Maximum a posteriori estimation

Assume that we want to estimate an unobserved population parameter \theta on the basis of observations x. Let f be the sampling distribution of x, so that f(x\mid\theta) is the probability of x when the underlying population parameter is \theta. Then the function: :\theta \mapsto f(x \mid \theta) \! is known as the likelihood function and the estimate: :\hat{\theta}_{\mathrm{MLE}}(x) = \underset{\theta}{\operatorname{arg\,max}} \ f(x \mid \theta) \! is the maximum likelihood estimate of \theta. Now assume that a prior distribution g over \theta exists. This allows us to treat \theta as a random variable as in Bayesian statistics. We can calculate the posterior density of \theta using Bayes' theorem: :\theta \mapsto f(\theta \mid x) = \frac{f(x \mid \theta) \, g(\theta)}{\displaystyle\int_{\Theta} f(x \mid \vartheta) \, g(\vartheta) \, d\vartheta} \! where g is density function of \theta, \Theta is the domain of g. The method of maximum a posteriori estimation then estimates \theta as the mode of the posterior density of this random variable: :\begin{align} \hat{\theta}_{\mathrm{MAP}}(x) & = \underset{\theta}{\operatorname{arg\,max}} \ f(\theta \mid x) \\ & = \underset{\theta}{\operatorname{arg\,max}} \ \frac{f(x \mid \theta) \, g(\theta)} {\displaystyle\int_{\Theta} f(x \mid \vartheta) \, g(\vartheta) \, d\vartheta} \\ & = \underset{\theta}{\operatorname{arg\,max}} \ f(x \mid \theta) \, g(\theta). \end{align} \! The denominator of the posterior density (the marginal likelihood of the model) is always positive and does not depend on \theta and therefore plays no role in the optimization. Observe that the MAP estimate of \theta coincides with the ML estimate when the prior g is uniform (i.e., g is a constant function), which occurs whenever the prior distribution is taken as the reference measure, as is typical in function-space applications. In the context of Bayes estimators, the MAP can be recovered as the minimizer of the Bayes risk with risk function : L(\theta, a) = \begin{cases} 0, & \text{if } |a-\theta| in the limit as c goes to 0, provided that the distribution of \theta is quasi-concave. Generally, however, a MAP estimator is not a Bayes estimator unless \theta is discrete. == Computation ==

MAP estimates can be computed in several ways: • Analytically, when the mode(s) of the posterior density can be given in closed form. This is the case when conjugate priors are used. • Via numerical optimization such as the conjugate gradient method or Newton's method. This usually requires first or second derivatives, which have to be evaluated analytically or numerically. • Via a modification of an expectation-maximization algorithm. This does not require derivatives of the posterior density. • Via a Monte Carlo method using simulated annealing ==Limitations==

While only mild conditions are required for MAP estimation to be a limiting case of Bayes estimation (under the 0–1 loss function), In contrast, Bayesian posterior expectations are invariant under reparameterization. As an example of the difference between Bayes estimators mentioned above (mean and median estimators) and using a MAP estimate, consider the case where there is a need to classify inputs x as either positive or negative (for example, loans as risky or safe). Suppose there are just three possible hypotheses about the correct method of classification h_1, h_2 and h_3 with posteriors 0.4, 0.3 and 0.3 respectively. Suppose given a new instance, x, h_1 classifies it as positive, whereas the other two classify it as negative. Using the MAP estimate for the correct classifier h_1, x is classified as positive, whereas the Bayes estimators would average over all hypotheses and classify x as negative. ==Example==

Suppose that we are given a sequence (x_1, \dots, x_n) of IID N(\mu,\sigma_v^2 ) random variables and a prior distribution of \mu is given by N(\mu_0,\sigma_m^2 ). We wish to find the MAP estimate of \mu. Note that the normal distribution is its own conjugate prior, so we will be able to find a closed-form solution analytically. The function to be maximized is then given by :g(\mu) f(x \mid \mu)=\pi(\mu) L(\mu) = \frac{1}{\sqrt{2 \pi} \sigma_m} \exp\left(-\frac{1}{2} \left(\frac{\mu-\mu_0}{\sigma_m}\right)^2\right) \prod_{j=1}^n \frac{1}{\sqrt{2 \pi} \sigma_v} \exp\left(-\frac{1}{2} \left(\frac{x_j - \mu}{\sigma_v}\right)^2\right), which is equivalent to minimizing the following function of \mu: : \sum_{j=1}^n \left(\frac{x_j - \mu}{\sigma_v}\right)^2 + \left(\frac{\mu-\mu_0}{\sigma_m}\right)^2. Thus, we see that the MAP estimator for μ is given by :\hat{\mu}_\mathrm{MAP} = \frac{\sigma_m^2\,n}{\sigma_m^2 \,n+ \sigma_v^2 } \left(\frac{1}{n} \sum_{j=1}^n x_j \right) + \frac{\sigma_v^2}{\sigma_m^2 \,n+ \sigma_v^2 } \,\mu_0 =\frac{\sigma_m^2\left(\sum_{j=1}^n x_j\right) + \sigma_v^2 \,\mu_0}{\sigma_m^2\,n + \sigma_v^2 }. which turns out to be a linear interpolation between the prior mean and the sample mean weighted by their respective covariances. The case of \sigma_m \to \infty is called a non-informative prior and leads to an improper probability distribution; in this case \hat{\mu}_\mathrm{MAP} \to \hat{\mu}_\mathrm{MLE}. == References ==