# MAP estimation for multiple parameters

Consider $N$ observed data points $x_i$ ($i=1,..,N$), and a likelihood that depends on $p$ parameters: $f(x_i|\theta_n)$ ($n=1,..p$). From Bayes' theorem

$$p(\theta_n|x_i) = \frac{f(x_i|\theta_n)g(\theta_n)}{h(x_i)}$$

I apply an MCMC sampler and obtain as a result the (sampled) posterior $p(\theta_n|x_i)$, and a (sampled) probability distribution for each $\theta_n$ individually (from where I can obtain for example the mean and the median for each $\theta_n$).

By definition, the maximum a posteriori (MAP) is obtained as:

$$\hat{\theta}_{\mathrm{MAP}}(x_i) = \underset{\theta_n}{\operatorname{arg\,max}} \ p(\theta_n|x_i)$$

This is where I get lost. Are the MAP estimates of each $\theta_n$:

1. the values associated to the maximum value of the (sampled) full posterior $p(\theta_n|x_i)$?, or
2. the values obtained finding the maximum (ie: the mode) in the (sampled) distribution of each single $\theta_n$ parameter?

To illustrate what I mean, here's the result of an MCMC run showing the sampled distribution of a single $\theta_n$ parameter:

You can see that the mode (cyan line) of the distribution for this parameter is very different from the MAP (red line) value that I obtain from the maximum value of the sampled posterior $p(\theta_n|x_i)$.

There is a unique* maximum a posteriori estimate (MAP) associated with the parameter vector $\theta=(\theta_1,\ldots,\theta_p)$. The MAP is the value that maximizes the posterior distribution, i.e.

$$\hat{\theta}_{MAP} = \text{argmax}_\theta p(\theta|x) = \text{argmax}_\theta f(x|\theta)g(\theta)$$

where $f(x|\theta)$ is the joint density/mass function of all of your observations $x=(x_1,\ldots,x_N)$ conditional on the parameter vector $\theta$ and $g(\theta)$ is the prior density for $\theta$.

You have not provided enough information in the problem to understand how to construct $f(x|\theta)$ given the individual $f(x_i|\theta_n)$. This notation actually doesn't make much sense as there doesn't seem to be a correspondence between $i$ and $n$.

*Technically the MAP does not need to be unique, but it often is.

• There is no correspondence between $i$ and $n$. The former runs through all my observed data, and the latter through the parameters of my model. Perhaps I over-complicated the notation? – Gabriel Apr 13 '18 at 21:54
• Why does the form of the likelihood ($f(x|\theta)$) matter? – Gabriel Apr 13 '18 at 21:57
• Am I understanding correctly then that the MAP is obtained maximizing the full posterior distribution ant not the distribution of each parameter separately? – Gabriel Apr 16 '18 at 15:13