  Hidden Markov Models, HMM's, are mathematical models of Markov processes with
state that is hidden, but from which information can leak. They are typically
represented as 3-way joint-probability distributions.
  We use HMM's as denotations of probabilistic hidden-state sequential
programs: for that, we recast them as `abstract' HMM's, computations in the
Giry monad $\mathbb{D}$, and we equip them with a partial order of increasing
security. However to encode the monadic type with hiding over some state
$\mathcal{X}$ we use $\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ rather
than the conventional $\mathcal{X}{\to}\mathbb{D}\mathcal{X}$ that suffices for
Markov models whose state is not hidden. We illustrate the
$\mathbb{D}\mathcal{X}\to \mathbb{D}^2\mathcal{X}$ construction with a small
Haskell prototype.
  We then present uncertainty measures as a generalisation of the extant
diversity of probabilistic entropies, with characteristic analytic properties
for them, and show how the new entropies interact with the order of increasing
security. Furthermore, we give a `backwards' uncertainty-transformer semantics
for HMM's that is dual to the `forwards' abstract HMM's - it is an analogue of
the 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, one
can see that the Dalenius desideratum for statistical databases is actually an
issue in compositionality. We propose a means for taking it into account.
