I am a scientist fitting some binomial data, and I have been using maximum likelihood.
My model gives a probability for each datum, $L = P(y|\theta)$. The model likelihood function is not a simple logistic model, but rather it estimates $P$ over a range of hyperparameters, and returns the marginal.
I was very surprised to find that I get different estimates of theta if I maximise the likelihood of outcome 1 occurring
$\hat{\theta}_1=argmax(L_y) $
or if I minimise the estimated probability of the opposite outcome occurring
$\hat{\theta}_0=argmin(1-L_y)$.
And through experimenting I realised this is because in general,
$\underset{\theta}{\mathrm{argmax}} [ \underset{y}{\sum}\mathrm{log}P(y|\theta) ] \ne \underset{\theta}{\mathrm{argmax}} [ - \underset{y}{\sum}\mathrm{log}(1-P(y|\theta)) ] $
I thought my estimate should be symmetrical for the two outcomes. I'm sure it's something simple, so I was looking for an explanation for this online. But I did not know where to start looking; googling for "minimum unlikelihood" and suchlike did not get me very far!
edit
It seems like $\theta_1$ overweights outcome 2, and $\theta_2$ overweights outcome 1, is that right?