This week's post has more math than most, so I wrote in it LaTeX and translated it to HTML using HeVeA. Some of the formulas are not as pretty as they could be. If you prefer, you can read this
article in PDF.
Abstract:
My two favorite topics in probability and statistics are
Bayes’s theorem and logistic regression. Because there are
similarities between them, I have always assumed that there is
a connection. In this note, I demonstrate the
connection mathematically, and (I hope) shed light on the
motivation for logistic regression and the interpretation of
the results.
1 Bayes’s theorem
I’ll start by reviewing Bayes’s theorem, using an example that came up
when I was in grad school. I signed up for a class on Theory of
Computation. On the first day of class, I was the first to arrive. A
few minutes later, another student arrived. Because I was expecting
most students in an advanced computer science class to be male, I was
mildly surprised that the other student was female. Another female
student arrived a few minutes later, which was sufficiently
surprising that I started to think I was in the wrong room. When
another female student arrived, I was confident I was in the wrong
place (and it turned out I was).
As each student arrived, I used the observed data to update my
belief that I was in the right place. We can use Bayes’s theorem to
quantify the calculation I was doing intuitively.
I’ll us
H to represent the hypothesis that I was in the right
room, and
F to represent the observation that the first other
student was female. Bayes’s theorem provides an algorithm for
updating the probability of
H:
Where
 P(H) is the prior probability of H before the other
student arrived.
 P(HF) is the posterior probability of H, updated based
on the observation F.
 P(FH) is the likelihood of the data, F, assuming that
the hypothesis is true.
 P(F) is the likelihood of the data, independent of H.
Before I saw the other students, I was confident I was in the right
room, so I might assign
P(
H) something like 90%.
When I was in grad school most advanced computer science classes were
90% male, so if I was in the right room, the likelihood of the
first female student was only 10%. And the likelihood of three
female students was only 0.1%.
If we don’t assume I was in the right room, then the likelihood of
the first female student was more like 50%, so the likelihood
of all three was 12.5%.
Plugging those numbers into Bayes’s theorem yields
P(
H
F) = 0.64
after one female student,
P(
H
FF) = 0.26 after the second,
and
P(
H
FFF) = 0.07 after the third.
[UPDATE: An earlier version of this article had incorrect values in the previous sentence. Thanks to David Burger for catching the error.]
2 Logistic regression
Logistic regression is based on the following functional form:
logit(p) = β_{0} + β_{1} x_{1} + ... + β_{n} x_{n} 
where the dependent variable,
p, is a probability,
the
xs are explanatory variables, and the βs are
coefficients we want to estimate. The
logit function is the
logodds, or
logit(p) = ln  ⎛
⎜
⎜
⎝ 
  ⎞
⎟
⎟
⎠ 
When you present logistic regression like this, it raises
three questions:
 Why is logit(p) the right choice for the dependent
variable?
 Why should we expect the relationship between logit(p)
and the explanatory variables to be linear?
 How should we interpret the estimated parameters?
The answer to all of these questions turns out to be Bayes’s
theorem. To demonstrate that, I’ll use a simple example where
there is only one explanatory variable. But the derivation
generalizes to multiple regression.
On notation: I’ll use
P(
H) for the probability
that some hypothesis,
H, is true.
O(
H) is the odds of the same
hypothesis, defined as
I’ll use
LO(
H) to represent the logodds of
H:
I’ll also use
LR for a likelihood ratio, and
OR for an odds
ratio. Finally, I’ll use
LLR for a loglikelihood ratio, and
LOR for a logodds ratio.
3 Making the connection
To demonstrate the connection between Bayes’s theorem and
logistic regression, I’ll start with the odds form
of Bayes’s theorem. Continuing the previous example,
I could write
O(HF) = O(H) LR(FH)
(1) 
where
 O(H) is the prior odds that I was in the right room,
 O(HF) is the posterior odds after seeing one female student,
 LR(FH) is the likelihood ratio of the data, given
the hypothesis.
The likelihood ratio of the data is:
where ¬
H means
H is false.
Noticing that logistic regression is expressed in terms of
logodds, my next move is to write the logodds form of
Bayes’s theorem by taking the log of Eqn
1:
LO(HF) = LO(H) + LLR(FH)
(2) 
If the first student to arrive had been male, we would write
LO(HM) = LO(H) + LLR(MH)
(3) 
Or more generally if we use
X as a variable to represent
the sex of the observed student, we would write
LO(HX) = LO(H) + LLR(XH)
(4) 
I’ll assign
X=0 if the observed student is female and
X=1 if male. Then I can write:
LLR(XH) =  ⎧
⎨
⎩ 
LLR(FH)  if X = 0 
LLR(MH)  if X = 1 


(5) 
Or we can collapse these two expressions into one by using
X as a multiplier:
LLR(XH) = LLR(FH) + X [LLR(MH) − LLR(FH)]
(6) 
4 Odds ratios
The next move is to recognize that
the part of Eqn
4 in brackets is the logodds ratio
of
H. To see that, we need to look more closely at odds ratios.
Odds ratios are often used in medicine to describe the association
between a disease and a risk factor. In the example scenario, we
can use an odds ratio to express the odds of the hypothesis
H if we observe a male student, relative to the odds if we
observe a female student:
I’m using the notation
OR_{X} to represent the odds ratio
associated with the variable
X.
Applying Bayes’s theorem to
the top and bottom of the previous expression yields
OR_{X}(H) = 
O(H) LR(MH) 

O(H) LR(FH) 
 =


Taking the log of both sides yields
LOR_{X}(H) = LLR(MH) − LLR(FH)
(7) 
This result should look familiar, since it appears in
Eqn
4.
5 Conclusion
Now we have all the pieces we need; we just have to assemble them.
Combining Eqns
4 and
5 yields
LLR(HX) = LLR(F) + X LOR(XH)
(8) 
Combining Eqns
3 and
6 yields
LO(HX) = LO(H) + LLR(FH) + X LOR(XH)
(9) 
Finally, combining Eqns
2 and
7 yields
LO(HX) = LO(HF) + X LOR(XH) 
We can think of this equation as the logodds form of Bayes’s theorem,
with the update term expressed as a logodds ratio. Let’s compare
that to the functional form of logistic regression:
logit(p) = β_{0} + X β_{1} 
The correspondence between these equations suggests the following
interpretation:
 The predicted value, logit(p), is the posterior log
odds of the hypothesis, given the observed data.
 The intercept, β_{0}, is the logodds of the
hypothesis if X=0.
 The coefficient of X, β_{1}, is a logodds ratio
that represents odds of H when X=1, relative to
when X=0.
This relationship between logistic regression and Bayes’s theorem
tells us how to interpret the estimated coefficients. It also
answers the question I posed at the beginning of this note:
the functional form of logistic regression makes sense because
it corresponds to the way Bayes’s theorem uses data to update
probabilities.
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.
Wow, this was interesting! While I don't have a background in logistic regression (yet), this was a fantastic first exposure to the versatility and usefulness of Bayes's theorem.
ReplyDeleteI loved your example about the college students  we really do use statistics and probability in everyday situations.
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