Examples of hidden Markov models problems? I read quite a bit of hidden Markov models and was able to code a pretty basic version of it myself.
But there are two main ways I seem to learn. One is to read and implement it into code (which is done) and the second is to understand how it applies under different situations (so I can better understand how it relates to problems I might be working on). All the examples I have done so far have involved either some kind of DNA prediction or coin tossing.
I'm wondering if there are any resources to get other Markov problems (language doesn't matter but hopefully with the answers as well so I can know if I'm right or wrong)?
 A: I've used HMM in a demand / inventory level estimation scenario, where we had goods being purchased from many stores that might or might not be out of inventory of the goods.  The sequence of daily demands for these items thus contained zeroes that were legitimate zero demand days and also zeroes that were because the store was out of stock.  You would think you'd know whether the store was out of stock from the inventory level, but errors in inventory records propagate and it is not at all uncommon to find a store that thinks it has a positive number of items on hand, but actually has none; the hidden state is, more or less, whether the store actually has any inventory, and the signal is the (daily demand, nominal inventory level).  No references for this work, though; we were not supposed to publish the results for competitive reasons.
Edit:  I'll add that this is especially important because, with zero demands, the store's nominal on hand inventory doesn't ever decrease and cross an order point, triggering an order for more inventory - therefore, a zero on hand state due to erroneous inventory records doesn't get fixed for a long time, until somebody notices something is wrong or a cycle count occurs, which may be many months after the problem has started.
A: I pretty much experienced the same thing and didn't find much beyond the weather. Areas that come to mind include: speech recognition, change point detection, tagging parts of speech in text, aligning overlapping items/text, and recognizing sign language.
One example I found and did some exploration of was in Section 8 of this introduction, which is one of the references for HMM's in Wikipedia. (It's actually pretty fun: your analysis discovers that there are vowels and consonants.) This also introduces you to working with a text corpus, which is useful.
(If you want to play with generation with HMMs, you could train on Shakespeare text and then generate faux-Shakespeare.)
A: Most speech recognition software uses Hidden Markov Models. You can experiment with natural language processing if you want to get a feel for HMM applications.
Here's a good source: Probabilistic Graphical Models, by Koller and Friedman.
A: Hidden markov models are very useful in monitoring HIV.  HIV enters the blood stream and looks for the immune response cells.  It then sits on the protein content of the cell and gets into the core of the cell and changes the DNA content of the cell and starts proliferation of virions until it burst out of the cells.  All these stages are unobservable and called latent.  An ideal example for hidden markovian modelling. 
A: For me, very nice application of HMM is chord identification in musical composition. See for example this lecture. 
A: Markov models may be useful in analyzing the interactions of a user with a website - For Example on Amazon.com where figuring out what series of interactions lead to a checkout to give recommendations in the future.
A fun example showing the use of Markov Model is the following-
http://freakonometrics.blog.free.fr/index.php?post/2011/12/20/Basic-on-Markov-Chain-(for-parents)
