Credal set comparisons

Goal Identify and compare the credal sets of a Dempster-Shafer structure (DSS), probability box (PBOX), and fuzzy number (FN).

Hypothesis The intersection of the credal sets of a pbox and a FN yields a better/exact/... representation of the DSS.

Ratcheting Both a PBOX and a FN can be used to find an outer approximation of the credal set of a DSS with focal sets A_i = [l_i, r_i] and focal masses m_i.

• 1. DSS to PBOX: Use cumulative belief and cumulative plausibility as PBOX bounds.

     • Interpretation: Sort endpoints. Let l_(i) and r_(i) be s.t. l_(1) <= l_(2) <= ... and r_(1) <= r_(2) <= ... The new focal sets are A_(i) = [l_(i), r_(i)] if m_1 = m_2 = ....

  2. DSS to FN: Generally governed by IP2Pos-Transformation, i.e. depends on subjective plausibility and is not unique.

     • universal "melting" approach: subjective plausibility from DSS, i.e. pl(x) = sum_{i: x \in A_i} m_i

     • This may lead to bad results, though.

     • "table cloth" Interpretation: Make focal sets consonant, i.e. fill in the cavities. The new focal sets are of the form A_i = [ min( l_i, l_{i-1} ), max( r_i, r_{i-1} ) ].

Examples

• 1. [1,2] (true INTERVAL)

  2. [1,3], [2,4] (true PBOX)

  3. [1,4], [2,3] (true FN)

  4. [1,3], [2,4], [2,2.5] (mixed PBOX and FN)

  5. [1,2], [3,4] (disjunct focal sets)

  6. more weird stuff...

ToDo

• • Find distributions P which are in both PBOX and FN but not in DSS; find characteristics.

  • Find distributions P which are either in PBOX or FN (but not both) and, therefore, cannot be in DSS; find characteristics.

  • Find a way to quantify the effect of the intersection, e.g. by Hausdorff distances.?.