But if I know that there are external factors, I know the bullet will deviate for sure. I don’t know where but I know it will.
You assume that blur kernel is non-monotonic, and this is our entire disagreement. I guess that different tasks have different noise structure (for instance, if somehow noise geometrically increased - ±1,±2,…,±2i - we wouldn’t ever return to an exact point we had left).
However, if noise is composed from many i.i.d. small parts, then it has normal distribution which is monotonic in the relevant sense.
I mentioned this in my comment above, but I think it might be worthwhile to differentiate more explicitly between probability distributions and probability density functions. You can have a monotonically-decreasing probability density function F(r) (aka the probability of being in some range is the integral of F(r) over that range, integral over all r values is normalized to 1) and have the expected value of r be as large as you want. That’s because the expected value is the integral of r*F(r), not the value or integral of F(r).
I believe the expected value of r in the stated scenario is large enough that missing is the most likely outcome by far. I am seeing some people argue that the expected distribution is F(r,θ) in a way that is non-uniform in θ, which seems plausible. But I haven’t yet seen anyone give an argument for the claim that the aimed-at point is not the peak of the probability density function, or that we have access to information that allows us to conclude that integrating the density function over the larger-and-aimed-at target region will not give us a higher value than integrating over the smaller-and-not-aimed-at child region
You assume that blur kernel is non-monotonic, and this is our entire disagreement. I guess that different tasks have different noise structure (for instance, if somehow noise geometrically increased - ±1,±2,…,±2i - we wouldn’t ever return to an exact point we had left).
However, if noise is composed from many i.i.d. small parts, then it has normal distribution which is monotonic in the relevant sense.
I mentioned this in my comment above, but I think it might be worthwhile to differentiate more explicitly between probability distributions and probability density functions. You can have a monotonically-decreasing probability density function F(r) (aka the probability of being in some range is the integral of F(r) over that range, integral over all r values is normalized to 1) and have the expected value of r be as large as you want. That’s because the expected value is the integral of r*F(r), not the value or integral of F(r).
I believe the expected value of r in the stated scenario is large enough that missing is the most likely outcome by far. I am seeing some people argue that the expected distribution is F(r,θ) in a way that is non-uniform in θ, which seems plausible. But I haven’t yet seen anyone give an argument for the claim that the aimed-at point is not the peak of the probability density function, or that we have access to information that allows us to conclude that integrating the density function over the larger-and-aimed-at target region will not give us a higher value than integrating over the smaller-and-not-aimed-at child region