I can't believe I can't find any information on this, but do variables in Bayesian Networks have to be boolean? Every example I've found in my textbook or online uses T/F variables, but how do I represent a variable that has more than two possible values in a Bayesian network?
For example, I was given the following problem:
We have a bag of three biased coins a, b, and c with probabilities of coming up heads of 20%, 60%,
and 80%, respectively. One coin is drawn randomly from the bag (with equal likelihood of drawing
each of the three coins), and then the coin is flipped three times to generate the outcomes X1, X2, and
X3.
Draw the Bayesian network corresponding to this setup and define the necessary CPTs (Conditional
Probability Table).
Can anyone help point me in a direction to get started with this?
Bayesian networks support variables that have more than two possible values. Koller and Friedman's "Probabilistic Graphical Models" has examples with larger variable domains.
Usually BNs have discrete random variables (with a finite number of different values). But it's also possible to define them with either countably infinite, or continuous variables. In the latter case, the inference algorithms change considerably, though.
Now that I tried finding some examples online, I have to admit you're correct. They're hard to find. Here is an example that is taken from above book. The variable Grade can take on three different values.
Excellent question. Someone already pointed you in the right direction in terms of the specific homework problem, so I won't repeat that; I'll try to add some intuition that might be helpful.
The intuition you need here is that a Bayesian network is nothing more than a visual (graphical) way of representing a set of conditional independence assumptions.
So, for example, if X and Z are conditionally independent variables given Y, then you could draw the Bayesian network X → Y → Z. And conversely, the one and only thing that the Bayes net X → Y → Z tells you is that there are three variables (X, Y, Z) and that X and Z are conditionally independent given Y.
Once you understand this, then you realize that anything you could write a conditional independence assumption for, you can draw a Bayes net for, and vice-versa.
i.e., they need not be Boolean at all.
Usually Bayesian Networks are modeled with discrete values for each node, and when these values are known (or values get set by the modeler) then people say a probability distibution factorizes over these values.
I think theoretical frameworks for bayesian networks with continuous values also exist, but they are mathematically more difficult than discrete (maybe only suited for PhDs?)
Furthermore I cannot solve your problem off the top of my head, but maybe
try this in R:
library(dplyr) # loads mutate(), %>% (pipe operator)
Model <- c("Coin a", "Coin b", "Coin c")
Prior <- c(0.2, 0.6, 0.8)
Likelihood <- c(1/3, 1/3, 1/3)
bayes_df <- data.frame(Model=Model, Prior=Prior, Likelihood=Likelihood)
# posterior probabilities
bayes_df %>%
mutate(Product = Likelihood * Prior, Posterior = Product/sum(Product))
Result
Model Prior Likelihood Product Posterior
1 Coin a 0.2 0.3333 0.06667 0.125
2 Coin b 0.6 0.3333 0.20000 0.375
3 Coin c 0.8 0.3333 0.26667 0.500
I think the "network" is just 2 bubbles connected with an arrow coin -> pick and the CPT is the numbers from above, but I'm not sure.
Related
I have been working with R's isoreg function and have experienced a problem: the function is generating more knots than unique fitted values.
From the R help,
iKnots [is an] integer vector giving indices where the fitted curve jumps, i.e., where the convex minorant has kinks
I believe I have an idea about the cause of the problem, and I have a reproducible example:
# Demonstrating the problem
set.seed(100)
x<-runif(88000,0,1)
x<-x[order(x)]
y<- c(rep(c(0.1000001,0.1000000),11000),rep(c(0.1000002,0.1000003),11000),rep(c(0.2000002,0.2000003),11000),rep(1,22000))
plot(test<-isoreg(x,y))
length(unique(test$yf))
length(test$iKnots)
# Evidence of a floating point arithmetic problem
unique(test$yf)
print(c(unique(test$yf)[1],unique(test$yf)[2]),digits=18)
unique(test$yf)[1]==unique(test$yf)[2]
print(c(unique(test$yf)[4],unique(test$yf)[5]),digits=18)
unique(test$yf)[4]==unique(test$yf)[5]}
Here is the plot produced by this example:
You can see that R's isoreg function is identifying many more knots than it should (where there are a lot of red Xs in the plot). However at other places, it correctly uses only 2 knots (the black lines).
It is clear that the problem is connected to R's floating point arithmetic. I also note that isoreg uses .Call to call a C routine to actually do the isotonic regression, so perhaps the problem lies with differences between the C and R languages.
I am using isoreg to calibrate model probabilities, and I would like to be as precise as possible. Therefore, I have 2 questions:
1) Is there some way I could alter the x and y variables used in the isoreg function to avoid this problem while maintaining as high precision as possible?
2) I can manually find the unique fitted values and the respective knots. However, is this ok? Can I assume that the algorithm found the best fit or could this problem invalidate that assumption?
In a trained neural net the weight distribution will fall close around zero. So it makes sense for me to initiate all weights to zero. However there are methods such as random assignment for -1 to 1 and Nguyen-Widrow that outperformes zero initiation. How come these random methods are better then just using zero?
Activation & learning:
Additionally to the things cr0ss said, in a normal MLP (for example) the activation of layer n+1 is the dot product of the output of layer n and the weights between layer n and n + 1...so basically you get this equation for the activation a of neuron i in layer n:
Where w is the weight of the connection between neuron j (parent layer n-1) to current neuron i (current layer n), o is the output of neuron j (parent layer) and b is the bias of current neuron i in the current layer.
It is easy to see initializing weights with zero would practically "deactivate" the weights because weights by output of parent layer would equal zero, therefore (in the first learning steps) your input data would not be recognized, the data would be negclected totally.
So the learning would only have the data supplied by the bias in the first epochs.
This would obviously render the learning more challenging for the network and enlarge the needed epochs to learn heavily.
Initialization should be optimized for your problem:
Initializing your weights with a distribution of random floats with -1 <= w <= 1 is the most typical initialization, because overall (if you do not analyze your problem / domain you are working on) this guarantees some weights to be relatively good right from the start. Besides, other neurons co-adapting to each other happens faster with fixed initialization and random initialization ensures better learning.
However -1 <= w <= 1 for initialization is not optimal for every problem. For example: biological neural networks do not have negative outputs, so weights should be positive when you try to imitate biological networks. Furthermore, e.g. in image processing, most neurons have either a fairly high output or send nearly nothing. Considering this, it is often a good idea to initialize weights between something like 0.2 <= w <= 1, sometimes even 0.5 <= w <= 2 showed good results (e.g. in dark images).
So the needed epochs to learn a problem properly is not only dependent on the layers, their connectivity, the transfer functions and learning rules and so on but also to the initialization of your weights.
You should try several configurations. In most situations you can figure out what solutions are adequate (like higher, positive weights for processing dark images).
Reading the Nguyen article, I'd say it is because when you assign the weight from -1 to 1, you are already defining a "direction" for the weight, and it will learn if the direction is correct and it's magnitude to go or not the other way.
If you assign all the weights to zero (in a MLP neural network), you don't know which direction it might go to. Zero is a neutral number.
Therefore, if you assign a small value to the node's weight, the network will learn faster.
Read Picking initial weights to speed training section of the article. It states:
First, the elements of Wi are assigned values from a uniform random distributation between -1 and 1 so that its direction is random. Next, we adjust the magnitude of the weight vectors Wi, so that each hidden node is linear over only a small interval.
Hope it helps.
I would like to use a genetic program (gp) to estimate the probability of an 'outcome' from an 'event'. To train the nn I am using a genetic algorithm.
So, in my database I have many events, with each event containing many possible outcomes.
I will give the gp a set of input variables that relate to each outcome in each event.
My questions is - what should the fitness function be in the gp be ????
For instance, right now I am giving the gp a set of input data (outcome input variables), and a set of target data (1 if outcome DID occur, 0 if outcome DIDN'T occur, with the fitness function being the mean squared error of the outputs and targets). I then take the sum of each output for each outcome, and divide each output by the sum (to give the probability). However, I know for sure that this is not the right way to be doing this.
For clarity, this is how I am CURRENTLY doing this:
I would like to estimate the probability of 5 different outcomes occurring in an event:
Outcome 1 - inputs = [0.1, 0.2, 0.1, 0.4]
Outcome 1 - inputs = [0.1, 0.3, 0.1, 0.3]
Outcome 1 - inputs = [0.5, 0.6, 0.2, 0.1]
Outcome 1 - inputs = [0.9, 0.2, 0.1, 0.3]
Outcome 1 - inputs = [0.9, 0.2, 0.9, 0.2]
I will then calculate the gp output for each input:
Outcome 1 - output = 0.1
Outcome 1 - output = 0.7
Outcome 1 - output = 0.2
Outcome 1 - output = 0.4
Outcome 1 - output = 0.4
The sum of the outputs for each outcome in this event would be: 1.80. I would then calculate the 'probability' of each outcome by dividing the output by the sum:
Outcome 1 - p = 0.055
Outcome 1 - p = 0.388
Outcome 1 - p = 0.111
Outcome 1 - p = 0.222
Outcome 1 - p = 0.222
Before you start - I know that these aren't real probabilities, and that this approach does not work !! I just put this here to help you understand what I am trying to achieve.
Can anyone give me some pointers on how I can estimate the probability of each outcome ? (also, please note my maths is not great)
Many thanks
I understand the first part of your question: What you described is a classification problem. You're learning if your inputs relate to whether an outcome was observed (1) or not (0).
There are difficulties with the second part though. If I understand you correctly you take the raw GP output for a certain row of inputs (e.g. 0.7) and treat it as a probability. You said this doesn't work, obviously. In GP you can do classification by introducing a threshold value that splits your classes. If it's bigger than say 0.3 the outcome should be 1 if it's smaller it should be 0. This threshold isn't necessarily 0.5 (again it's just a number, not a probability).
I think if you want to obtain a probability you should attempt to learn multiple models that all explain your classification problem well. I don't expect you have a perfect model that explains your data perfectly, respectively if you have you wouldn't want a probability anyway. You can bag these models together (create an ensemble) and for each outcome you can observe how many models predicted 1 and how many models predicted 0. The amount of models that predicted 1 divided by the number of models could then be interpreted as a probability that this outcome will be observed. If the models are all equally good then you can forget weighing between them, if they're different in quality of course you could factor these into your decision. Models with less quality on their training set are less likely to contribute to a good estimate.
So in summary you should attempt to apply GP e.g. 10 times and then use all 10 models on the training set to calculate their estimate (0 or 1). However, don't force yourself to GP only, there are many classification algorithms that can give good results.
As a sidenote, I'm part of the development team of a software called HeuristicLab which runs under Windows and with which you can run GP and create such ensembles. The software is open source.
AI is all about complex algorithms. Think about it, the downside is very often, that these algorithms become black boxes. So the counterside to algoritms, such as NN and GA, are they are inherently opaque. That is what you want if you want to have a car driving itself. On the other hand this means, that you need tools to look into the black box.
What I'm saying is that GA is probably not what you want to solve your problem. If you want to solve AI types of problems, you first have to know how to use standard techniques, such as regression, LDA etc.
So, combining NN and GA is usually a bad sign, because you are stacking one black box on another. I believe this is bad design. An NN and GA are nothing else than non-linear optimizers. I would suggest to you to look at principal component analysis (PDA), SVD and linear classifiers first (see wikipedia). If you figure out to solve simple statistical problems move on to more complex ones. Check out the great textbook by Russell/Norvig, read some of their source code.
To answer the questions one really has to look at the dataset extensively. If you are working on a small problem, define the probabilities etc., and you might get an answer here. Perhaps check out Bayesian statistics as well. This will get you started I believe.
Let's say you have 3 inputs: A, B, C. Can an artificial neural network (not necessarily feed forward) learn this pattern?
if C > k
output is A
else
output is B
Are there curtain types of networks, which can or which are well suited for this type of problem?
Yes, that's a relatively easy pattern for a feedforward neural network to learn.
You will need at least 3 layers I think assuming sigmoid functions:
1st layer can test C>k (and possibly also scale A and B down into the linear range of the sigmoid function)
2nd layer can calculate A/0 and 0/B conditional on the 1st layer
3rd (output) layer can perform a weighted sum to give A/B (you may need to make this layer linear rather than sigmoid depending on the scale of values you want)
Having said that, if you genuinely know the structure of you problem and what kind of calculation you want to perform, then Neural Networks are unlikely to be the most effective solution: they are better in situations when you don't know much about the exact calculations required to model the functions / relationships.
If the inputs can be only zeros and ones, then this is the network:
Each neuron has a Heaviside step function as an activation function. The neurons y0 and z have bias = 0.5; the neuron y1 has a bias = 1.5. The weights are shown above the corresponding connections. When s = 0, the output z = d0. When s = 1, the output z = d1.
If the inputs are continuous, then Sigmoid, tanh or ReLU can be used as the activation functions of the neurons, and the network can be trained with the back-propagation algorithm.
I've read about neural network a little while ago and I understand how an ANN (especially a multilayer perceptron that learns via backpropagation) can learn to classify an event as true or false.
I think there are two ways :
1) You get one output neuron. It it's value is > 0.5 the events is likely true, if it's value is <=0.5 the event is likely to be false.
2) You get two output neurons, if the value of the first is > than the value of the second the event is likely true and vice versa.
In these case, the ANN tells you if an event is likely true or likely false. It does not tell how likely it is.
Is there a way to convert this value to some odds or to directly get odds out of the ANN. I'd like to get an output like "The event has a 84% probability to be true"
Once a NN has been trained, for eg. using backprogation as mentioned in the question (whereby the backprogation logic has "nudged" the weights in ways that minimize the error function) the weights associated with all individual inputs ("outside" inputs or intra-NN inputs) are fixed. The NN can then be used for classifying purposes.
Whereby the math (and the "options") during the learning phase can get a bit thick, it is relatively simple and straightfoward when operating as a classifier. The main algorithm is to compute an activation value for each neuron, as the sum of the input x weight for that neuron. This value is then fed to an activation function which purpose's is to normalize it and convert it to a boolean (in typical cases, as some networks do not have an all-or-nothing rule for some of their layers). The activation function can be more complex than you indicated, in particular it needn't be linear, but whatever its shape, typically sigmoid, it operate in the same fashion: figuring out where the activation fits on the curve, and if applicable, above or below a threshold. The basic algorithm then processes all neurons at a given layer before proceeding to the next.
With this in mind, the question of using the perceptron's ability to qualify its guess (or indeed guesses - plural) with a percentage value, finds an easy answer: you bet it can, its output(s) is real-valued (if anything in need of normalizing) before we convert it to a discrete value (a boolean or a category ID in the case of several categories), using the activation functions and the threshold/comparison methods described in the question.
So... How and Where do I get "my percentages"?... All depends on the NN implementation, and more importantly, the implementation dictates the type of normalization functions that can be used to bring activation values in the 0-1 range and in a fashion that the sum of all percentages "add up" to 1. In its simplest form, the activation function can be used to normalize the value and the weights of the input to the output layer can be used as factors to ensure the "add up" to 1 question (provided that these weights are indeed so normalized themselves).
Et voilà!
Claritication: (following Mathieu's note)
One doesn't need to change anything in the way the Neural Network itself works; the only thing needed is to somehow "hook into" the logic of output neurons to access the [real-valued] activation value they computed, or, possibly better, to access the real-valued output of the activation function, prior its boolean conversion (which is typically based on a threshold value or on some stochastic function).
In other words, the NN works as previously, neither its training nor recognition logic are altered, the inputs to the NN stay the same, as do the connections between various layers etc. We only get a copy of the real-valued activation of the neurons in the output layer, and we use this to compute a percentage. The actual formula for the percentage calculation depends on the nature of the activation value and its associated function (its scale, its range relative to other neurons' output etc.).
Here are a few simple cases (taken from the question's suggested output rules)
1) If there is a single output neuron: the ratio of the value provided by the activation function relative to the range of that function should do.
2) If there are two (or more output neurons), as with classifiers for example: If all output neurons have the same activation function, the percentage for a given neuron is that of its activation function value divided by the sum of all activation function values. If the activation functions vary, it becomes a case by case situation because the distinct activation functions may be indicative of a purposeful desire to give more weight to some of the neurons, and the percentage should respect this.
What you can do is to use a sigmoid transfer function on the output layer nodes (that accepts data ranges (-inf,inf) and outputs a value in [-1,1]).
Then by using the 1-of-n output encoding (one node for each class), you can map the range [-1,1] to [0,1] and use it as probability for each class value (note that this works naturally for more than just two classes).
The activation value of a single output neuron is a linearly weighted sum, and may be directly interpreted as an approximate probability if the network is trained to give outputs a range from 0 to 1. This would tend to be the case if the transfer function (or output function) in both the preceding stage and providing the final output is in the 0 to 1 range too (typically the sigmoidal logistic function). However, there is no guarantee that it will but repairs are possible. Moreover unless the sigmoids are logistic and the weights are constrained to be positive and sum to 1, it is unlikely. Generally a neural network will train in a more balanced way using the tanh sigmoid and weights and activations that range positive and negative (due to the symmetry of this model). Another factor is the prevalence of the class - if it is 50% then a 0.5 threshold is likely to be effective for logistic and a 0.0 threshold for tanh. The sigmoid is designed to push things towards the centre of the range (on backpropogation) and constrain it from going out of the range (in feedforward). The significance of the performance (with respect to the Bernoulli distribution) can also be interpreted as a probability that the neuron is making real predictions rather than guessing. Ideally the bias of the predictor to positives should match the prevalence of positives in the real world (which may vary at different times and places, e.g. bull vs bear markets, e.g. credit worthiness of people applying for loans vs people who fail to make loan payments) - calibrating to probabilities has the advantage that any desired bias can be set easily.
If you have two neurons for two classes, each can be interpreted independently as above, and the halved difference between them can also be. It is like flipping the negative class neuron and averaging. The differences can also give rise to a probability of significance estimate (using the T-test).
The Brier score and its Murphy decomposition give a more direct estimate of the probability that an average answer is correct, while Informedness gives the probability the classifier is making an informed decision rather than a guess, ROC AUC gives the probability a positive class will be ranked higher than a negative class (by a positive predictor), and Kappa will give a similar number that matches Informedness when prevalence = bias.
What you normally want is both a significance probability for the overall classifier (to ensure that you are playing on a real field, and not in an imaginary framework of guestimates) and a probability estimate for a specific example. There are various ways to calibrate, including doing a regression (linear or nonlinear) versus probability and using its inverse function to remap to a more accurate probability estimate. This can be seen by the Brier score improving, with the calibration component reducing towards 0, but the discrimination component remaining the same, as should ROC AUC and Informedness (Kappa is subject to bias and may worsen).
A simple non-linear way to calibrate to probabilities is to use the ROC curve - as the threshold changes for the output of a single neuron or the difference between two competing neurons, we plot the results true and false positive rates on a ROC curve (the false and true negative rates are naturally the complements, as what isn't really a positive is a negative). Then you scan the ROC curve (polyline) point by point (each time the gradient changes) sample by sample and the proportion of positive samples gives you a probability estimate for positives corresponding to the neural threshold that produced that point. Values between points on the curve can be linearly interpolated between those that are represented in the calibration set - and in fact any bad points in the ROC curve, represented by deconvexities (dents) can be smoothed over by the convex hull - probabilistically interpolating between the endpoints of the hull segment. Flach and Wu propose a technique that actually flips the segment, but this depends on information being used the wrong way round and although it could be used repeatedly for arbitrary improvement on the calibration set, it will be increasingly unlikely to generalize to a test situation.
(I came here looking for papers I'd seen ages ago on these ROC-based approaches - so this is from memory and without these lost references.)
I will be very prudent in interpreting the outputs of a neural networks (in fact any machine learning classifier) as a probability. The machine is trained to discriminate between classes, not to estimate the probability density. In fact, we don't have this information in the data, we have to infer it. For my experience I din't advice anyone to interpret directly the outputs as probabilities.
did you try prof. Hinton's suggestion of training the network with softmax activation function and cross entropy error?
as an example create a three layer network with the following:
linear neurons [ number of features ]
sigmoid neurons [ 3 x number of features ]
linear neurons [ number of classes ]
then train them with cross entropy error softmax transfer with your favourite optimizer stochastic descent/iprop plus/ grad descent. After training the output neurons should be normalized to sum of 1.
Please see http://en.wikipedia.org/wiki/Softmax_activation_function for details. Shark Machine Learning framework does provide Softmax feature through combining two models. And prof. Hinton an excellent online course # http://coursera.com regarding the details.
I can remember I saw an example of Neural network trained with back propagation to approximate the probability of an outcome in the book Introduction to the theory of neural computation (hertz krogh palmer). I think the key to the example was a special learning rule so that you didn't have to convert the output of a unit to probability, but instead you got automatically the probability as output.
If you have the opportunity, try to check that book.
(by the way, "boltzman machines", although less famous, are neural networks designed specifically to learn probability distributions, you may want to check them as well)
When using ANN for 2-class classification and logistic sigmoid activation function is used in the output layer, the output values could be interpreted as probabilities.
So if you choosing between 2 classes, you train using 1-of-C encoding, where 2 ANN outputs will have training values (1,0) and (0,1) for each of classes respectively.
To get probability of first class in percent, just multiply first ANN output to 100. To get probability of other class use the second output.
This could be generalized for multi-class classification using softmax activation function.
You can read more, including proofs of probabilistic interpretation here:
[1] Bishop, Christopher M. Neural networks for pattern recognition. Oxford university press, 1995.