where Kx = Pr (u)Pr (e | x, u) and Kx = Pr (u)Pr (e | x, u**) are constants in terms of both**
θx|u and θx|u. By substituting y, e and z, e for e **in Theorem 2.1, we get:**

$$\begin{array}{r c l}{{\alpha_{y,\mathbf{e}}={\frac{\partial P r(y,\mathbf{e})}{\partial\tau_{x|\mathbf{u}}}}}}&{{=}}&{{{\frac{P r(y,\mathbf{e},x,\mathbf{u})}{\theta_{x|\mathbf{u}}}}-{\frac{P r(y,\mathbf{e},x,\mathbf{u})}{\theta_{x|\mathbf{u}}}}}}\\ {{\alpha_{z,\mathbf{e}}={\frac{\partial P r(z,\mathbf{e})}{\partial\tau_{x|\mathbf{u}}}}}}&{{=}}&{{{\frac{P r(z,\mathbf{e},x,\mathbf{u})}{\theta_{x|\mathbf{u}}}}-{\frac{P r(z,\mathbf{e},z,\mathbf{u})}{\theta_{x|\mathbf{u}}}}}\end{array}$$

Pr (y, e, x, u)

θx|u

$${\bf\Sigma},\overline{{{x}}},{\bf u})\overleftarrow{\bf\Sigma};\qquad\qquad\qquad\qquad(2)$$ $${\bf\Sigma},\overline{{{x}}},{\bf u})\overleftarrow{\bf\Sigma}.\qquad\qquad\qquad\qquad(3)$$ $$\overline{{\bf\Sigma}}{\bf u}$$

Pr (z, e, x, u)

θx|u

Now, if we want to enforce the Difference constraint, Pr (y | e) − Pr (z | e) ≥ ǫ**, it**
suffices to ensure that Pr (y, e) − Pr (z, e) ≥ ǫPr (e**). Suppose that the previous constraint**
does not hold, and we wish to establish it by applying a change of δ **to the meta parameter**
τx|u. Such a change leads to a change of αeδ in Pr (e). It also changes Pr (y, e**) and** Pr (z, e)
by αy,eδ and αz,eδ**, respectively. Hence, to enforce the** Difference **constraint, we need to**
solve for δ **in the following inequality:**
[Pr (y, e) + αy,eδ] − [Pr (z, e) + αz,eδ] ≥ ǫ[Pr (e) + αeδ].

Rearranging the terms, we get the following result.

Corollary 2.1 **To satisfy the** Difference constraint, we need to change the meta parameter τx|u by δ**, such that:**

$$1\geq\delta[-\alpha_{y,\mathbf{e}}+\alpha_{z,\mathbf{e}}+\epsilon\alpha_{\mathbf{e}}],$$

Pr (y, e) − Pr (z, e) − ǫPr (e) ≥ δ[−αy,e + αz,e + ǫαe],
where the α **constants are defined by Equations 1, 2 and 3.**
We can similarly solve for parameter changes δ **that enforce the** Ratio **constraint,**
Pr (y | e)/Pr (z | e) ≥ ǫ**, in the following inequality:**

$$[P r(y,{\bf e})+\alpha_{y,{\bf e}}\delta]/[P r(z,{\bf e})+\alpha_{z,{\bf e}}\delta]\geq\epsilon.$$
$$\leq\,0\,|\,1$$

Rearranging the terms, we get the following result.

Corollary 2.2 **To satisfy the** Ratio **constraint, we need to change the meta parameter** τx|u by δ**, such that:**
Pr (y, e) − ǫPr (z, e) ≥ δ[−αy,e + ǫαz,e],
where the α **constants are defined by Equations 2 and 3.**
For both the Difference and Ratio constraints, the solution of δ**, if any, is always in**
one of two forms:

- δ ≤ q, for some computed q < **0, in which case the new value of meta parameter** τx|u must be in the interval [0, p + q].

- δ ≥ q, for some computed q > **0, in which case the new value of meta parameter** τx|u must be in the interval [p + q, 1].