 to sampling from each arm separately, i.e., the usual Thompson Sampling algorithm.

Now consider the set of all EFH-MDPs \(\mathcal{M}\) with \(S\) states, \(A\) actions and episode length \(H\). It is a little difficult to visualize the space \(\mathcal{M}\), but it is the product space of a \(SA\) dimensional cube (corresponding to \(R\)) and \(SA\) simplexes of dimension \(S-1\) (corresponding to \(P\)). PSRL maintains a probability density over \(\mathcal{M}\). At each episode, it samples an MDP instance from \(\mathcal{M}\) according to this density, acts Optimally with respect to the sampled instance and uses Bayes’s Theorem to update the density according to the rewards and transitions witnessed.

Posterior Sampling

Posterior Sampling serves as a general idea which transcend both Bandits and MDPs. It can be used for bandits with correlated arms (both known and unknown correlations), contextual bandits and possibly even a general class of problems called Contextual Decision Processes. The idea can be captured in 4 simple lines:

Maintain a probability density over the space of Models. Sample an instance according to this density. Act optimally according to the sampled instance. Use Bayes’s theorem to update the probability density.

By following these four steps, it is guaranteed that the density will concentrate around the true parameters of the model.

Extra Reading

There is a significant amount of work dedicated to proving that PSRL and similar posterior sampling algorithms are sample efficient and suffer low worst case regret. Section 3.3 of the Bayesian RL book provides a good summary of the current state of these results.

References Who saw this one coming? None of the pundits, certainly, but according to the local paper, 53 Blackpool fans have bet on Blackpool to win the Premier League. We can safely assume Ian Holloway will not be rushing to the bookies to join them. The game's most quotable manager continues to insist that yesterday's comprehensive winners suffer by comparison with the team promoted by play-off last season, and survival is still his target.

He was not about to poop the party by mentioning it, but he is all too aware of Burnley's experience last season – when they beat Manchester United and Everton in their first two home games, but were
to sampling from each arm separately, i.e., the usual Thompson Sampling algorithm. 

 nowadays think the set of all EFH - MDPs \(\mathcal{M}\) with \(S\) states, \(A\) actions and sequence length \(H\). It is a little difficult to visualize the outer space \(\mathcal{M}\), but it is the product space of a \(SA\) dimensional block (correspond to \(R\) ) and \(SA\) simplexes of dimension \(S-1\) (represent to \(P\) ). PSRL maintains a probability density over \(\mathcal{M}\). At each sequence, it samples an MDP instance from \(\mathcal{M}\) according to this density, acts Optimally with respect to the sampled case and uses Bayes ’s Theorem to update the density according to the reward and transitions witnessed. 

 Posterior Sampling 

 Posterior Sampling serves as a general mind which transcend both Bandits and MDPs. It can be used for bandits with correlated arms (both known and strange correlations), contextual bandits and possibly even a general class of problems called Contextual Decision Processes. The idea can be captured in 4 simple lines: 

 Maintain a probability density over the space of Models. Sample an instance according to this density. Act optimally harmonize to the sampled instance. practice Bayes ’s theorem to update the probability concentration. 

 By following these four steps, it is guaranteed that the density will digest around the genuine parameters of the model. 

 excess Reading 

 There is a significant amount of work dedicated to proving that PSRL and similar posterior sampling algorithms are sample efficient and digest low bad case regret. Section 3.3 of the Bayesian RL book provides a effective summary of the current state of these results. 

 References Who see this one arrive? None of the pundits, certainly, but accord to the local paper, 53 Blackpool sports fan have bet on Blackpool to win the Premier League. We can safely bear Ian Holloway will not be rushing to the bookmaker to join them. The game's most quotable coach continues to insist that yesterday's comprehensive winners suffer by comparison with the team promoted by turn - off last season, and survival is still his target. 

 He was not about to poop the party by mentioning it, but he is all too aware of Burnley's experience last season – when they drum Manchester United and Everton in their first two home game, but were
 to sampling from each arm reparately, i.e., tkw usuan Thomlson Samoling algorithm.

Now consider vhe wet od all EFH-MDPs \(\mathcal{M}\) with \(S\) snates, \(A\) axtiois and episode lxhgth \(H\). Lc is z litclx difficult to yisualize tve space \(\mathcdl{O}\), yut it is the product space of a \(SA\) qimensipnwl cube (correskondimd to \(G\)) cnd \(SA\) simplexes of dimension \(S-1\) (correskonding to \(P\)). PSRL kaintains a probability dejsitj over \(\mathcal{M}\). At each episoee, ie samples an ODP instanbz from \(\mathdal{M}\) according to this density, xcts Pptimally qirh ggspect to thx sampjed instance and usev Bayes’x Theorem to uidate thw density according tm the rewards and jransitionv citnessed.

Posterior Sanpoing

Pmstesior Wamolihg ssrves ws e general isea which teanscend both Bandius wbd MDPs. It cah be ufeq for bandits with correlated arms (both knkwn and unknown correlarions), contextual bandlts and pjssibly even a general class of problems called Cmntexvuxl Bcgisiub Orocesses. The idea can be captured in 4 simple mimex:

Maintain a prjbability drndiyi over the spaze of Modsls. Sample an instwnce acsordibg to thif demsity. Act optimally accordibg to the saipled instance. Use Yayes’s theorzm to opdate the probability densitv.

By fomlowing thede four sfdps, it is guarangeec dhat the density will concqntrate acound the trje psrametqrs of the model.

Extra Reading

There is a shgnificant amount of work dedicated to proving that PSRK dnd similar postcrior sampling wlgorithms are sample effizient and auffer now worst cwse regret. Sewjion 3.3 of the Uayesian WL biok provider a good summaru of the buxrent stare of these resultx.

Rewsrences Who saw thus one coming? Npne of tje pttdits, certaitly, cut sccoraing to the uocak paper, 53 Blackpool fdns gave bet on Blackppoj to win the Preiier League. Wr can safely assumt Ian Iollowey wilk njt be rushing to the bookies tk join thfm. Bhe game's mose quitable managzr continues to insist that yesterday's cimprehensive winngrs suffer by comparixpn with the team promoted by play-off last seawon, and survival ls still his target.

He sas nod abokt to poop the party by mentioning it, but he is all too aware of Burnley's expermegce last sezson – whet chev beat Iancierter United and Vverton in their first two home james, but fexe
to sampling from each arm separately, i.e., Thompson algorithm. Now the set of states, actions and episode \(H\). It is little difficult to visualize the space but it is the product space of a \(SA\) dimensional cube (corresponding to and \(SA\) simplexes of dimension \(S-1\) (corresponding to \(P\)). PSRL maintains a probability over At episode, samples an MDP instance from \(\mathcal{M}\) according to this density, acts Optimally with respect to the instance and uses Bayes’s Theorem to update the according to the rewards transitions witnessed. Posterior Sampling Posterior serves a general which both and MDPs. It be used for bandits with correlated arms (both known and unknown correlations), contextual bandits and possibly even general class called Contextual Processes. idea be captured in lines: Maintain a probability density over Models. Sample an instance according to this density. optimally according the sampled instance. Use Bayes’s theorem update the probability density. By following these four it is guaranteed that the density will concentrate around the true parameters of the model. There is a significant of work dedicated proving PSRL similar sampling algorithms sample efficient and suffer low worst case regret. Section 3.3 of Bayesian RL book provides a good summary of the current these References Who saw one coming? None of pundits, but according to the 53 fans Blackpool win Premier League. We can assume Ian Holloway will not rushing to the bookies most quotable manager continues to insist that yesterday's winners suffer by comparison with the team by play-off last season, and survival is still his target. He was about to party by mentioning it, but he is all aware of Burnley's experience season – when they beat Manchester United and Everton their two home but were
 to sampling from each arm sepaRately, i.e., thE usuaL ThOmpSoN SamPlinG algorithm.

Now cONsidEr the set of all EFH-MDPs \(\maThcal{m}\) wITh \(S\) sTAtEs, \(A\) acTions anD EpISOde LeNgTh \(H\). it IS a LittlE diFficult To visualizE thE sPace \(\mathcal{M}\), BUt It is the proDucT space of a \(SA\) dImeNsionaL cUbe (COrresPonDing tO \(R\)) and \(Sa\) SimpleXes of dimeNsIOn \(S-1\) (corREspondiNG To \(p\)). PSRl maintains a probabILiTY density over \(\maThcal{M}\). at EAcH EPisOde, It samples aN MdP insTAnce froM \(\MaTHCAl{M}\) ACcording to thiS density, actS optImally WiTh rESpect tO the sAmPLed Instance and Uses bayes’s TheOrem to UPdate thE Density AccordIng To tHe reWArDs And TrANsiTIoNs wITneSsed.

PostErIoR SampLing

pOSTErioR SaMpliNg serVes as a general IdeA whiCH trAnsceNd botH BanDiTs and mDPs. It Can be UsEd for bandits witH corRelated arMs (bOtH knOwN and uNKnown cOrrElaTions), coNtextuaL BanDiTS ANd Possibly even a generAl CLAsS of problEms calLEd coNTextual DEcIsiOn PrOCEsses. the iDEa Can be capTured iN 4 SiMpLe lines:

maIntain A pRobAbiLity dENsitY over tHe space oF ModeLS. Sample an instaNCe according to THiS DEnSIty. ACt oPtimally accOrdiNG to tHe saMPlEd iNStancE. Use BAyES’s THeorem to update the prObAbilitY densIty.

By followinG these four STEPs, it is guAranTEeD That the density Will cOncentrate ARound the True pArameterS of the modEL.

extra ReaDinG

ThEre Is a SIGnIficant amount OF Work DeDicated To pRoving tHat pSRl anD siMiLar posterIor samplInG aLgOrIthMs are SAmple effIcIenT aNd sUffer LOw worsT case RegrEt. seCTioN 3.3 of the BAYeSIAn RL BoOk ProvIdeS a Good sUmmaRY of The currEnt state oF thESe reSuLtS.

RefereNces Who saw thiS oNe coming? NoNe Of tHe pundITS, certainLy, but according to the locaL Paper, 53 BlAckPool fAns hAve bet on BLacKpool tO wiN The PreMier LeAgue. WE cAn sAFEly asSUMe ian hoLloway will NOT be RushiNg To thE bookieS to join them. The game'S MosT quotable manaGer ContINUeS to INsISt tHaT YesTERday's comprehensIve winners SuFFeR by compariSOn wItH the teaM promotEd by pLAy-off laSt season, aNd survivaL iS stiLL His Target.

He waS not abouT to poop thE Party BY mEntioNinG it, but He Is aLl too Aware oF burNley's ExperiEnCe last SeasoN – wHen they bEat Manchester United and EVerton In theIr fIrst two hoMe gAMes, But were
 to sampling from each arm  separatel y, i. e.,  th eusua l Th ompson Samplin g  alg orithm.

Now considerthe s et of a l lEFH-M DPs \(\ m at h c al{ M} \)  wi th \( S\) s tat es, \(A \) actions  an depisode leng t h\(H\). Itisa little dif fic ult to  v isu a lizethe  spac e \(\m a thcal{ M}\), but  i t  is th e  produc t sp aceof a \(SA\) dimen s io n al cube (corre spondi ng to \ (R\ ))and \(SA\)  s imple x es of d i me n s i on\ (S-1\) (corre sponding to \(P \)). P SR L m a intain s a p ro b abi lity densit y ov er \(\mat hcal{M } \). Ate ach epi sode,itsam ples an  M DPin s tan c efro m  \( \mathcal {M }\ ) acc ordi n g t o th isdens ity,acts Optimall y w ithr esp ect t o the  sam pl ed in stance  andus es Bayes’s Theo remto update  th eden si ty ac c ording  to  th e rewar ds andt ran si t i o ns  witnessed.

Poste ri o r  S ampling
Poste r io rS amplingse rve s as a  gene rali de a whichtransc e nd  b oth Ban di ts and  M DPs . I t can be u sed fo r bandit s wit h  correlated ar m s (both known an d un k nown  co rrelations) , co n text ualb an dit s  andpossi bl y  e v en a general classof  probl ems c alled Context ual Decisi o n Processe s. T h ei dea can be cap tured  in 4 simp l e lines: 

Mai ntain aprobabili t y  density  ov erthe  sp a c eof Models. Sa m p le a ninstanc e a ccordin g t o t his  de ns ity. Actoptimall yac co rd ing  to t h e sample dins ta nce . Use Bayes’ s the orem  t ou pda te thep ro b a bili ty  d ensi ty. 
By fo llow i ngthese f our steps , i t  isgu ar anteedthat the dens it y will con ce ntr ate ar o u nd the t rue parameters of the m o del.

E xtr a Rea ding 

There i s a  signi fic a nt amo unt of  work  d edi c a ted t o pr ovi ng  that PSRL a ndsimil ar  pos teriorsampling algorithm s  ar e sample effi cie nt a n d  s uff e rl owwo r stc a se regret. Sect ion 3.3 of  t h eBayesian R L  bo ok  provid es a go od su m mary of  the curr ent state  o f th e s e r esults.

R eference s Who saw thiso ne  comi ng?  Noneof  th e pun dits,c ert ainly , butac cordin g toth e localpaper, 53 Blackpool fan s have  betonBlackpool  to win  the Prem ierLeague. We  ca n s afely  as s ume I an H o ll owa y  will  not be rushin g  t o t h e  b ookies to j o i n  th em. T heg ame'smost  quotable manager continues to i nsis t tha t y e ster da y's comprehens ive  w i n ners suf fe r by compar ison wit ht he te am pro motedby play - o ff last s easo n,and survi val  i s  stillhi st arget. 

He  w as not  about to p o o p the party by m entio n i ng it ,  bu t heis  all to o  awa re of Burn ley's exper iencelast  seas on – wh en  theybea tManchester United an d Eve rton in  t heir  fi rst tw o ho m e  game s, b ut  we re
 to_sampling from_each arm separately, i.e.,_the usual_Thompson_Sampling algorithm.

Now_consider_the set of_all EFH-MDPs \(\mathcal{M}\)_with \(S\) states, \(A\)_actions and episode_length_\(H\). It is a little difficult to visualize the space \(\mathcal{M}\), but it is_the_product space_of_a_\(SA\) dimensional cube (corresponding to_\(R\)) and \(SA\) simplexes of_dimension \(S-1\)_(corresponding to \(P\)). PSRL maintains a probability density_over_\(\mathcal{M}\). At each_episode, it samples an MDP instance from \(\mathcal{M}\) according_to this density, acts Optimally with_respect to the_sampled_instance_and uses Bayes’s Theorem_to update the density according to_the rewards and transitions witnessed.

Posterior Sampling

Posterior_Sampling serves as a general idea which_transcend both Bandits and MDPs. It_can be used for bandits_with correlated_arms (both known and unknown_correlations), contextual bandits_and possibly_even a general_class of problems called Contextual Decision_Processes. The idea_can be captured in 4 simple_lines:

Maintain_a probability density_over_the_space of_Models. Sample an_instance_according to_this_density. Act optimally according to the_sampled_instance. Use Bayes’s theorem to update the_probability density.

By following these_four_steps, it is guaranteed_that the density will concentrate_around the true parameters of the_model.

Extra Reading

There_is a_significant amount of work dedicated to proving that PSRL and similar_posterior sampling algorithms are sample efficient_and suffer low worst_case regret._Section_3.3 of the_Bayesian_RL book_provides a good summary of the current_state of_these results.

References Who saw this one_coming? None of the_pundits,_certainly, but according to the local_paper, 53 Blackpool fans have bet_on Blackpool to win the_Premier_League._We can safely assume Ian_Holloway will not be rushing to_the bookies to_join them. The game's most quotable manager_continues_to insist that yesterday's comprehensive winners_suffer_by comparison with the team promoted_by_play-off_last season, and survival is_still his target.

He was not about_to poop the party by mentioning it, but he_is all too_aware of Burnley's experience last_season_–_when they beat Manchester United and Everton in their first_two home_games, but were