# Is MLE with regularization a bayesian method?

It is usually said that priors on bayesian statistics can be regarded as regularization factors since they penalize solutions where the prior places low density of probability.

Then, given this simple model whose MLE parameters are: $$argmax_{\mu} \text{ } \mathcal{N}(y; \mu, \sigma)$$

and I add a prior: $$argmax_{\mu} \text{ } \mathcal{N}(y; \mu, \sigma) \mathcal{N}(\mu; 0, \sigma_0)$$ the parameters are not the MLE parameters but the MAP parameters.

Question: Does this mean that if I introduce some regularization in my model I am doing a bayesian analysis (even if only use point-estimates)?

Or this just makes no sense making this "ontological" distinction at this point since the method to find either MLE or MAP is the same (isn't it?)?