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This is a brief technical note to clarify some of the issues with applyingthe application of the algorithm posterior sampling for reinforcement learning(PSRL) in environments without fixed episodes. In particular, this paper aimsto:  - Review some of results which have been proven for finite horizon MDPs(Osband et al 2013, 2014a, 2014b, 2016) and also for MDPs with finite ergodicstructure (Gopalan et al 2014).  - Review similar results for optimistic algorithms in infinite horizonproblems (Jaksch et al 2010, Bartlett and Tewari 2009, Abbasi-Yadkori andSzepesvari 2011), with particular attention to the dynamic episode growth.  - Highlight the delicate technical issue which has led to a fault in theproof of the lazy-PSRL algorithm (Abbasi-Yadkori and Szepesvari 2015). Wepresent an explicit counterexample to this style of argument. Therefore, wesuggest that the Theorem 2 in (Abbasi-Yadkori and Szepesvari 2015) be insteadconsidered a conjecture, as it has no rigorous proof.  - Present pragmatic approaches to apply PSRL in infinite horizon problems. Weconjecture that, under some additional assumptions, it will be possible toobtain bounds $O( \sqrt{T} )$ even without episodic reset.  We hope that this note serves to clarify existing results in the field ofreinforcement learning and provides interesting motivation for future work.

Posterior Sampling for Reinforcement Learning Without Episodes