# Original Question
Suppose we have a sequence of Bernoulli trials $X_1, X_2, \cdots X_T$ which are ordered in time and may or may not be independent. I am interested in understanding the probability of success.. The way I'm thinking about it, I have two primary options.

1. Consider the trials, $X_t \stackrel{iid}{\sim} Bern(\theta)$ and estimate $\theta$.
2. Treat the observations as a markov chain (does success/failure at time $t$ have an effect on probability of success at time $t+p$?

**How do i determine which model is best?** Or in otherwords, how do I estimate the order of the markov chain.

To test the validity of the first model, I was considering using the approach of Andrew Gelman in his *Bayesian Data Analysis* book, where he constructs an "empirical p-value" via simulation using the test statistic $T = \text{longest streak of successes}$. Beyond this, I am unsure how to appropriately compare the different models.

# Partial Answer

It appears that the PACF (partial autocorrelation function) can be used to estimate the order of these sequences.

I generated 3 large sequences $(T=1000)$ of Bernoulli trials, under independence, first order Markov dependence and second order Markov dependence. We plot the ACF for each case.
[![enter image description here][1]][1]

Moreover, for the first order Markov Chain, the ACF seems to satisfy the property of an AR(1) model that $\rho_k = \rho_1^k$.

[![enter image description here][2]][2]

I can't seem to find any formal justification for why this works however. As I understand it, if the Bernoulli trials satisfy a $p$ order Markov property, this does not imply the chain is an $AR(p)$ process, ecause we cannot write 
$$X_t = \phi X_{t-1} + \epsilon_t$$ 
for $\epsilon_t$ iid. (If somebody can confirm or correct me on this, that would be great).

So if the $p$ order Markov process is not an $AR(p)$ process, why does this work?


  [1]: https://i.sstatic.net/ryLOm.png
  [2]: https://i.sstatic.net/wmH4p.png