# Conditional Independence Assumption

In Bayesian inference, but also in MLE, the conditional independence assumption (CIA) is a usual assumption.

Typically, it is used to simplify the likelihood of the model, and to allow the factorization of the joint conditional distribution:

$$p(\mathbf{y}|\mathbf{\theta}) = \prod_{n=1}^{N}p(y_n|\mathbf{\theta})$$

where, $$\mathbf{y}$$ is the vector ($$\mathtt{Nx1}$$) of observations (measurements) and $$\mathbf{\theta}$$ contains the model parameters.

Questions:

1. When CIA is violated, it means that new realizations of $$y$$ depend on past observations. And clearly, if we assume independence, we can't trust the results. The data-generating model is wrong. Is this reasoning correct?

2. How to test whether this is a reasonable assumption to make? Is it simply by looking at the auto-correlation function for the observations?

3. In BDA 3, by Gelman et al., there is an Example: Checking the assumption of independence in binomial trials at page 147. How does that generalize to multiple regressors, so that $$p(\mathbf{y}|\mathbf{\theta})$$ is actually $$p(\mathbf{y}|\mathbf{\theta}, \mathbf{X})$$? and when $$\mathbf{y}$$ is a continuous variable?

4. If there is high dependence (e.g. in time series), how can I get rid of it? One idea which comes to mind is to treat the problem as time-dependent, and model somehow the dependence.

5. Do I understand it right that state-space models disguise this CIA under the Markovian process model construction?

• If in reality $$y$$'s a are dependent (as it is usually the case in time-series), is it right to say that the dependence is "outsourced" to the hidden state?

If there are some other CIA-related aspects, which one should be aware of, and often overlooked, please share.

Thank you!