unif[f_]:=1;central[f_]:=6 f(1-f);broad[f_]:=Cosh[3(f-1/2)]/(2/3 Sinh[3/2]);

Setting priors :

priorProbHH=1;priorProbTT=1;priorProfFair=1;priorProbBiased = 1; phtf[f_]:=central[f]

Boilerplate :

HH[data_]:=priorProbHH If[MemberQ[data,t],0,1]

TT[data_]:=priorProbTT If[MemberQ[data,h],0,1]

Fair[data_]:=priorProfFair (1/2)^Length[data]

BiasedF[data_,f_]:=f^Count[data,h](1-f)^Count[data,t]phtf[f]

Biased[data_]:=priorProbBiased Integrate[BiasedF[data,f],{f,0,1}]

p[data_]:={HH[data],TT[data],Fair[data],Biased[data]}

pNorm[data_]:=p[data]/Sum[p[data][[i]],{i,1,4}]

pNorm[{}]//N

Go :

ddata={h,h,h,t};
ans=pNorm[ddata]//N   ;
BarChart[{priorProbHH,priorProbTT,priorProfFair,priorProbBiased}/(priorProbHH +priorProbTT +priorProfFair+priorProbBiased), ChartLabels->{"Prior HH","Prior TT","Prior Fair","Prior Biased"},ColorFunction->Red]    
BarChart[ans, ChartLabels->{"HH","TT","Fair","Biased"}]    
ans
Plot[BiasedF[ddata,f],{f,0,1},(* Fills->RGBColor[1,0,0], *) PlotRange->All,Filling->Axis]

ans

Integrate[unif[x],{x,0,1}]

Integrate[central[x],{x,0,1}]

Integrate[broad[x],{x,0,1}]

realBias=Join[Table[h,{i,1,35}],Table[t,{i,1,35}]];

strongBias=Join[Table[h,{i,1,350}],Table[t,{i,1,250}]];