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]  *)

(* fDepPriorBiased[f_] := central[f]   *)

Boilerplate :

HH[data_,priors_]:=(priorProbHH /.priors)If[MemberQ[data,t],0,1]

TT[data_,priors_]:=(priorProbTT/.priors) If[MemberQ[data,h],0,1]

Fair[data_,priors_]:=(priorProfFair/.priors) (1/2)^Length[data]

BiasedF[data_,f_,priors_]:=f^Count[data,h](1-f)^Count[data,t](fDepPriorBiased/.priors)[f]

Biased[data_,priors_]:=(priorProbBiased/.priors) Integrate[BiasedF[data,f,priors],{f,0,1}]

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

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

pNorm[{}]//N

Make Plots :

plotsFor[data_,priors_]:=(

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

plotsFor[{h,t,h,t,h,t,h,t,h},{
priorProbHH->1,priorProbTT->1,priorProfFair->1,priorProbBiased->1,
fDepPriorBiased->broad   (* centra, broad, unif *)
}]//MatrixForm

plotsFor[{h,t,h,t,h,t,h,t,h},{
priorProbHH->1,priorProbTT->1,priorProfFair->1,priorProbBiased->1,
fDepPriorBiased->unif   (* centra, broad, unif *)
}]//MatrixForm

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}]];