# Relation between bootstrap mean and parameter value estimated via maximum likelihood

I have an observed data set $O$ and a synthetic model $S(\theta)$ which attempts to describe it.

By fixing $\theta$ to different values I can generate $M$ synthetic realizations of the model:

$$S_k = S(\theta_k),\, k=1..M$$

I want to find the $S_k$ that best matches my observed data set $O$.

I use a genetic algorithm to find the maximum of a likelihood equation $L$ between the $S_k$ and $O$. The $S_k$ with the maximum likelihood value is:

$$S_{max} = S(\theta_{max}) = max[L(S_k, O), \, k=1..M]$$

$\theta_{max}$ is thus the value of $\theta$ that I associate with my observed data set $O$.

To obtain the uncertainty of $\theta_{max}$, I repeat the above process $n$ times on random samplings with replacement of the true data set $O$, ie: I apply a bootstrap with replacement process.

The bootstrap returns $\theta_i, \,i=1..n$ values of the model variable.

I obtain the uncertainty on $\theta_{max}$ by calculating the standard deviation of the $\theta_i$ values:

$$\theta_{max} \pm std[\theta_i, \,i=1..n]$$

But I can also use the $\theta_i, \,i=1..n$ values to obtain a mean:

$$\theta_m = mean[\theta_i, \,i=1..n]$$

My question: what is the relation between this $\theta_m$ and the $\theta_{max}$ value obtained via the maximum likelihood?