B69.DAT.562.24 COMPLETE Quiz #3 AFTER VIDEO 2: The Dirichlet Distribution

To induce sparseness into draws from the Dirichlet distribution we need the parameters of the Dirichlet distribution to be:

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