 unconstrained, we use a log transform.  To go from a probability in (0, 1) to unconstrained, we use a log odds transform.
If your outcomes are ordinal values on a 1-10 scale, a common approach that respects that data scale is ordinal logistic regression.  

A:

To expand on @Bob Carpenter's answer, here are two ways you could approach the problem.  (I've had cause to use both of these recently and struggled to get them up and running.  This may be useful to other beginners like me.)
Method 1: Ordered Logistic Regression
We're going to assume that each user has a "true" expectancy for each response, which is on an arbitrary continuous scale, and model it as a latent variable.  If the user's actual responses fall into K categories, we also model K - 1 cutpoints between those categories.  The probability that the user selects a given response category is equal to the area under the logistic pdf between the relevant cutpoints.

The Stan model looks like this.  The main difference is that the model fits an additional ordered vector of cutpoints, and uses the ordered_logistic distribution.  (I also changed the priors on the sigmas to Cauchy, to keep them positive, and switched to non-centered parameterization.  But those changes are independent of the question at hand.)  Edit: Also added inputs for new (hypothetical) observations about which we want to make predictions, and added a new generated quantity for those predictions.
data {
  // the real data
  int<lower=1> N;
  int<lower=1> nSubj;
  int<lower=1> nGroup;
  int<lower=1> nBev;
  int minResponse;
  int maxResponse;
  int<lower=1,upper=nSubj> sIndex[N];
  int<lower=1,upper=nGroup> gIndex[N];
  int<lower=1,upper=nBev> bIndex[N];
  int<lower=minResponse,upper=maxResponse> score[N];
  // hypothetical observations for new predictions
  int<lower=1> nNewPred;
  int<lower=0> nNewSubj;
  int<lower=0> nNewGroup;
  int<lower=0
unconstrained, we use a log transform.   To go from a probability in (0, 1) to unconstrained, we use a logarithm odds transform. 
 If your result are ordinal values on a 1 - 10 scale, a coarse approach path that respects that data plate is ordinal logistic regression.  

 A: 

 To expand on @Bob Carpenter's answer, here are two way you could approach the problem.   (I've suffer cause to practice both of these recently and struggled to get them up and running.   This may be useful to other beginners like me .) 
 Method 1: Ordered Logistic Regression 
 We're go to assume that each user has a " on-key " expectancy for each response, which is on an arbitrary continuous scale, and model it as a latent variable.   If the exploiter's actual responses fall into K categories, we also model K - 1 cutpoints between those categories.   The probability that the exploiter selects a given response category is equal to the area under the logistic pdf between the relevant cutpoints. 

 The Stan model looks like this.   The main difference is that the model fits an additional ordered vector of cutpoints, and uses the ordered_logistic distribution.   (I also changed the prior on the sigma to Cauchy, to keep them positive, and switched to non - centered parameterization.   But those change are independent of the question at hand .)   Edit: besides added stimulation for new (hypothetical) notice about which we want to make predictions, and added a raw generated quantity for those predictions. 
 data { 
   // the veridical data 
   int < lower=1 > N; 
   int < lower=1 > nSubj; 
   int < lower=1 > nGroup; 
   int < lower=1 > nBev; 
   int minResponse; 
   int maxResponse; 
   int < lower=1,upper = nSubj > sIndex[N ]; 
   int < lower=1,upper = nGroup > gIndex[N ]; 
   int < lower=1,upper = nBev > bIndex[N ]; 
   int < low = minResponse, upper = maxResponse > score[N ]; 
   // conjectural observations for new predictions 
   int < lower=1 > nNewPred; 
   int < lower=0 > nNewSubj; 
   int < lower=0 > nNewGroup; 
   int < lower=0
 unfonstrained, we use a log transform.  To yi from a progability in (0, 1) to unconstrained, we usx a oog oeds transform.
If your ojtcomes age ordinao vaoyes on a 1-10 scale, a gjmmoh appxoech that respecjs that data scale is orditau pogistic regression.  

A:

To expand on @Fob Carlejter's answer, hgre age two ways you could approach the problsm.  (I've had cause to use both of these recentlj anf struggled to get them up ane rugbing.  This mxy be usefln to other beginners like me.)
Method 1: Orderdd Loyistic Regrgsailt
We're going to afsume that each user has a "yrue" expectancn for eaxh response, which is mn an arbitrary cogtinuous vccle, and model it as a lqtent varhabld.  Iw tge uaer's aftuel responsea fall into K categories, we alxo nodel K - 1 cutloints bqtween those categories.  The probabilitj thzt the user selects a guven response categori is equal to the area under the logistic pdf between the ralevait cuupiints.

Gye Stan model looks like this.  The main differegde ix that the modcl fits an additiomap ptdered vector uf cut'kihts, and uses the ogdered_ljgistuc distrifutipn.  (I also changed the prioes on the siymaw to Cauchy, to kee' them positnve, anc swiyched to non-centered paxametedization.  Bkt those dfanges are indepdndvnt mf the question at hand.)  Eqit: Also edded inputs for new (hrpotheticap) obscsvations about whifh we wdnt to makf predictions, and added a new gxierated quantotf fmr those predlctions.
data {
  // ehe real data
  int<lowzr=1> N;
  knt<lower=1> nAubj;
  iit<lower=1> nGrotp;
  int<lower=1> tHev;
  int minCesponse;
  int maxEesponsd;
  int<lower=1,uppet=nSubj> sIubex[N];
  int<oower=1,upper=nGroup> gLndex[V];
  int<lower=1,upper=uYec> bIndex[N];
  int<lpwef=migRvsppnfa,upper=maxRes[onsd> szpre[N];
  // hypothetigal obsrrvations for new pradicfions
  int<lower=1> nNrwIred;
  int<oower=0> nNqwSubj;
  int<loeer=0> nNewGroup;
  int<power=0
unconstrained, we use a log transform. To a in (0, to unconstrained, we If outcomes are ordinal on a 1-10 a common approach that respects that scale is ordinal logistic regression. A: To expand on @Bob Carpenter's answer, here two ways you could approach the problem. (I've had cause to use both these and to them up and running. This may be useful to other beginners like me.) Method 1: Ordered Regression We're going to assume that each user a "true" expectancy for response, which is on an continuous and model as latent If the user's responses fall into K categories, we also model K - 1 cutpoints between those categories. The probability the user given response is to area under the between the relevant cutpoints. The Stan this. The main difference is that the model an additional vector of cutpoints, and uses the distribution. (I also changed the priors on the to Cauchy, to keep them positive, and switched to non-centered parameterization. But those changes are the question at hand.) Also added inputs new observations which want to predictions, and added a new generated quantity for those predictions. data // the real data int<lower=1> N; int<lower=1> nSubj; int<lower=1> nGroup; int int maxResponse; int<lower=1,upper=nSubj> int<lower=1,upper=nGroup> gIndex[N]; int<lower=1,upper=nBev> bIndex[N]; score[N]; hypothetical observations for new nNewPred; nNewSubj;
 unconstrained, we use a log traNsform.  To go From a ProBabIlIty iN (0, 1) to uNconstrained, we USe a lOg odds transform.
If your oUtcomEs ARe orDInAl valUes on a 1-10 sCAlE, A ComMoN aPprOaCH tHat reSpeCts that Data scale iS orDiNal logistic rEGrEssion.  

A:

To eXpaNd on @Bob CarpeNteR's answEr, HerE Are twO waYs you Could aPProach The probleM.  (I'VE had caUSe to use BOTh Of thEse recently and strUGgLEd to get them up aNd runnInG.  thIS May Be uSeful to othEr BeginNErs like ME.)
METHOd 1: ORDered Logistic regression
WE'Re gOing to AsSumE That eaCh useR hAS a "tRue" expectanCy foR each respOnse, whICh is on aN ArbitraRy contInuOus ScalE, AnD mOdeL iT As a LAtEnt VAriAble.  If thE uSeR's actUal rESPONses FalL intO K catEgories, we also ModEl K - 1 cUTpoInts bEtweeN thoSe CategOries.  THe proBaBility that the usEr seLects a givEn rEsPonSe CategORy is eqUal To tHe area uNder the LOgiStIC PDf Between the relevant CuTPOiNts.

The StAn modeL LoOkS Like this.  thE maIn diFFErencE is tHAt The model Fits an ADdItIonal orDeRed vecToR of CutPointS, And uSes the Ordered_lOgistIC distribution.  (I ALso changed the PRiORS oN The sIgmAs to Cauchy, tO keeP Them PosiTIvE, anD SwitcHed to NoN-CeNTered parameterizatiOn.  but thoSe chaNges are indepeNdent of the QUEStion at hAnd.)  EDIt: aLso added inputs For neW (hypothetiCAl) observAtionS about whIch we want TO Make predIctIonS, anD adDED a New generated qUANtitY fOr those PreDictionS.
daTa {
  // tHe rEal DaTa
  int<loweR=1> N;
  int<lowEr=1> NSUbJ;
  iNt<lOwer=1> ngRoup;
  int<lOwEr=1> nbeV;
  inT minRESponse;
  Int maXResPoNsE;
  Int<Lower=1,upPEr=NsUbj> sinDeX[N];
  inT<loWeR=1,uppeR=nGrOUp> gindex[N];
  iNt<lower=1,upPer=Nbev> binDeX[N];
  int<loWer=minResponsE,uPper=maxResPoNse> Score[N];
  // HYPotheticAl observations for new preDIctions
  Int<Lower=1> NNewpred;
  int<loWer=0> NNewSuBj;
  iNT<lower=0> NNewGrOup;
  inT<lOweR=0
 unconstrained, we use a l og transfo rm. Togofr om a  pro bability in (0 ,  1)to unconstrained, we u se alo g  odd s  t ransf orm.
If yo u r  ou tc om esar e  o rdina l v alues o n a 1-10 s cal e,  a common ap p ro ach that r esp ects that da tascaleis  or d inallog istic  regre s sion. 

A:

To  e x pand o n  @Bob C a r pe nter 's answer, here a r et wo ways you co uld ap pr o ac h the  pr oblem.  (I 'v e had cause t o  u s e bot h  of these rec ently and s t rug gled t oget them u p and  r u nni ng.  This m ay b e usefulto oth e r begin n ers lik e me.) 
Me tho d 1: Or de red  L o gis t ic  Re g res sion
We' re  g oingto a s s u m e th ateach  user  has a "true"  ex pect a ncy  foreachresp on se, w hich i s onan  arbitrary cont inuo us scale,  an dmod el  it a s  a lat ent  va riable.   If th e  us er ' s ac tual responses fal li n to  K categ ories, we  a l so model  K  -1 cu t p oints  bet w ee n thosecatego r ie s.   The p ro babili ty  th atthe u s er s elects  a given  resp o nse category i s  equal to the ar e a  u n derthe  logistic p df b e twee n th e  r ele v ant c utpoi nt s .The Stan model look slike t his. The main dif ference is t h at the m odel fi t s an additiona l ord ered vecto r  of cutp oints , and us es the or d e red_logi sti c d ist rib u t io n.  (I also c h a nged  t he prio rson thesig mas  to  Ca uc hy, to ke ep thempo si ti ve , a nd sw i tched to  n on- ce nte red p a ramete rizat ion.  Bu t  th ose cha n ge s arein de pend ent  o f the  que s tio n at ha nd.)  Edi t:A lsoad de d input s for new (hy po thetical)ob ser vation s about wh ich we want to make pre d ictions , a nd ad deda new gen era ted qu ant i ty for  those  pred ic tio n s .
dat a {  / /the real d a t a
 int< lo wer= 1> N;
 int<lower=1> nSub j ;
 int<lower=1>  nG roup ;  int < lo w er= 1> nBe v ; 
  int minRespo nse;
  int  m a xR esponse;
int <l ower=1, upper=n Subj> sIndex[ N];
  int <lower=1, up per= n G rou p> gIndex[ N];
  in t<lower=1 , upper = nB ev> b Ind ex[N]; 
 in t<low er=min R esp onse, upper= ma xRespo nse>sc ore[N];  // hypothetical obser vation s for  ne w predict ion s 
 int<lower =1>nNewPred;  i nt< lower =0> nNewS ubj;  int < lower =0>n NewGroup;  int < l ow er=0
 unconstrained,_we use_a log transform. _To go_from_a probability_in_(0, 1) to_unconstrained, we use_a log odds transform.
If_your outcomes are_ordinal_values on a 1-10 scale, a common approach that respects that data scale is_ordinal_logistic regression.__

A:

To_expand on @Bob Carpenter's answer,_here are two ways you_could approach_the problem.  (I've had cause to use_both_of these recently_and struggled to get them up and running. _This may be useful to other_beginners like me.)
Method_1:_Ordered_Logistic Regression
We're going to_assume that each user has a_"true" expectancy for each response, which_is on an arbitrary continuous scale, and_model it as a latent variable._ If the user's actual_responses fall_into K categories, we also_model K -_1 cutpoints_between those categories._ The probability that the user_selects a given_response category is equal to the_area_under the logistic_pdf_between_the relevant_cutpoints.

The Stan model_looks_like this.__The main difference is that the_model_fits an additional ordered vector of cutpoints,_and uses the ordered_logistic_distribution._ (I also changed_the priors on the sigmas_to Cauchy, to keep them positive,_and switched_to non-centered_parameterization.  But those changes are independent of the question at_hand.)  Edit: Also added inputs_for new (hypothetical) observations_about which_we_want to make_predictions,_and added_a new generated quantity for those predictions.
data_{
 _// the real data
  int<lower=1>_N;
  int<lower=1> nSubj;
__int<lower=1> nGroup;
  int<lower=1> nBev;
 _int minResponse;
  int maxResponse;
 _int<lower=1,upper=nSubj> sIndex[N];
  int<lower=1,upper=nGroup> gIndex[N];
__int<lower=1,upper=nBev>_bIndex[N];
  int<lower=minResponse,upper=maxResponse> score[N];
 _// hypothetical observations for new predictions
_ int<lower=1> nNewPred;
_ int<lower=0> nNewSubj;
  int<lower=0> nNewGroup;
 _int<lower=0