# Probabilistic Coherence

#### Tommy Jones

#### 2023-07-14

Coherence measures seek to emulate human judgment. They tend to rely
on the degree to which words co-occur together. Yet, words may co-occur
in ways that are unhelpful. For example, consider the words “the” and
“this”. They co-occur very frequently in English-language documents. Yet
these words are statistically independent. Knowing the relative
frequency of the word “the” in a document does not give you any
additional information on the relative frequency of the word “this” in a
document.

Probabilistic coherence uses the concepts of co-occurrence and
statistical independence to rank topics. Note that probabilistic
coherence has not yet been rigorously evaluated to assess its
correlation to human judgment. Anecdotally, those that have used it tend
to find it helpful. Probabilistic coherence is included in both the
*textmineR* and *tidylda* packages in the R language.

Probabilistic coherence averages differences in conditional
probabilities across the top \(M\)
most-probable words in a topic. Let \(x_{i,
k}\) correspond the \(i\)-th
most probable word in topic \(k\), such
that \(x_{1,k}\) is the most probable
word, \(x_{2,k}\) is the second most
probable and so on. Further, let

\[\begin{align}
x_{i,k} =
\begin{cases}
1 & \text{if the } i\text{-th most probable word appears in a
randomly selected document} \\
0 & \text{otherwise}
\end{cases}
\end{align}\]

Then probabalistic coherence for the top \(M\) terms in topic \(k\) is calculated as follows:

\[\begin{align}
C(k,M) &= \frac{1}{\sum_{j = 1}^{M - 1} M - j}
\sum_{i = 1}^{M - 1} \sum_{j = i+1}^{M}
P(x_{j,k} = 1 | x_{i,k} = 1) - P(x_{j,k} = 1)
\end{align}\]

Where \(P(x_{j,k} = 1 | x_{i,k} =
1)\) is the fraction of contexts containing word \(i\) that contain word \(j\) and \(P(x_{j,k} = 1)\) is the fraction of all
contexts containing word \(j\).

This brings us to interpretation:

- If \(P(x_{j,k} = 1 | x_{i,k} = 1) -
P(x_{j,k} = 1)\) is zero, then \(P(x_{j,k} = 1 | x_{i,k} = 1) = P(x_{j,k} =
1)\), the definition of statistical independence.
- If \(P(x_{j,k} = 1 | x_{i,k} = 1) >
P(x_{j,k} = 1)\), then word \(j\) is more present than average in
contexts also containing word \(i\).
- If \(P(x_{j,k} = 1 | x_{i,k} = 1) <
P(x_{j,k} = 1)\), then word \(j\) is less present than average in
contexts that contain word \(i\). LDA
is unlikely to find strong negative co-occurrences. In practice,
negative values tend to be near zero.