Definition of a probability in forward-backward method for HMM I am confused about the way that the Jurafsky and Martin book (Appendix A, page 6) explains the relationship between the observations and hidden states:

Each cell of the forward algorithm trellis $α_t(j)$ represents the
probability of being in state $j$ after seeing the first $t$
observations, given the automaton $λ$

Then they define $\alpha_t(j)$ as $P(o_1,o_2,\dots,o_t,q_t = j|λ)$.
I am not sure why the quoted text refers to the joint probability between the observations and the state at time $t$. I'd rather define it as $\alpha t(j) = P(q_t = j|o_1,o_2,\dots,o_t,λ)$. I think the conditional distribution makes more sense to use here, as we first see the first $t$ observations then we find the probability of being in state $j$.
A separate question: Although I am not convinced that the joint distribution is what we should use here, I think they marginalized over all previous states:
\begin{align}
\alpha t(j) &= \sum_{q_1,\dots,q_{t-1}}P(q_1, q_2,\dots,q_t=j,o_1,o_2,\dots,o_t|λ) \\
&=P(o_1,o_2,\dots,o_t,q_t = j|λ)
\end{align}
Is this correct?

One reason why it is more intuitive to define $α_t(j)$ as conditional on just $\lambda$ and not also the observations $o_t$ is that ultimately we are building up to finding the model parameterization $\lambda$ which gives the highest probability of generating the observations.
See for example the final equation on page 7 which lays out some pseudocode for the algorithm. The quantity we are interested in at the end is $P(O|\lambda)$.
