A=Sqrt[2/3]{Cos[#],Sin[#],Sqrt[1/2]}&/@Table[Pi/2+2Pi/3+2k Pi/3,{k,0,2}]//Transpose;
B=Inverse[A];
(* Bounds to center the simplex *)
{p1,p2,p3}=Transpose[A];
e=1/20;
x1=First[p1]-e/2;
x2=First[p2]+e/2;
w=x2-x1;
h=p3[[2]]-p2[[2]];
marg=(w-h+e)/2;
y1=p2[[2]]-marg;
y2=p3[[2]]+marg;
(* map 3d probability vector into 2 vector *)
p2r[x_,y_,z_]:=Most[A.{x,y,z}];
(* map 2-vector in 3d probability vector *)
r2p[u_,v_]:=B.{u,v,Sqrt[1/3]};
valid[x_,y_]:=Min[r2p[x,y]]≥0&&Max[r2p[x,y]]≤1;

KL[q_,p_]:=Total[MapThread[#1 Log[#1/#2]&,{q,p}]];
(* generates expansion of KL(q,p) around p as function of q *)
Clear[q1,q2,q3,p1,p2,p3];
Fisher1[p_]:=Function@@{{q1,q2,q3},
hess=D[KL[{q1,q2,q3},p],{{q1,q2,q3},2}]/.{q1->p[[1]],q2->p[[2]],q3->p[[3]]};
1/2({q1,q2,q3}-p).hess.({q1,q2,q3}-p)
};

getplot:=(
p=RandomReal[{0,1},3]//Normalize[#,Total]&;
fish=Fisher1[p];
p0=RegionPlot[valid[x,y],{x,x1,x2},{y,y1,y2},PlotStyle->None];
p1=RegionPlot[valid[x,y]&&(fish@@(r2p[x,y])>KL[r2p[x,y],p]),{x,x1,x2},{y,y1,y2},PlotPoints->100];
p2=Graphics[Disk[p2r@@p,.04]];
Show[p0,p1,p2,Frame->None]
);
plots=Table[getplot,{i,1,16}];
GraphicsGrid[Partition[plots,4]]