equation solving - Why is Solve returning an empty list?

I evaluate

Solve[{-((2 (4 θ0 + θ1))/σ^2) - (2 (4 θ0 + θ2))/σ^2, 
       -((4 θ0 + θ1)/(2 σ^2)), -((4 θ0 + θ2)/(2 σ^2)),
       -(2/σ) + (4 θ0 + θ1)^2/(2 σ^3) 
       + (4 θ0 + θ2)^2/(2 σ^3)} == {0, 0, 0, 0} && σ > 0, 
      { θ0, θ1, θ2, σ}]

and Solve returns {}. Why? Should it be able to solve the equation?

Answer

Solve works correctly, when returning {} it means there are no solutions.

To demonstrate it let's rewrite your system:

system = Thread[{-((2 (4 θ0 + θ1))/σ^2) - (2 (4 θ0 + θ2))/σ^2,
                 -((4 θ0 + θ1)/(2 σ^2)), 
                 -((4 θ0 + θ2)/(2 σ^2)), 
                 -(2/σ) + (4 θ0 + θ1)^2/(2 σ^3) + (4 θ0 + θ2)^2/(2 σ^3)} ==
                  {0, 0, 0, 0}] 

{-2 (4 θ0 + θ1)/σ^2 - 2 (4 θ0 + θ2)/σ^2 == 0,
 -(4 θ0 + θ1)/(2 σ^2) == 0, 
 -(4 θ0 + θ2)/(2 σ^2) == 0, 
 -(2/σ) + (4 θ0 + θ1)^2/(2σ^3) + (4 θ0 + θ2)^2/(2 σ^3) == 0}

One can see from the second and third equations that we can define appropriate rules to simplify the system:

rules = { θ1 -> -4 θ0, θ2 -> -4 θ0};

Now we have:

system /. rules

{True, True, True, -2/σ == 0}

Thus the system cannot be satisfied because σ > 0 in your assumptions supplementing the system.