EPBP claims etc

[tlienart]

Should be careful with:

• if the mixture components are normalised, then this is correct but it's not SAPBP as in Briers and co, in fact it's just sampling from a mixture
• if the mixture components are NOT normalised, then this is indeed SAPBP (ish) but it's NOT proper sampling

should reflect a bit more, connect with the discussion for smoothing (since it's effectively exactly the same thing).

[tlienart]

The particular claim to investigate is:

the algorithm still provides consistent estimators if we consider the evaluation of unbiased estimators of the messages instead

This may be correct in that we're using approximate belief as a proposals. We don't actually care if the proposals are consistent estimators of the beliefs. But the sentence is misleading. We don't actually care about the unbiasedness of the estimators of the messages. Additionally we do not have unbiased estimators of the messages.

So this sentence may be best left aside.

[tlienart]

The way it's now in the thesis should be ok.

[tlienart]

Interesting to note that in Ihler