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Unconstrained Optimization with combining likelihood

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Hi all, I'm a little frustrated about one situation. I am doing an optimization using fminunc. I have two likelihood function l1=f1(b,x1,tu), l3=f3(b,x3,tl), in which b is a common input and tu, tl are parameters to be estimated. If I do optimization based on each likelihood separately, I got stable results which means each time with random inputs of tu or tl returns almost identical estimations. However, If I combine these likelihood into one function l13=f13(b,x1,x3,tu,tl)=f1(b,x1,tu)+f3(b,x3,tl), the results for tu and tl are close but not identical to the previous results and the results differ each time with random inputs. Does anyone have encountered similar situation or know what is the cause? Thanks in advance.
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jgg on 18 Jul 2016
First of all, what do you mean by "random inputs" - do you mean you vary the values of x or starting points? Both of these can cause changes; the starting point value means you just need to do a more complete search, while the x changes are expected.
More importantly, why are you combining your likelihood functions like that? If they are likelihoods, you need to look at the probabilistic model which is generating your data to see how they should be combined; even if they're independent x variables you should be multiplying most likely.

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