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Suppose that $x\in R^1$, $y\in R^1$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Also, the MAP estimate should have an analytical expression.

Much thanks in advance!! Also, the more options the better!

edit:
This should work for any value of $y$ and the prior needs to be chosen before seeing $y$

Suppose that $x\in R^1$, $y\in [-30,30]$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Also, the MAP estimate should have an analytical expression.

Much thanks in advance!! Also, the more options the better!

edit:

1. This should work for any value of $y$ and the prior needs to be chosen before seeing $y$.
   
2. sorry made a mistake,instead of $y\in R^1$, $y\in [-30,30]$. As long as it's bounded by a negative number and a positive number it would work fine.

Suppose that $x\in R^1$, $y\in R^1$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Also, the MAP estimate should have an analytical expression.

Much thanks in advance!! Also, the more options the better!

Suppose that $x\in R^1$, $y\in R^1$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Also, the MAP estimate should have an analytical expression.

Much thanks in advance!! Also, the more options the better!

edit:
This should work for any value of $y$ and the prior needs to be chosen before seeing $y$

Suppose that $x\in R^1$, $y\in R^1$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Much thanks in advance!! Also, the more options the better!

Suppose that $x\in R^1$, $y\in R^1$ and $P(y|x) \sim N(y; x,\sigma_x)$. If I were to choose $P(x) \sim N(x;0,\sigma_p)$ and observe a single $y$ value, my MAP estimate of $x$ would be $$\frac{\frac{y}{\sigma_x}}{\frac{1}{\sigma_x}+\frac{1}{\sigma_p}}$$ or $$ \frac{y}{1+\frac{\sigma_x}{\sigma_p}}$$ since the denominator is always a constant greater than 1, the estimate is always the $y$ value divided by a constant that biases it towards 0. Is there a way for me to choose a prior such that the MAP estimate would be some form of $y$ that is biased away from 0?

e.g. If $y$ is -5, MAP estimate of $x$ should be a bit more negative than -5. If $y$ is 5, MAP estimate of $x$ should be a bit more positive than 5.

Also, the MAP estimate should have an analytical expression.

Much thanks in advance!! Also, the more options the better!