The thing that settles down is the probability distribution（P(v,h)） over configurations（給定可見層和隱層單元的一個指派）. That's a difficult concept the first time you meet it, and so I'm going to giveyou an example. The probability distribution settles to a particular distribution called the Stationary Distribution. The stationary distribution is determined by the energy function of the system.(當然包括權重W了,且W是固定的) And, in fact,in the stationary distribution, the probability of any configuration is proportional to each of the minus its energy. A nice intuitive way to think about thermal equilibrium is to imagine a huge ensemble ofidentical systems that all have exactly the same energy function. So, imagine a very large number of stochastic Hopfield nets all with the same weights. Now, in that huge ensemble(全體), we can define the probability of configuration as the fraction of the systems that are in that configuration.
So, now we canunderstand what's happening as we approach thermal equilibrium.
We can start with any distribution we like
over all these identical systems. We could
make them all,be in the same
configuration.So, that's the distribution
with a property of one on one
configuration,and zero on everything
else. Or we could start them off, with an
equal number ofsystems in each possible
configuration.
So that's auniform distribution.
And then, we're going to keep applying our
stochastic update rule.
Which, in the case of a stochastic
Hopfield net would mean,
You pick a unit, and you look at its
energy gap.
And you make a random decision based on
that energy gap about whether to turn it
on orturn it off.
Then, you go and pick another unit, and so
on.
We keep applying that stochastic rule.
And after we've run systems stochastically
in this way,
We may eventually reach a situation where
the fraction of the systems in each
configuration remains constant.
We apply the update rule,
And the states of its units will keep
flipping between zero and one.
But, the fraction of systems in any
particular configuration doesn't change.
And that's because we have many, many more
systems than we have configurations.
In general, we're interested in reaching
equilibrium for systems where some
configurations have lower energy than
others.
如果理解了上面的概念，那試着看看下面的說法能否理解：
We want to maximize the product of the probabilities that the Boltzmann machine assigns to the binary vectors in the training set.
– This is equivalent to maximizing the sum of the
log probabilities that the Boltzmann machine
assigns to the training vectors.
It is also equivalent to maximizing the probability that we would obtain exactly the N training cases if we did the following
– Let the network settle to its stationary distribution N
different times with no external input.
– Sample the visible vector once each time.
第一個概念很容易理解，第二個是什麼意思呢，通過從stationary distribution抽樣得到的visible vector和我們的 training data visible vector用於學習BM網絡效果是一樣的，也就是最後得到的網絡是一致的
下面讓我們看一個有趣的事
權重更新有兩項，第一項固定V可見單元達到熱平衡狀態，第二項不固定v達到熱平衡狀態，爲什麼要減去第二項呢，因爲第二項與V無關，即使沒有V,si和sj同時爲1的期望，不減掉的話就多學習了
An inefficient way to collect the statistics required for learning
Positive phase: Clamp a data
vector on the visible units and set
the hidden units to random
binary states.
– Update the hidden units one
at a time until the network
reaches thermal equilibrium
at a temperature of 1.
– Sample for every
connected pair of units.
– Repeat for all data vectors in
the training set and average
Negative phase: Set all the
units to random binary states.
– Update all the units one at
a time until the network
reaches thermal
equilibrium at a
temperature of 1.
– Sample for every
connected pair of units.
– Repeat many times (how
many?) and average to get
good estimates.