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Hidden Markov Models, HMM's, are mathematical models of Markov processes withstate that is hidden, but from which information can leak. They are typicallyrepresented as 3-way joint-probability distributions.  We use HMM's as denotations of probabilistic hidden-state sequentialprograms: for that, we recast them as `abstract' HMM's, computations in theGiry monad $\mathbb{D}$, and we equip them with a partial order of increasingsecurity. However to encode the monadic type with hiding over some state$\mathcal{X}$ we use $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ ratherthan the conventional $\mathcal{X}{\to}\mathbb{D}\mathcal{X}$ that suffices forMarkov models whose state is not hidden. We illustrate the$\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ construction with a smallHaskell prototype.  We then present uncertainty measures as a generalisation of the extantdiversity of probabilistic entropies, with characteristic analytic propertiesfor them, and show how the new entropies interact with the order of increasingsecurity. Furthermore, we give a `backwards' uncertainty-transformer semanticsfor HMM's that is dual to the `forwards' abstract HMM's - it is an analogue ofthe duality between forwards, relational semantics and backwards,predicate-transformer semantics for imperative programs with demonic choice.  Finally, we argue that, from this new denotational-semantic viewpoint, onecan see that the Dalenius desideratum for statistical databases is actually anissue in compositionality. We propose a means for taking it into account.

Abstract Hidden Markov Models: a monadic account of quantitative information flow