So today, we will go on with the uncertainty analysis, which

we have started yesterday.

And so the basic idea was the reason for the uncertainty

is coming from the lack of possibilities

to find a unique reconstruction of the underlying hidden

variables.

And then I have claimed in the next slide

here that it's nice to have an estimation of model uncertainty,

but everybody at the end is interested in forecast

uncertainty.

And my claim yesterday was that the forecast uncertainty

is equal to the model uncertainty

if you speak about large-scale neural networks.

With the augmentation, in principle,

because of the universal approximation argument,

whatever is the complexity of the world,

large-scale neural network is able to reproduce it,

at least in a finite time horizon.

And then the second point is, if you

study such distributions here, you

will find that it contains information which

are standalone and not directly dependent on the model itself.

We have seen that if the models are large,

the distribution becomes independent of the model.

And if the ensemble is large, the details,

which means the moments, first moment, second moment,

third moment, fourth moment, become

more and more independent of the size of the ensemble.

And for this, I do not have a formal proof,

but I have shown you a lot of experiments.

For the first and the second moment here,

my statements are practically perfect.

If you go to the higher moments here,

then you see first and second moment are fine with the third.

And the fourth moment here, you really

need large ensembles to have this stability there.

Same thing with the meta parameters,

with the special design of the neural networks.

If they become larger and larger,

then the whole story is more and more

going here in the direction of a stable description

for the higher moments.

But we have discussed this experience

with the nature versus nature in a small room, which

means a whole year.

And it never worked in such a good way

that in a small hall, you can keep

the balance between all the variables there,

which is in a large nature, no topic.

And so therefore, in the future, this