# Statistical machine translation word alignment for FR-ENG and ENG-FR: what is p(e) and p(f)?

I'm currently trying to implement this paper, but am struggling to understand some of the math here. I'm pretty sure I understand how to implement the E-step, but for the M-step, I'm confused on how to compute the M-step. It says just before section 3.1 that $$p_1(x, z; \theta_1) = p(e)p(a, f|e; \theta_1)$$, and then the same for $$p_2$$ but with $$e$$ and $$f$$ swapped. The second part of this makes sense to me, but what is $$p(e)$$ or $$p(f)$$? From my understanding, $$e, f$$ are sentences in the bi-text. So how would we compute the probability of a sentence?

It says earlier that $$p(e)$$ and $$p(f)$$ are arbitrary distributions that don't affect the optimization problem, but then how do we compute $$p_1(x, z; \theta_1)$$?

Thanks!

You are right that $$p(e)$$ is the probability of the English sentence. Estimating the probability of a sentence is achieved by a language model.
This kind of machine translation model is known as the noisy channel model. The noisy channel model says that given a french sentence $$f$$, its best English translation is
$$e^* = \arg\max_{e\in E} p(e)p(f|e)$$
In this equation the $$p(e)$$ is the language model. Back in the era of IBM models (which are built upon the noisy channel approach), it is usually an n-gram based language model, calculated as (assuming bigram) $$p(e_1e_2...e_n)=p(e_1|)p(e_2|e_1)p(e_3|e_2)...p(|e_n)$$
And $$p(f|e)$$ is the translation model where you need to use the EM algorithm to solve. Inside the EM algorithm you do not update the language model parameters, so yes, $$p(e)$$ and $$p(f)$$ don't affect the optimization problem.