[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81727-en":3,"doc-seo-81727-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81727,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Bayesian Updates from Coalgebraic Determinisation","Bayesian updates from coalgebraic determinisation connects classical determinisation of nondeterministic automata with a coalgebraic framework for structured coalgebras. For stochastic Moore machines over the distribution monad, determinisation yields semantics that maps each finite input word to a distribution over current outputs, suitable only when intermediate observations are irrelevant. When agents must condition on all realized observations, the paper shows unifilarisation corresponds to coalgebraic determinisation, producing causal stochastic behaviours consistent with observation histories.","Bayesian updates from coalgebraic determinisation  \nManuel Baltieri Araya Inc., Japan*  \n[manuel_baltieri@araya.org](manuel_baltieri@araya.org)  \nNathaniel Virgo University of Hertfordshire, UK*  \n[n.virgo@herts.ac.uk](n.virgo@herts.ac.uk)  \narXiv :2607 .00034v1 [ cs .LO] 24 Jun 2026  \nThe powerset construction is the classical determinisation procedure for nondeterministic finite automata. In the coalgebraic setting, this construction has been generalised to structured coalgebras, which are coalgebras equipped with extra data. For stochastic Moore machines over the distribution monad, a type of structured coalgebra, the determinisation construction induces a semantics assigning to each finite input word a distribution on the current output. This semantics is appropriate when only the current output matters, but it is too coarse for settings in which intermediate observations must also be taken into account, as is typical for agents solving POMDPs in control theory and reinforcement learning. In these contexts, agents need to condition on all realised observations, not just the final one, so to better plan for the future. This has been addressed from a category theoretic perspective through a procedure called unifilarisation, which (in our context) takes a stochastic Mealy machine and produces a machine whose states are priors over the original state space and whose transitions are given by Bayesian filtering. Here we show that unifilarisation is an instance of coalgebraic determinisation.  \nWe work with Mealy machines over monads equipped with extra structure generalising the notion of the support of a distribution. We show that in this setting, unifilarisation arises from the general determinisation procedure. We then compare the resulting final coalgebra semantics with the Moorestyle one. Instead of assigning only a distribution on current outputs to each finite input word, it yields causal stochastic behaviours, that is, families mapping input words to distributions on output words compatible with the “causality” constraint that outputs cannot depend on future inputs.  \n1 Introduction  \nDeterminisation is classically introduced in automata theory as the move from nondeterministic finite automata to deterministic finite automata via the powerset (or subset) construction [RS59]: given anondeterministic finite automata with state space S and input space I, the determinised machine has state space P fin(S), with a transition function mapping a subset of states S′ ∈ P fin(S) and an input symbol i ∈ Ito the set of all i-successors of states in S′, i.e. all the states that can be reached from S′ after consuming an input symbol i. This construction yields an equivalent deterministic automaton that accepts the same language of finite words.  \nThis construction has previously been generalised to a coalgebraic setup in [SBBR10, SBBR13, JSS15, BSS21] . Specifically, the construction in [SBBR10, SBBR13] starts with a structured coalgebra, which is a coalgebra (S, f : S → FT(S)) of FT, where T is a monad, together with some extra data. The monad T is used to model nondeterminism or stochasticity, with the precise type of nondeterminism (or “branching”) determined by the monad. The determinisation procedure then turns a structured coalgebra into an algebra 􀀀T (S), f ♯ : T(S) → FT(S)􀀁, which is a coalgebra of F (and can therefore be seen as deterministic), whose carrier is T (S) . The traditional powerset construction is a special case of this framework, where the transition type is that of a Moore machine of a specific kind and the branching is given by the finite powerset monad T = P fin. For this monad, determinisation groups states (by union) that can be  \n*This work was funded by the Advanced Research + Invention Agency (ARIA) through project code MSAI-SE01-P011 .  \n© M. Baltieri & N. Virgo  \nThis work is licensed under the Creative Commons Attribution License.  \n2 1 INTRODUCTION  \nreached by consuming the same input. Its extension t","cbCainwyX86VZ414","https://ap.wps.com/l/cbCainwyX86VZ414","pdf",350880,5,1,15,"English","en",105,"# Introduction\n## Classical determinisation and powerset construction\n## Coalgebraic generalisation with monads\n## Semantics from determinised coalgebras\n## Unifilarisation and Bayesian belief updates","[{\"question\":\"What does coalgebraic determinisation provide for stochastic Moore machines?\",\"answer\":\"It induces a semantics that maps each finite input word to a distribution over the current output, obtained by propagating an initial distribution through transitions.\"},{\"question\":\"Why is the usual Moore-style semantics too coarse in POMDP-like settings?\",\"answer\":\"It records only the distribution of the current output after a finite input history, but agents also need how the belief over states updates for each realized intermediate observation.\"},{\"question\":\"How does unifilarisation relate to coalgebraic determinisation in the paper?\",\"answer\":\"Unifilarisation is shown to be an instance of coalgebraic determinisation: it constructs a machine whose states are priors over the original state space, with transitions given by Bayesian filtering, yielding causal stochastic behaviours.\"}]",1784175668,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"bayesian-updates-from-coalgebraic-determinisation","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/bayesian-updates-from-coalgebraic-determinisation/81727/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does coalgebraic determinisation provide for stochastic Moore machines?","Question",{"text":76,"@type":77},"It induces a semantics that maps each finite input word to a distribution over the current output, obtained by propagating an initial distribution through transitions.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is the usual Moore-style semantics too coarse in POMDP-like settings?",{"text":81,"@type":77},"It records only the distribution of the current output after a finite input history, but agents also need how the belief over states updates for each realized intermediate observation.",{"name":83,"@type":74,"acceptedAnswer":84},"How does unifilarisation relate to coalgebraic determinisation in the paper?",{"text":85,"@type":77},"Unifilarisation is shown to be an instance of coalgebraic determinisation: it constructs a machine whose states are priors over the original state space, with transitions given by Bayesian filtering, yielding causal stochastic 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