How Alpha Shapes the Dirichlet Distribution?

In Python, sampling from a Dirichlet distribution looks deceptively simple:

np.random.dirichlet(alpha, size)

At first, I treated it just like another random generator — specify an alpha, draw some probability vectors, and move on. But as I used it more, I realized that alpha is not just a parameter. It fundamentally changes how the samples behave — shifting the Dirichlet distribution from sparsity to uniformity, from unpredictable peaks to smooth, balanced vectors.

This blog is not about coding tricks. It’s about understanding what alpha really does, how it mathematically shapes randomness, and how even a small change in alpha leads to dramatically different outcomes.

Introduction to the Dirichlet Distribution

The Dirichlet distribution models probability vectors — lists of non-negative numbers that sum to 1. Formally, if:

\[\mathbf{p} = (p_1, p_2, \dots, p_K)\]

then \(\mathbf{p}\) is drawn from a Dirichlet distribution with parameter vector \(\boldsymbol{\alpha} = (\alpha_1, \alpha_2, \dots, \alpha_K)\) as:

\[\mathbf{p} \sim \text{Dirichlet}(\boldsymbol{\alpha})\]

Each \(\alpha_k\) influences the corresponding component \(p_k\), controlling how large or small it tends to be across different samples.

When I first used the Dirichlet function, I focused mainly on generating random compositions. But once I started varying alpha and observing the output, deeper patterns became clear.

The Role of Alpha

When all \(\alpha_k\) are set to the same value (for example, \(\alpha = 1\) for each component), the behavior of the Dirichlet distribution depends strongly on the size of alpha:

When \(\alpha < 1\): Samples tend to be sparse. Most of the probability mass concentrates in a few categories.
When \(\alpha = 1\): Samples are uniform over the simplex — any combination is equally likely.
When \(\alpha > 1\): Samples become dense and balanced, with values clustering around equal proportions.

Through experimentation, I noticed that when alpha was very small, a few elements dominated, while others were nearly zero. As alpha increased, samples became smoother, and no single element stood out sharply.

Mathematical Properties

The expected value for each component \(p_k\) is:

\[\mathbb{E}[p_k] = \frac{\alpha_k}{\sum_{i=1}^K \alpha_i}\]

If all components have the same alpha, each expected probability is simply \(\frac{1}{K}\).

More importantly, the variance of each \(p_k\) tells how much each component can fluctuate:

\[\text{Var}[p_k] = \frac{\alpha_k(\alpha_0 - \alpha_k)}{\alpha_0^2 (\alpha_0 + 1)} \quad \text{where} \quad \alpha_0 = \sum_{i=1}^K \alpha_i\]

When all \(\alpha_k\) are equal, this simplifies to:

\[\text{Var}[p_k] \propto \frac{1}{\alpha (K\alpha + 1)}\]

From this equation, the trend is clear: as \(\alpha\) increases, the variance decreases. Samples cluster closer to the expected uniform distribution. When \(\alpha\) is small, variance is large, and extreme values are much more likely.

This mathematical behavior aligned exactly with what I observed in my Python simulations.

Why Alpha = 1 Matters

During my tests, I noticed that alpha = 1 seemed like a turning point. When alpha was less than 1, samples were sparse and uneven. When alpha was greater than 1, samples became more uniform and balanced.

Mathematically, alpha = 1 corresponds to a uniform distribution over the probability simplex. No particular configuration is favored — every combination of probabilities is equally likely. This makes alpha = 1 a meaningful boundary between randomness dominated by few categories and randomness evenly spread across all categories.

Practical Implications

This behavior has real-world consequences. In machine learning, for example, using small alpha values in topic models or clustering algorithms tends to produce “hard” assignments — most weight is assigned to a few topics or clusters. Larger alpha values lead to “softer” assignments, where each sample belongs to multiple groups more evenly.

In simulation experiments, adjusting alpha allowed me to control how diverse or focused the generated samples were. It wasn’t just about generating numbers — it was about tuning how much variation I wanted to model.

Lessons Learned

What started as a simple Python function turned into a deeper exploration of how one parameter can control the shape of randomness itself. Understanding \(\alpha\) isn’t just about getting the math right. It’s about seeing how small changes in our assumptions ripple into the patterns we create.