Feedback May 11 - Githubissues

HMM Evidence Likelihood

[x] Given X = {..} -> Given observation sequence X =
[x] \theta = ... -> GIven HMM parameters \theta =
[x] Head(H) -> Heads (H)
[ ] Rewrite as: Before we compute this likelihood, let us work out a simpler problem. What is the probability of sequence X = {HHH} given that the state sequence is {z1= B}. Then, say that the following Trellis diagram shows this state sequence. Then, write (color coded). We can calculate this as the product of prior probability of observing B, emission probability of heads given state B at time 1, a transition from B to B between time 1 and 2, a ..... Then, mention we have thus far considered one possible path for observing HHH. But, there are in total 8 such paths shown below..
[x] For all animation, do not autostart the animation. Best left to the user to press play.
[x] Hence the time complexity of .. -> Hence, the time complexity...also choose some suitable values of K and T and show this scales poorly in comparison to say number of atoms in the universe.
[x] Can we do better? Consider the following paths? Which paths? I think this needs to be removed.
[x] Directly write - let us see if we can efficiently compute .. and then say that for the coin example, we can write L(X|\theta) as ..
[x] The above is the tree -> The above tree does not calcuklate the same node multiple times.
[x] So, we need to store -> From the optimized tree, we can see that we can efficiently compute.. if we can store P(X_{1:t}..)
[x] Let us not call Forward Algorithm Problem 2. It is a solution to the problem. Either we have algorithms in this section or problems, but, not mix both. Best would be to have subsections on: Forward algorithm where we start the discussion on finding the likelihood evidence, then on backward algorithm, then Viterbi, then MLE for observed case, then EM for unobserved case
[x] Aftet mentioning that this is what forward algorithm does, direclty state that we previously discussed that we need to compute P(X_{1:t}|..), just sya we denote this quantiry by alpha_t.
[x] in the definition of alpha_t say being in state i at time t given the observation sequence till time t.
[x] There is little gap after alpha_t and alpha_t+1 caption and the following text
[x] The collapsible in Apoorv's bayesian article have a + towards the right. can you add that as well?
[x] In the collapsible on Forward algorithm it would be good to work out 2 examples. Fix the \theta and show the calculation of two sequences. You'd need to show alpha_1(B), alpha_2(B) and so on..
[x] Time complexity of forward?

Backward algorithm

[x] Start backward algorithm by stating that you want to compute the probability of future state sequence given that you are in a particular state currently. Say, we will see that this is required later for HMM parameter learning. This entire discussion could be 3-4 lines. Currently, it is being repeated several times.
[x] Baum Welch or elsh?
[x] No need ot give the rainy sunny example.
[x] No need to show the equation involving alpha and beta at this stage.
[x] When you start writing Backward Algorithm, see if repetitions can be avoided. In my first comment in this section, I suggest upfront mentioning what you are trying to calculate and then you say that similar to forward algorithm, you just need to compute beta..
[x] You have written of of
[x] Use the same structure for all algorithms in terms of problem, example, algorithm, collapsible, example..In the beta collapsible, like forward collapsible give a concrete worked out example.

Optinal sequence of hidden states

[x] you are using capital Z here!?
[x] I think we can give the intuition better in this section. First, I would suggest you draw a Figure (say in latex) or table which shows the 8 paths (z1=..) and show that we can compute these K^T paths and find that for X = {hhh}, the most optimal sequence is z1=B, z2=B..But, this is exponential..could we do better like we did in forward algorithm? Yes, we can! the idea, like in the case of forward algorithm is to use DP and reuse computations.

The key idea is to store Best score (highest prob) along a single path at time , which accounts for the first observations and ends in .. We can compute the same quantity for the next timestamp by considering all K transitions from each of the states. Then define the quantity delta_t and colour code the above text..

Trellis.pdf

[x] See the attachment pdf to build the story for motivating Viterbi. you can reuse many of your existing graphics..just use the intuition I have presented in drawing and rleate your diagrams and any maths in the equations accordingly.

Parameter learning

[x] Use either i or j for indexing example, both both?!
[x] Clarify that the sequences can be of different lenghts
[x] Before wrigint the formula for \Pi_k etc, just mention that we use the principle of MLE to estimate the params..
[x] Missing full stop in Calculating A and Phi
[x] We know the procedure to calculate optimal parameters, when we are provided with hidden state sequences. So, if we are not provided the hidden state sequences, we try to "estimate" the "expected" state sequence. Now, we have the "expected hidden state sequences", so we can find parameters.

Replace the above with something like:

We have seen the procedure to calculate the optimal parameters given the hidden state sequence. However, it is common that the hidden state sequence is unknown. In such a case, we first try to "estimate" the "expected" state sequence based on some initial estimates of parameters. Then, we use the principles of MLE for observed state sequence to refine the parameters. We apply these two steps iteratively via an algorithm called Expectation maximization..

[x] Expectation step: what is ESS? Better to say that we can compute the expected state sequence. But, from our earlier MLE computations, we know that we only care about the number of transitions from i to j and number of emissions from state i..thus, we compute e_t..which is the ESS..
[ ] while defining e_t you forgot to mention the observed seqwuence.. and you have not yet colour coded.
- [ ] For the EM worked out example, you could mention you have taken some samples from HMM with given parameters. You would like to see if you can recover the parameters from the observations without looking at the state sequence..you could use some good Python etc. implementation and store images for these calculation instead of computing on your own in JS. essentially you could generate the TEX code with filled out numbers from Python..

nipunbatra / hmm

Feedback May 11 #6

HMM Evidence Likelihood

Backward algorithm

Optinal sequence of hidden states

Parameter learning