I have a model such that after a $\textit{single}$ experiment, I gain a string of $N \in \mathbb{N}$ binary points $\mathbf{y} = y_1, y_2, \dots, y_N;$ that is to say that a single observation is $\textbf{y}$ and each $y_i \in \{0,1\}$.
Given an $unknown$ parameter vector $\boldsymbol{\theta} \in \mathbb{R}^{15}$, I can derive and compute the distribution $\mathbb{P}(\mathbf{y} | \boldsymbol{\theta})$. This distribution depends on $\boldsymbol{\theta}$ in a $highly$ non-linear way so that gaining an ML estimate of $\boldsymbol{\theta}$ can only be done through numerical methods. I am unable to explicitly differentiate $\mathbb{P}(\mathbf{y} | \boldsymbol{\theta})$ wrt $\boldsymbol{\theta}$.
I was wondering how I can attempt to show that the mapping from the parameter space to the space of distributions is injective; i.e. that the parameters of this model are identifiable? I'm not entirely convinced that the injectivity condition can be easily proved since the space of observations is discrete. Presumably I need some constraint on $N$ at the very least?
