Here is the problem description.
Given the following generative model, p(x|y)~N(y,1), where x is the observation and y is the state/category.
Given that the a-priori distribution of y is y ~ N(z, 1/s) (where s and z are parameters).Calculate the MAP estimator $\ y_1$ as a function of the Maximum- Likelihood estimator $\ y_2$
And my guess for the answer is as follows,
Let’s say $\ y_2$ is the maximum likelihood estimator, such that $\ y_2$=$\ argmax_y$ $p(x|y)$. Given that $p(x|y) ~ N(y,1)$, then we have $\ y_2=y$.
Let’s say $\ y_1$ is the MAP estimator, such that $\ y_1$ = $argmax_y$ $\frac{p(y)p(x|y)}{p(x)}$ =$argmax_y$ $p(y)p(x|y)$= $argma_y$ $N(z,1/s)N(y,1)$ =??.
Therefore, $y_1$=??.
But I can not figure it out for the final answer . And I am not sure if it's right this way? Anyone knows pls help me.
Without giving away the exact answer ...
In what you've written for both the maximum likelihood estimator and the MAP estimator, replace your "$\operatorname{p}(v) \sim \operatorname{N}(\mu,\sigma^2)$" notation for the normal distributions with the explicit Gaussian probability density functions: "$\operatorname{p}(v) = C\exp(-\frac{(v - \mu)^2}{2\sigma^2})$", where $C$ is the normalizing factor. Then find $\underset{y}{\operatorname{argmax}}$.
For the MAP solution, this reference by P.A. Bromiley may be helpful: "Products and Convolutions of Gaussian Probability Density Functions".