Introduce `final_states_cont` and `root_states_cont` about cellrank HOT 6 CLOSED

I'm not sure if i follow you correctly for probs in GPCCA - you want so sum up the lineages
for the main states except for rest, or do you want to do this only for some restricted cells?
If you want them sum up, I don't think it will work, see below (Alpha, Beta, Epsilon):

Max seems to work better:

And for Alpha and Epsilon only:

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Note that I haven't normalized so that it sums to 1, but with sum over the lineages, the picture is almost the same.
Maybe more robust way instead of max could be var - if a cell is in a metastable state, the variance of probs should be big (not uniform). Here's how it would look (now normalized to sum to 1):

If I did misunderstand, please correct me.

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Thanks Mike, you are right, simply summing them up is not a good idea, as you show above.

What do you mean by max? Can you post your code?

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@VolkerBergen @michalk8 should we go for root_states_probs and final_states_probs, or simple root_probs and final_probs? Simpler and shorter, no?

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Just for reference, I've plotted how P(A u B u C) = P(A) + P(B) + P(C) - P(A & B) - ... would look like (in code using the max as discussed). Code for reference:

from itertools import combinations

x = adata.obsm['to_final_states']
x = x['Alpha', 'Beta', 'Epsilon']

current = x.sum(1)
for c in combinations(range(x.shape[1]), 2):
    val = np.prod(x[:, c], axis=1)
    current -= val
current += np.prod(x, axis=1)

adata.obs['P(A u B u C)'] = current.X.squeeze()
adata.obs['diff'] = adata.obs['P(A u B u C)'] - adata.obs['final_states_probs']
scv.pl.scatter(adata, color=['final_states_probs', 'P(A u B u C)', 'diff'])

I'd keep it root_state_probs - it's a bit more descriptive and not much longer (+ we have to_final_states and final_states).

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Just for clarification: you compare P(XOR(A, B, C)) with using the max? Or how it final_states_probs computed above?

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