Determing in a survey which people have likely lied You have data for each person on their BMI, age, weight, etc... (it doesnt actually matter, just a number of variables on their characteristics). 
You then ask people to fill out a survey whether they have diabetes or not, and you know that 20% of them have lied to you.
What are the common techniques to determine which people have lied? Ie whats the good data and bad data.
 A: Let $X_i$ denote the covariates for the $i^{th}$ person and let $Y_i$ denote their observed response ($1$ if they say they have diabetes and $0$ otherwise). The true unobserved (latent) responses can be modeled using logistic regression (or something similar).
$$Z_i \stackrel{\text{ind}}{\sim} Bern(p_i)$$
where $p_i = (1+\exp(-\beta X_i))^{-1}.$ The observed responses are then equal to 
$$Y_i = \xi_iZ_i + (1-\xi_i)(1-Z_i)$$
where $\xi_i$ is an indicator which is equal to $1$ if they told the truth and $0$ if they told a lie. You also have the constraint
$$\frac{1}{n} \sum_{i=1}^n \xi_i = 0.8.$$
The likelihood function conditional on the latent responses is straightforward
$$L(\beta | Z, X) = \prod_{i=1}^n p_i^{Z_i}(1-p_i)^{1-Z_i},$$
but the likelihood function for the observed data $Y$ is much less straightforward
$$L(\beta | Y, X) = \prod_{i=1}^n \left(p_i^{Y_i}(1-p_i)^{1-Y_i}\right)^{\xi_i}\left(p_i^{1-Y_i}(1-p_i)^{Y_i}\right)^{1-\xi_i}$$ This is exactly the time of problem that the EM algorithm was designed for. I would look into the EM algorithm to try to estimate the model parameters as well as the latent responses. Intuitively however, I would not be surprised if the parameters are unidentifiable, requiring additional modeling assumptions to be made. 
