Inflation expectations. Whose expectations?

#economics #braindump Reading Simon Wren-Lewis made me want to look at econometric studies of Phillips curve models. Now, IANAE, so I might be misinterpreting a lot of things here, but anyway, as far as I can understand, one interesting question in macroeconimics is inflation and inflation expectations.

Using the notation from a paper by Nason and Smith here and whatever info I could dig out. If we look at a discrete-time model with inflation rate $\pi_t$, the basic equation is \[ \pi_t = \gamma_f \E_t \pi_{t+1} + \lambda x_t, \] where $E_t$ is an expectation operator; I choose to interpret the underlying probability space as that of beliefs over states of the world. Meanwhile $\gamma_f, \lambda$ are scalars and $x_t$ is "marginal costs" which I interpret as some external variable.

One interesting thing in that paper is that it explicitly mentions the alternative model \[ \pi_t = \gamma_b \pi_{t-1} \gamma_f \E_t \pi_{t+1} + \lambda x_t. \] Now I don't see how this helps things, because one would expect the expectations to be dependent on $\pi_{t-1}$ in any case. And why are we only looking at $\pi_{t-1}$ and not $\pi_t$ or maybe $\pi_{t-k}$? The rationale seems to be that a very simple (apparently, didn't look at the details) price-setting model ends up having the above form.

The thing that is strange about all this is that the $\E_t$ operator is not clearly defined yet. At the bottom of page 366 the authors talk about iterated expectations. This works out symbolically, but... let's say our information state at time $t$ is $z_t$. Then we have the probability distribution over the states of the world $P(\omega \mid z_t)$, with each world state $\omega \in \Omega$ corresponding to a particular $\pi_t(\omega0)$ sequence and the corresponding expectation \[ \E_P(\pi_{t+1} \mid z_t) = \int_{\Omega} \pi_{t+1}(\omega) dP(\omega \mid z_t). \] But that is not necessarily the same expectation as that of another. Say we have a probability measure $Q$ over the possible beliefs $B \in \cal B$ the other forecaster has, conditioned on the evidence. Then \[ \E_Q(\pi_{t+1} \mid z_t) = \int_{\cal B} \int_{\Omega} \pi_{t+1}(\omega) dB(\omega) dQ(B \mid z_t). \] So I don't see how it follows that "our effort to predict what someone with better information will forecast simply gives us our own best forecast". For example it might be that the evidence tells us that the other forecaster's belief is that the next period inflation is going to be equal to the current period inflation. So generally $\E_P \neq \E_Q$. In addition, even if the expectations are equal, the distribution of possible future inflation may not be. I am either missing something obvious, or is this rather sloppy? What kind of statistically meaningful conclusions can one obtain in this way, if any?

Any enlightening comments?