Common parameters for conditional likelihood

Question

I am trying to understand the concept of conditional likelihood in the context of logistic regression.

One paper I am reading defines $L(\theta; y|x) = f(y|x; \theta)$, then goes on to point out that there may be different distributions $f(y|x; \theta)$ for every different $x$, and also that these different distributions share the same parameters $\theta$.

The first part of this makes sense that there may be different distributions for each $x$, however, I am failing to understand why $\theta$ must be the same for each of these distributions. Can anyone shed some light on this for me?

Answer 1

The point is not that $\theta$ must be the same, only that that is the usual situation in applications! Each x typically indexes a different observation unit, and if each unit has its own parameters you will not be able to do pooling of information over units ... typically, in that situation the mle need not be consistent, for instance. One simple example is paired comparisons: We have $n$ independent pairs $(X_i, Y_i)$, say, the pair is correlated, expectation is given by $$ \text{E}(X_i) = \theta_i $$, $$ \text{E}(Y_i) = \theta_i + \Delta $$ You can write down a likelihood function (for example, completing the specification to be bivariate normal) with each $\theta_i$ included, but there will be no consistent estimator of the $\theta_i$. Worse, the mle for $\Delta$ will be inconsistent, too! ( it will converge in probability to $\Delta/2$). If the interest is only in $\Delta$, as is typical, you can reduce the data by taking differences $D_I = Y_i - X_i$, that way obtaining a likelihood function only depending on $\Delta$, which will give a consistent mle.