# Can i fold in the data detection probabililtes in Maximum likelihood fitting?

I have a set of data points ($x_1,x_2,x_3 ...$) and would like to use the MLE estimator to fit a density function $f(x,a_1,a_2)$ to $x$, with some parameter $a_1, a_2$.

Normally, to do MLE I calculate the probablility of each data points in the function given specific parameter values, and minimize the joint likelihood function, say, $L= \prod p(x_i)$.

Imagine now for each of these data points, I also have a detection probability $p_{det}(x_i)$ to indicate whether they should be included in this dataset, high value indicates that it is a very reliable data point. This probability is pre-determined elsewhere.

How should I incorporate this probability into my MLE function fitting? I cannot just mulitply this with the above, say $L= \prod ( p(x_i) p_{det}(x_i)$, because then the two probability will be completely separable and the minimum of the likelihood function will be identical.

Is there a way to do this? Thanks a lot!

• Would you provide more mathematical details as to what the detection/inclusion/reliability probabilities are? If the "inclusion" probabilities are known, then under the model you're considering is there a positive probability that all data points will be selected and also that no data points are selected? – JimB Oct 16 '17 at 5:47