“Intensity” channels

Consider a random variable x taking discrete values $$x \in \{x_1,x_2,\ldots,x_n\},$$ with probability mass function $$p(x_i).$$ If all the possible values of x are positive, it turns out that the following is also a valid probability mass function (all positive, sums to 1): $$q(x_i) = p(x_i) \frac{x_i}{E[X]},$$ where $$E[X] = \sum_i x_i p(x_i)$$ is the expected value of x.

In fact q(x) is a posterior distribution for an “intensity” channel. Suppose we have a binary channel output $$y \in \{0,1\},$$ with x as the channel input. Further suppose, given x, we have $$\mathrm{Pr}(y = 1 \:|\: x_i) = k x_i$$ for some constant k; that is, the probability of turning y “on” is proportional to the input x. Then $$\mathrm{Pr}(y = 1) = \sum_i p(x_i) k x_i = k E[X]$$ and $$\Pr(x_i \:|\: y = 1) = \frac{p(x_i) k x_i}{k E[X]} = q(x_i).$$

I call this an “intensity” channel because the intensity of the input x is the key parameter in the input-output relation. A physical example would be a light sensor, where x represents the intensity of the light source (i.e., density of photons) and y = 1 represents the arrival of a photon at the sensor.

I feel like distributions of the form of q(x) must have a name and must be well studied. Drop me a line if you know more.

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