Should the average prediction for a given attribute value equal the rate for that value? Let's say I'm predicting the likelihood that someone will buy a widget, using their age, eye color, and gender as input attributes.
I split my data into a training set and a test set, and train up my predictive model using the (large) training set.
Let's say that 15% of people in the test set with blue eyes buy the widget, should I expect the average predicted buy probability of people with blue eyes would be around 0.15 also (assuming a good predictive model)?
Intuitively it seems like it should, but I don't want to assume that it's true.
 A: If the phrase  

Let's say that 15% of people in the test set with blue eyes buy the widget 

means "15% of those having blue eyes in the test sample", then the answer is "yes", this is the magnitude to which your model should compare.
Let's abstract from the existence of the other explanatory variables. In the very general terms you put it, you have a model to predict a conditional probability
$$\hat P(W =1 \mid X_1) = g(X_1)$$
and you wonder whether
$$\hat P_\text{model}(W =1 \mid X_1 = B)=g(X_1=B)\; ?\approx?\;\frac {n_\text{test}\{W=1,X_1=B\}}{n_\text{test}\{X_1=B\}} \tag{1}$$
I believe it is obvious how notation maps to the description you gave of your case.
Denote $N_\text{test}$ the test sample size. Then the right-hand-side of $(1)$ can be identically written
$$\frac {n_\text{test}\{W=1,X_1=B\} / N_\text{test}}{n_\text{test}\{X_1=B\}/N_\text{test}}$$
The numerator is the relative empirical frequency of the joint event "have blue eyes and bought the widget", while the denominator is the relative empirical frequency of blue-eyed people in the test sample. Under various assumptions, we usually accept this as estimates of the corresponding probabilities. In other words, 
$$\frac {n_\text{test}\{W=1,X_1=B\} / N_\text{test}}{n_\text{test}\{X_1=B\}/N_\text{test}} = \frac {\hat P_\text{test}(W=1,X_1=B)}{\hat P_\text{test}(X_1=B)} = \hat P_\text{test}(W =1 \mid X_1 = B)$$
the last equality from the theoretical relationship between joint, marginal and conditional probabilities.
So you want to compare
$$g(X_1=B) = \hat P_\text{model}(W =1 \mid X_1 = B) =\;\;?\approx ? \;\;\frac {\hat P_\text{test}(W=1,X_1=B)}{\hat P_\text{test}(X_1=B)}$$
$$\Rightarrow \hat P_\text{model}(W =1 \mid X_1 = B) \;\;?\approx?\;\; \hat P_\text{test}(W =1 \mid X_1 = B)$$
which is indeed what your model should tend to satisfy
