# Dirichlet distribution: Normalization of alpha values

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among others, a vector of alpha values as parameters - the same $\alpha$ values that determine the Dirichlet distribution.

So, if we specifiy $k$, we have a vector of $k$ $\alpha$-values. For $k = 3$ we may choose our $\alpha$ values like this: https://en.wikipedia.org/wiki/Dirichlet_distribution#mediaviewer/File:Dirichlet_distributions.png.

If I understood it correctly, a "flat"/neutral Dirichlet distribution can be achieved by choosing $\{a_k\}=1$. First question: Is that correct? I've read this in chapter 2.2.1 ("The Dirichlet Distribution") of "Pattern Recognition and Machine Learning" by Christopher M. Bishop, if that's of any interest.

The problem is that the library I use takes the alpha values only in normalized form, which means that all $\alpha$ values have to sum up to 1. Therefore my second question is: How do I normalize $\alpha$ values? I've found a normalization constant on https://en.wikipedia.org/wiki/Dirichlet_distribution#Probability_density_function, but it seems to apply to the probability density function only; otherwise it doesn't quite work out (the $\alpha$ values don't sum up to 1).

I am asking the second question because if, for example, $\alpha = \{1, 1, 1\}$ generates a "flat" Dirichlet distribution and $\alpha = \{5, 5, 5\}$ does not, I can't see how to normalize these values without the former having the same normalized values as the latter (e.g. $\alpha_{normalized} = \{\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\}$), which would mean that they'd generate the same distribution.

Again, I'm not a mathematician, and I'm pretty sure this probably seems like a pretty stupid question to you. Don't be too harsh, please :-)

I'd appreciate any help!

Best Regards & thanks for your time, MG

Edit: Unfortunately I don't seem to be allowed to neither comment nor accept answers since I posted this question without an account, so in response to...

• @tristan: Yeah, that's what I thought. Something like a concentration parameter would be necessary.
• @whuber: Well, turns out I just misunderstood the library's documentation. The library allows for normalized parameter, but also accepts non-normalized data (e.g. {1, 1, 1}).

• It has been suggested that stats.SE is a better venue for this question, and I am inclined to agree. Migrating now...
– Todd Trimble
Oct 20, 2014 at 2:04
• Are you perhaps confusing normalization of the variates with normalization of the parameters? Any library that insists the parameters sum to unity cannot be a full implementation of the Dirichlet distribution (and it would be strange for the library to be so limited).
– whuber
Oct 20, 2014 at 16:00
• This is a great question and I think you worded it well. I wish it had more discussion answers. Nov 8, 2016 at 18:09

As others noticed in the comments, it wouldn't make much sense to have normalized parameters for Dirichlet. Notice that for $$\alpha = (1/3, 1/3, 1/3)$$, $$\alpha = (1, 1, 1)$$, or $$\alpha = (100, 100, 100)$$, the results of $$\alpha' = \alpha / \sum_i \alpha_i$$ would be in each case the same, i.e. $$\alpha' = (1/3, 1/3, 1/3)$$.
You can check the What exactly is the alpha in the Dirichlet distribution? thread to learn more about parameters of Dirichlet, but if all the values of $$\alpha_i$$ are the same than the distribution is symmetric, but only for $$\alpha_1 = \alpha_2 = \dots = \alpha_k = 1$$ it is uniform. For high values of $$\alpha$$'s it is more concentrated, while for low values, the values are pushed more to the extremes, so those are very different distributions. You can find some examples below, check also the linked thread.
In answer to your first question, yes: if $a_i = 1$ for all $i$ you get the uniform distribution. For second question I suspect you need to find a concentration parameter (call it $r$ for sake of argument), so that you would specify $r=3$ and $a_1=a_2=a_3=1/3$.
• I disagree that you get Uniform distributions on the margins of a Dirichlet distribution for $\alpha_i = 1$ when $i > 2$. The mean of a Dirichlet marginal is $\alpha_ i / \sum_i \alpha_i$ so the mean of a marginal with 3 margins is 1/3. That is not the mean of a Uniform distribution on (0,1). I'm commenting 5 years later because other's have cited this answer incorrectly. Jul 12, 2020 at 19:13
• @RCarnell Yes I think I've been overly imprecise here. You are correct that the marginal distributions will not be uniform on (0,1). Once k > 2 you will not get the marginal distributions of all variates to be uniform on (0,1). When $\alpha_i = 1$ you have that the joint probability density function is constant (i.e. uniform). As per Wiki "When α=1, the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open standard (K − 1)-simplex, i.e. it is uniform over all points in its support." Jul 21, 2020 at 19:58