Some data

Maybe this is easiest to explain by going straight with the data. Here is how much money Bob has at the end of each day.

Worked?   Ate food? Gambled?  Cash left
--------  --------  --------  --------
   0         0         0         100
   1         0         0        1100
   0         1         0          95
   1         1         0        1095
   0         0         1         200
   0         0         1           0

Assume we have a whole bunch of observations they follow the pattern:

$$Left = 100 + 1000Worked -5Ate + X(-100, 100)Gambled + \epsilon $$

$X$ is some random variable, let's just say it is an unbiased coin flip, either +100 or -100.

Normal regression

If you plug this into Excel and run a regression you will get:

$$Left = 100 + 1000Worked -5Ate + \epsilon $$

This is because the arithmetic average of $Gambled$ is zero and the randomness of that variable is accounted for in $\epsilon$.

Regression with variance per input

Instead of the model just spitting out an estimated value for each output row, I want it to give me the estimated value and a variance.

But specifically, each variance should be based on the uncertainty each input brings. (I understand this might only make sense for categorical inputs, not scalar inputs.)

What type of modeling do I need to do to get the actual model I want, which is:

$$Left = 100 + Worked(1000 ± 0) + Ate(-5 ± 0) + Gambled(0 ± 100) + (0 ± 0) $$

(The actual model might have numbers close to zero instead of zero because of limited sampling.)


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