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Sufficient Reasons

Formally, an abductive explanation $t$ for an instance $x$ given a classifier $f$ (that is binary or not) is a subset $t$ of the characteristics of $x$ such that every instance $x’$ sharing this set t of characteristics is classified by $f$ as $x$ is. A sufficient reason $t$ for $x$ given $f$ is an abductive explanation for $x$ given $f$ such that no proper subset $t’$ of $t$ is an abductive explanation for $x$ given $f$ (i.e., minimal w.r.t. set inclusion).

Considering the boosted tree of the Building Models page consisting of $4$ features ($A_1$, $A_2$, $A_3$ and $A_4$), we can derive as an example the sufficient reason $t$ = ($A_1 = 4$, $A_4 = 1$) for the instance $x$ = ($A_1=4$, $A_2 = 3$, $A_3 = 1$, $A_4 = 1$). In this figure, this sufficient reason is shown in red:


As you can see on the figure, some leaves of this sufficient reason corresponds to positive or negative predictions. However, all the instances $x’$ extending $t$ can be gathered into four categories, obtained by considering the truth values of the Boolean conditions over the two remaining attributes ($A_2$ and $A_3$) as encountered in the trees of $BT$. This table shows that, in every case, we have $W(F, x’) > 0$, showing that $f(x’) = 1$.

$A_1 = 4$ $A_2 = 3$ $A_3 = 1$ $A_4 = 4$ $w(T_1,x')$ $w(T_2,x')$ $w(T_3,x')$ $W(F,x)$
1 0 0 1 0.4 0.3 0.2 0.9
1 0 1 1 0.4 0.3 -0.4 0.3
1 1 0 1 -0.3 0.5 0.3 0.5
1 1 1 1 0.3 0.5 0.1 0.9

The algorithms to compute sufficient reasons are still under development and should be available in the next versions of PyXAI.

Calculating the sufficient reasons is a computationally difficult task (deciding whether $t$ is an abductive explanation for $x$ given $BT$ is ${\sf coNP}$-complete. To overcome this problem, we introduce another kind of abductive explanations easier to calculate, the Tree-Specific reasons.