Restricted Boltzman Monte Carlo Markov Chains

Question 1

Restricted Boltzman Machines

Answer 1

Finding the inference or probabilistic inference is realy hard, therefore in RBM we try to find the joint probability in order to estimate the prob. inference.
P(v|h) p(h|v)
Given training data we need to maximize
p(x:model)
easier to maximize log ligelihood
derivative of teta logp(x:model) = derivative of p unnormalized + derivative of teta logZ(model)

It is almost impossible to compute logZ(model)

Question 2

Monte Carlo Methods

Answer 2

The main problem of the probabilistic models is to compute the Derivative wrt problem logZ(model)
It is hard to compute it but we can make an estimation to it using monte carlo methods.
The expected value for the finite n is equal to the correct value
E[s] = 1/n TOPLAM E[f(xi)} = 1/n TOPLAM s = s
The value for any spesific n wil cause some random erorr but the other values will cancel it out
IMPORTANCE SAMPLING:
p(x)f(x) = q(x) p(x)f(x) / q(x)
Q is our new distribution P which will help to draw samples from
The ratio means that it willl be evaluated at eachsample.

q*(x) = p(x) f(x) / Z MINIMUM VARIANCA OCCURS WHEN Q IS OPTİMAL

Question 3

Markov Chains

Answer 3

So far all the things we have done was used on DIRECTED MODELS.
In order to make a SAMPLING from undericet models it is harder.
Markov Chain: Use a monte carlo algotihm, that incrementally updates samples, comes closer to sampling from the right distribution each step.

A markov Chain is:
random state x
transition distribution T(x’ |x): probabilty that a random state x’ to go x
Running the Markov Chain means that repeadetly update state x’ to x

Largest eigen value is 1. converges to stationary distribution

Question 4

Simple Markov Chains (Gibbs Sampling)

Answer 4

Repeadetly cycle from all variables:
For each variable randomly sample that variable given its Markov Blanket
For an undirected graph, the Markov blanket is just the neighbours.
Block Gibbs Trick: conditionally indepentedt variables may be sampled simultaneously

Question 5

RBM Based Deep Models

Answer 5

Deep Bolztman machine, in the first layer we have visible units connected densely to the hidden units. it is called as phase1
In phase two, different hidden units are conected to each other
In phase three to n: freeze the weights layer l-1 and train RBM
It ca also be used as the pre training for deep belief networks

Question 6

Deep Belief Networks

Answer 6

Mixed model, top is undirected and the rest is directed.
Binarry hidden units.
between directed and undirected parts gibbs sampling or anchestore sammpling is done. The direction is from the hidden to the visiblle units