The thing that settles down is the probability distribution（P(v,h)） over configurations（给定可见层和隐层单元的一个指派）. 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.(当然包括权重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(全体), 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，那试着看看下面的说法能否理解：
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.
第一个概念很容易理解，第二个是什么意思呢，通过从stationary distribution抽样得到的visible vector和我们的 training data visible vector用于学习BM网络效果是一样的，也就是最后得到的网络是一致的
下面让我们看一个有趣的事
权重更新有两项，第一项固定V可见单元达到热平衡状态，第二项不固定v达到热平衡状态，为什么要减去第二项呢，因为第二项与V无关，即使没有V,si和sj同时为1的期望，不减掉的话就多学习了
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.