I'm only familiar with online techniques used with Bayesian HMMs such as particle filtering, of which there are many variations. But I'm sure there are non-Bayesian methods as well.

I don't know Mathematica so I can't follow how you computed your expected number of heads. Care to explain that part? If we assume knowledge that coin B's bias is drawn from a uniform distribution on [0,1], then I don't see how you can expect to beat 50/50.

It obviously depends a lot on what kind of random sequence it is and the nature of the alteration. What random sequence do you have in mind?

Yes i was referring to all the NN research which has been cleverly rebranded as 'deep belief networks,' @mbq. I agree with your criticisms, but it's hard to argue with the state of the art performance they've achieved recently in one task after the next, like computer vision, speech recognition, and a recent Kaggle competition. Not to mention google now uses "deep nets" for their speech recognition. They're still opaque and require an element of black magic to get them working, but it's clear that they work.

Not positive I understand correctly but perhaps if you instead used the logarithm of your current scale it would look better. Do you have a picture you could show?

(1) Acquiring more knowledge and experience is only going to improve your chances of getting into a PhD program. That said, a finance gig is not gonna be your most efficient path to a PhD program. The most efficient way into a PhD program is to demonstrate an ability to do original research, by publishing papers or getting great recc letters from advisors . (2) Again it could only help but unless your finance job is to do original research then I wouldn't expect any dramatic benefit in this regard, so I wouldn't give this factor much weight in your decision process, if i were you.

L2 regularization has a Bayesian interpretation under certain assumptions. Is that what you mean by "in the Bayesian setting"?

This had some truth when you said it, @mbq , but not anymore.

What do you mean by "new observations"?

Well without more information on your data and model specification it's hard to give advice. It usually requires a fair amount of experimentation to get samplers working in large models like this. The obvious adjustments, if you haven't already, would be your priors. Also, might be worth experimenting with the variance of that metropolis proposal.

It's not that the hierarchical model is hopeless; rather it appears that your sampling scheme needs some tweaking. Have you tried different proposal distributions for the metropolis step? What is the acceptance rate for these proposals? It should be around 15-30%. If it's not then it needs tuning. Also, i'd be very surprised if you needed anything like 500k iterations to determine if it's converging.

The conjugate prior to the Poisson distribution is the Gamma distribution. But the distribution you have written is not exactly a Poisson distribution since your outcome space is $[0,1]$ rather than all positive integers. Perhaps someone whose calculus is better than mine can clarify how that changes things.

If the package is able to fit an HMM then has to be computing the probability of sequences given a model, so it's got to be in there somewhere. And the Viterbi algorithm is not what you want; that computes the most likely state path. All you need is the probability of a sequence given a model, which only requires some multiplication and summation.

I don't see how a chain could be both Markovian and adaptive.

It's not obvious to me how latent Dirichlet allocation (topic modeling) could be applied here. The OPs data are continuous signals not categorical variables, so it's not clear what the "words" or the "documents" would be.