Uniqueness of the Level Two Bayesian Network Representing a Probability Distribution

<table>(a): fraction of a BN2 on <svg height="11.4625" id="M331" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 12.5875 11.4625" width="12.5875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" id="x1D434"></path></g>
</svg>. (b): BN2 obtained ater summing out <svg height="15.4875" id="M332" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 23.237499 15.4875" width="23.237499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.2)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g>
<g transform="matrix(.012,-0,0,-.012,4.7,15.275)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g><g transform="matrix(.012,-0,0,-.012,10.673,15.275)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,-0,0,-.012,17.658,15.275)"><path d="M363 114l21 -20q-62 -106 -161 -106q-61 0 -94 40t-33 138v56l-54 -40l-14 27l68 57v89q0 112 15.5 166t49.5 91q43 46 96 46q44 0 71 -32.5t27 -88.5q0 -41 -17.5 -81.5t-49.5 -76t-55.5 -57t-55.5 -46.5v-87q0 -138 81 -138q60 0 105 63zM177 421v-101q117 105 117 217
q0 38 -15 60.5t-41 22.5q-34 0 -47.5 -46.5t-13.5 -152.5z" id="x2113"></path></g>
</svg>.</table>

International Journal of Mathematics and Mathematical Sciences

fig9

Figure 9

Figure 9: Uniqueness of the Level Two Bayesian Network Representing a Probability Distribution 