Pointwise mutual information

Pointwise mutual information (PMI), or point mutual information, is a measure of association used in information theory and statistics. In contrast to mutual information (MI) which builds upon PMI, it refers to single events, whereas MI refers to the average of all possible events.

Definition

The PMI of a pair of outcomes x and y belonging to discrete random variables X and Y quantifies the discrepancy between the probability of their coincidence given their joint distribution and their individual distributions, assuming independence. Mathematically:

$\operatorname{pmi}(x;y) \equiv \log\frac{p(x,y)}{p(x)p(y)} = \log\frac{p(x|y)}{p(x)} = \log\frac{p(y|x)}{p(y)}.$

The mutual information (MI) of the random variables X and Y is the expected value of the PMI over all possible outcomes (with respect to the joint distribution $p(x,y)$).

The measure is symmetric ($\operatorname{pmi}(x;y)=\operatorname{pmi}(y;x)$). It can take positive or negative values, but is zero if X and Y are independent. Note that even though PMI may be negative or positive, its expected outcome over all joint events (MI) is positive. PMI maximizes when X and Y are perfectly associated (i.e. $p(x|y)$ or $p(y|x)=1$), yielding the following bounds:

$-\infty \leq \operatorname{pmi}(x;y) \leq \min\left[ -\log p(x), -\log p(y) \right] .$

Finally, $\operatorname{pmi}(x;y)$ will increase if $p(x|y)$ is fixed but $p(x)$ decreases.

Here is an example to illustrate:

x	y	p(x, y)
0	0	0.1
0	1	0.7
1	0	0.15
1	1	0.05

Using this table we can marginalize to get the following additional table for the individual distributions:

	p(x)	p(y)
0	0.8	0.25
1	0.2	0.75

With this example, we can compute four values for $pmi(x;y)$. Using base-2 logarithms:

pmi(x=0;y=0)	=	−1
pmi(x=0;y=1)	=	0.222392421
pmi(x=1;y=0)	=	1.584962501
pmi(x=1;y=1)	=	−1.584962501

(For reference, the mutual information $\operatorname{I}(X;Y)$ would then be 0.214170945)

Similarities to mutual information

Pointwise Mutual Information has many of the same relationships as the mutual information. In particular,

$\begin{align} \operatorname{pmi}(x;y) &=& h(x) + h(y) - h(x,y) \\ &=& h(x) - h(x|y) \\ &=& h(y) - h(y|x) \end{align}$

Where $h(x)$ is the self-information, or $-\log_2 p(X=x)$.

Normalized pointwise mutual information (npmi)

Pointwise mutual information can be normalized between [-1,+1] resulting in -1 (in the limit) for never occurring together, 0 for independence, and +1 for complete co-occurrence.

$\operatorname{npmi}(x;y) = \frac{\operatorname{pmi}(x;y)}{-\log \left[ p(x, y) \right] }$

PMI Variants

In addition to the above-mentioned npmi, PMI has many other interesting variants. A comparative study of these variants can be found in

Chain-rule for pmi

Like MI, PMI follows the chain rule, that is,

$\operatorname{pmi}(x;yz) = \operatorname{pmi}(x;y) + \operatorname{pmi}(x;z|y)$

This is easily proven by:

$\begin{align} \operatorname{pmi}(x;y) + \operatorname{pmi}(x;z|y) & {} = \log\frac{p(x,y)}{p(x)p(y)} + \log\frac{p(x,z|y)}{p(x|y)p(z|y)} \\ & {} = \log \left[ \frac{p(x,y)}{p(x)p(y)} \frac{p(x,z|y)}{p(x|y)p(z|y)} \right] \\ & {} = \log \frac{p(x|y)p(y)p(x,z|y)}{p(x)p(y)p(x|y)p(z|y)} \\ & {} = \log \frac{p(x,yz)}{p(x)p(yz)} \\ & {} = \operatorname{pmi}(x;yz) \end{align}$

Applications

In computational linguistics, PMI has been used for finding collocations and associations between words. For instance, countings of occurrences and co-occurrences of words in a corpus can be used to approximate the probabilities $p(x)$ and $p(x,y)$ respectively. The following table shows counts of pairs of words getting the most and the least PMI scores in the first 50 millions of words in Wikipedia (dump of October 2015) filtering by 1,000 or more co-ocurrences.

word 1	word 2	count word 1	count word 2	count of co-occurrences	PMI
puerto	rico	1938	1311	1159	10.0349081703
hong	kong	2438	2694	2205	9.72831972408
los	angeles	3501	2808	2791	9.56067615065
carbon	dioxide	4265	1353	1032	9.09852946116
prize	laureate	5131	1676	1210	8.85870710982
san	francisco	5237	2477	1779	8.83305176711
nobel	prize	4098	5131	2498	8.68948811416
ice	hockey	5607	3002	1933	8.6555759741
star	trek	8264	1594	1489	8.63974676575
car	driver	5578	2749	1384	8.41470768304
it	the	283891	3293296	3347	-1.72037278119
are	of	234458	1761436	1019	-2.09254205335
this	the	199882	3293296	1211	-2.38612756961
is	of	565679	1761436	1562	-2.54614706831
and	of	1375396	1761436	2949	-2.79911817902
a	and	984442	1375396	1457	-2.92239510038
in	and	1187652	1375396	1537	-3.05660070757
to	and	1025659	1375396	1286	-3.08825363041
to	in	1025659	1187652	1066	-3.12911348956
of	and	1761436	1375396	1190	-3.70663100173

Good collocation pairs have high PMI because the probability of co-occurrence is only slightly lower than the probabilities of occurrence of each word. Conversely, a pair of words whose probabilities of occurrence are considerably higher than their probability of co-occurrence gets a small PMI score.

External links

• Demo at Rensselaer MSR Server (PMI values normalized to be between 0 and 1)