# Bias in estimation of a latent / hidden variable drawn from a skewed distribution: what is it called?

I observe a bias effect in my measurement system that I can explain and correct using a simple latent or hidden variable model. I am sure this kind of effect has been described earlier in other fields but I cannot find the correct keywords to connect to the published state of the art.

A simplified description (to make things analytically solvabel):

• We have an unobserved variable $$x$$ that is drawn from an exponential distribution $$x \sim \pi_\lambda(x) =\lambda\exp(-\lambda x)$$

• The observed variable $$y$$, the measurement, conditioned on $$x$$ follows a normal distribution with mean $$x$$ $$y \sim \pi_{\sigma}(y|x) = N(y; x, \sigma^2)$$

• We can use Bayes' theorem to get the distribution of $$x$$ given measurement $$y$$ $$\pi_{\lambda, \sigma}(x|y) \propto \pi_{\sigma}(y|x) \pi_\lambda(x) \propto \exp\left(-\frac{1}{2\sigma^2}(y - x)^2 - \lambda x\right) \propto N\left(x; y - \lambda \sigma^2, \sigma^2\right)$$

• Thus $$E[X|y] = y - \lambda \sigma^2$$

We can assume $$\lambda$$ and $$\sigma^2$$ to be known.In reality, $$\lambda$$ would be inferred from a larger sample.

$$y$$ itself is commonly used as the estimator of $$x$$ in my field. This leads to a biased measurement in situations with a strongly skewed prior distribution (large $$\lambda$$ in this toy example) and low signal-to-noise ratio ($$\sigma$$ not $$\ll y$$).

$$y - \lambda \sigma^2$$ on the other hand would be an unbiased estimator of the unobserved variable $$x$$.

My questions are:

• Is the reasoning layed out in this toy example flawed?
• Do you know a name for this kind of effect or can you point to a treatment of a similar situation?

I have searched the net for any combination of the terms bias, skewed distribution, estimation, noisy measurement and hidden/latent/unobserved variable or state, but couldn't find a description that really matches this setup.

Edit: Naturally the correct posterior distribution in the above toy example is a truncated normal as $$x \geq 0$$. Thus $$E[X|y] = y - \lambda \sigma^2 + \frac{\phi\left( \frac{-(y - \lambda \sigma^2)}{\sigma} \right) - 0}{1 - \Phi\left( \frac{-(y - \lambda \sigma^2)}{\sigma} \right)}\cdot \sigma$$ with $$\phi$$ and $$\Phi$$ denoting the PDF and CDF of the standard normal, respectively (https://en.wikipedia.org/wiki/Truncated_normal_distribution). Here a plot for $$\lambda=\sigma=1$$