Confusion about the use of the MLE & the posterior in parameter estimation for logistic regression

In classification one usually computes $$C = \operatorname*{argmax}_k p(C=k\mid X)$$ where $$p(C=k\mid X)$$ is the posterior distribution.

In a simple logistic regression setting with $$C \in \{0, 1\}$$ and $$p(C=1\mid X)=\frac{\exp(\beta_0+\beta_1 x_i)}{1+\exp(\beta_0+\beta_1 x_i)}$$ and therefore $$p(C=0\mid X)=\frac{1}{1+\exp(\beta_0+\beta_1 x_i)}$$ with $$X=\{x_i\},\ i=1,\ldots,N$$.

we estimate the parameters $$\beta_0, \beta_1$$ via the maximum likelihood estimation. To do so one has to compute the product of the likelihood function of all $$N$$ observations. So far, so normal. However, in all text books the authors plug in the posterior instead of the likelihood (e.g., Bishop, p. 206, Hastie, et al., p. 120): \begin{align} \ell(\beta) &= \log\left(\prod_{i=1}^N p(C_i=k\mid x_i, \beta)\right) \\[8pt] &= \log\left(\prod_{i=1}^N p(C_i=1\mid x_i, \beta)^{C_i}(1-p(C_i=1\mid x_i, \beta))^{1-C_i}\right) \end{align} And even though these probabilities are now conditioned on $$\beta$$ as well, they are still no likelihood to the posterior $$p(C=k\mid X)$$. So

1. How come we plug in the MLE just the posterior conditioned on the parameter $$\beta$$?

2. Why is $$p(C=k\mid X)$$ a posterior anyway? To me a posterior is a distribution of over a parameter given the observed data. But the class $$C$$ is to me not a parameter but a target just like the observations $$y_i$$ in a linear regression setting.

• 1. See 2. 2. It isn't. It's a likelihood. Logistic regression is maximum likelihood. – AdamO Nov 19 '18 at 20:33
• In the above mentioned textbooks (pattern recognition and machine learning by bishop and elements of statistical leaning by Haste at al.) they explicitly call $p(C=k|X)$ the posterior distribution on several occasions... – guest1 Nov 19 '18 at 20:55
• It can be, but it's not in what you write. You write $p(C=k|X) = l(X, \beta)$. Where's the prior? Write $p(C|X) = l(X, \beta) p(C)$ and make life more interesting, or Bayesian at least. – AdamO Nov 19 '18 at 21:45
• Well in logistic Regression we learn the posterior directly since it is a discriminative algorithm. A prior would only be used in a generative algorithm like LDA, where we learn the likelihood and the prior. To me if $p(C=k|X)$ is the posterior, then the likelihood should look like $p(X|C)$ or similarly. – guest1 Nov 19 '18 at 21:58
• No. Logistic regression is maximum likelihood. A prior is used in any Bayesian analysis: no prior, not Bayesian. A likelihood is not a probability function. It has to be scaled by a prior. – AdamO Nov 19 '18 at 22:00

\begin{align} \ell(\beta) &= \log\prod_{i=1}^N p(C=1, x_i\mid\beta)^{t_i} \cdot p(C=0, x_i\mid\beta)^{1-t_i} \\[8pt] &= \log\prod_{i=1}^N \left( p(C=1\mid x_i,\beta)~p(x_i)\right)^{t_i} \cdot \left(p(C=0\mid x_i,\beta)~p(x_i)\right)^{1-t_i} \\[8pt] &= \log\prod_{i=1}^N p(C=1\mid x_i,\beta)^{t_i} \cdot p(C=0\mid x_i,\beta)^{1-t_i}~~ p(x_i) \\[8pt] &= \sum_{i=1}^N t_i\cdot \log (p(C=1\mid x_i,\beta))+ (1-t_i)\cdot \log(p(C=0\mid x_i,\beta))+\log(p(x_i)) \\[8pt] \end{align}
Now, as you minimize the expression (where the sigmoid is used as ansatz for the posterior $$p(C\mid x)$$) with respect to $$\beta$$ the marginal distribution $$p(x_i)$$ is dropped and the same minimum will be found independent if you start with the posterior probabilities $$p(C\mid x)$$ or joint probabilities $$p(C,x)$$ in the expression for the likelihood. I think this is equivalent to the fact that bigger $$p(C\mid x)$$ implies bigger $$p(C,x)$$ for a class.
• Thanks a lot for your answer. I like the approach but I think there is an error from the first to the second line. I think $p(C=1, x_i|\beta) = p(C=1|x_i, \beta) p(x_i|\beta)$, i.e., I think the prior should be conditioned on beta for that equality to be true. PS: I looked into the links provided in the topic that you opened, and to me they do not clairify this specific matter, what do you think? – guest1 May 2 at 16:49
• @guest1: Can you specify what is not clear yet. Hopefully sows residual misunderst. on my side too. I would summarize: We are not trying to model $p(x)$ at all, the $x_i$ are fixed. [Note, that the sampling of the $x_i$ apparently introduces bias on the logit MLE parameter: stats.stackexchange.com/questions/6067/…). So,$p(x)$ does not depend on the logit parameters, can proceed suggested. Intuition: We want to find the best logit fit to targets $t_i$ "above" given points $x_i$, hence the product of posteriors can be used. – bebissig May 8 at 11:49