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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.

Kai Xie
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  • Please format your question using MathJax; it is difficult to read as it is. https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – ekkilop Aug 03 '17 at 08:26
  • @ekkilop pls check it out – Kai Xie Aug 03 '17 at 08:45

1 Answers1

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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".

Michael T.
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