Assuming the world were deterministic, why would we still need to introduce stochasticity into our simulations?
In a nutshell, to simplify models.
Let’s go with your assumption, even though I don’t believe it. If the universe is completely deterministic, then in any given scenario you choose to model there is one and only one correct answer. Unless you include the complete state space of absolutely everything that determines that answer, your model is wrong. Wrong, wrong, wrong!!!
For instance, if you want to predict how long it will take to fly from New York to London, you need to know the vector sums of all forces acting on the aircraft, which means you need the complete state (down to the atomic level) of the aircraft itself, the passengers, the atmosphere, fluctuations in the magnetic field of the earth, cosmic rays that can trigger upper atmospheric events, etc, etc, ad nauseam. Exclusion of any aspect of the potential forces involved makes your answer wrong.
Clearly, there’s no way to measure it all, and even if there was, there’s no way to maintain so much state information in any computing device we can build. And so we simplify and acknowledge that there is some degree of uncertainty in our model’s predictions/solutions.
When you embrace the existence of uncertainty, it brings us directly to stochastic solutions. One view of probability is that it is a mathematical formalism for modeling uncertainty. Rather than try to model every physical aspect of an aircraft’s flight, we can characterize the likely outcomes based on what proportion of flights require less (or more) than any particular amount of time, i.e., describing the distribution of possible flight times.
Once you adopt distributional modeling, you can see how distributional behaviors propagate though other parts of a system—either analytically, if your system is sufficiently simple, or by generating realizations of the distributions and using replication and sampling via simulation.
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I think I just about grasp the basic understanding but it's still confusing me. In my statistics course I'm investigating a hypothetical bone-wasting disease dataset among different population groups, which has a number of geographic controls and the like. I got the question wrong and said that they weren't needed, but my teacher didn't elaborate. Like why do we need control for different groups in our regression - what do we gain by doing that? Thanks for any help!
This one is straightforward application of the OLS assumptions.
To take almost straight from undergraduate lecture slides. In OLS you make an assumption about the distribution of of errors - since the errors are joint normal and uncorrelated, which implies that (εi)i∈N is an i.i.d. set. If you don't have control variables it is likely that you are ignoring independent variables that are affecting your dependent variables. Implying that the unobservables between different populations are independent, which appears clearly unrealistic.
I have seen MICE implemented with different types of algorithms e.g. RandomForest or Stochastic Regression etc.
My question is that does it matter which type of algorithm i.e. does one perform the best? Is there any empirical evidence?
I am struggling to find any info on the web
Thank you
Yes, (depending on your task) it can matter quite a lot, which algorithm you choose.
You also can be sure, the mice developers wouldn't out effort into providing different algorithms, if there was one algorithm that anyway always performs best. Because, of course like in machine learning the "No free lunch theorem" is also relevant for imputation.
In general you can say, that the default settings of mice are often a good choice.
Look at this example from the miceRanger Vignette to see, how far imputations can differ for different algorithms. (the real distribution is marked in red, the respective multiple imputations in black)
The Predictive Mean Matching (pmm) algorithm e.g. makes sure that only imputed values appear, that were really in the dataset. This is for example useful, where only integer values like 0,1,2,3 appear in the data (and no values in between). Other algorithms won't do this, so while doing their regression they will also provide interpolated values like on the picture to the right ( so they will provide imputations that are e.g. 1.1, 1.3, ...) Both solutions can come with certain drawbacks.
That is why it is important to actually assess imputation performance afterwards. There are several diagnostic plots in mice to do this.
I am reading people's implementation of DCGAN, especially this one in tensorflow.
In that implementation, the author draws the losses of the discriminator and of the generator, which is shown below (images come from https://github.com/carpedm20/DCGAN-tensorflow):
Both the losses of the discriminator and of the generator don't seem to follow any pattern. Unlike general neural networks, whose loss decreases along with the increase of training iteration. How to interpret the loss when training GANs?
Unfortunately, like you've said for GANs the losses are very non-intuitive. Mostly it happens down to the fact that generator and discriminator are competing against each other, hence improvement on the one means the higher loss on the other, until this other learns better on the received loss, which screws up its competitor, etc.
Now one thing that should happen often enough (depending on your data and initialisation) is that both discriminator and generator losses are converging to some permanent numbers, like this:
(it's ok for loss to bounce around a bit - it's just the evidence of the model trying to improve itself)
This loss convergence would normally signify that the GAN model found some optimum, where it can't improve more, which also should mean that it has learned well enough. (Also note, that the numbers themselves usually aren't very informative.)
Here are a few side notes, that I hope would be of help:
if loss haven't converged very well, it doesn't necessarily mean that the model hasn't learned anything - check the generated examples, sometimes they come out good enough. Alternatively, can try changing learning rate and other parameters.
if the model converged well, still check the generated examples - sometimes the generator finds one/few examples that discriminator can't distinguish from the genuine data. The trouble is it always gives out these few, not creating anything new, this is called mode collapse. Usually introducing some diversity to your data helps.
as vanilla GANs are rather unstable, I'd suggest to use some version
of the DCGAN models, as they contain some features like convolutional
layers and batch normalisation, that are supposed to help with the
stability of the convergence. (the picture above is a result of the DCGAN rather than vanilla GAN)
This is some common sense but still: like with most neural net structures tweaking the model, i.e. changing its parameters or/and architecture to fit your certain needs/data can improve the model or screw it.
I'm creating an evolution-artificial-life-simulation game in 2D (purely for fun purposes). It combines neural networks (for behaviour controlling) and genetic algorithm (for breeding and mutations).
On input I give them X,Y position of nearest food (normalized) and X,Y position of the "look at" vector.
Currently they fly around and when they collide with food (let's call it "eating apples") their fitness index is increased by one and the apple's position is randomed - after 2000 turns the GA interrupts and does its magic.
After about 100 generations they learn that eating apples is good and try to fly to the nearest ones.
But my question, as a neural network newbie, is - if I created a room where apples spawn way more frequent than on the rest of the map, would they learn and understand that? Would they fly to that room more often? And is it possible to tell how many generations would it take for them to learn?
What they can learn and how fast depends a lot on the information you give them access to. For instance, if they have no way of knowing that they are in the room where food generates more frequently, then there is no way for them to evolve to go there more frequently.
It's not entirely clear from your question what the "look at" vector is. If it, for instance, shows them what's directly in front of them, then it might be enough information for them to figure out that they're in the room of plenty, particularly if that room "looks" distinctive somehow. A more useful input to give them might be their current X and Y coordinates. If you did that, then I would definitely expect them to evolve to be in the good room more frequently (in proportion to how good it is, of course), because it would be possible for them to take action to go to and stay in that room.
As for how many generations it will take, that is incredibly hard to predict (especially without knowing more about your setup). If it takes them 100 generations to learn to eat food, then I would expect it to be on the order of hundreds. But the best way to find out is just to try it.
If it's all about location, they may keep a state of the map in their mind and simple statistics will let them learn where the food may be located. Neural nets is an overkill there.
If there are other features of locations (for example color, smell, height etc...) to map those features to the label (food exists or not) is good for neural nets. Especially if some of features not available or not reliable randomly at the moment.
If they need many decisions to reach the goal, you will need reinforcement learning. Forexample, they may go to a direction which is good for a time, but make them away from resources they will need later.
I believe that a recurrent neural network could learn to expect apples to spawn in a certain region.
Seeing that as as far as we know, one half of your brain is logical and the other half of your brain is emotional, and that the wants of the emotional side are fed to the logical side in order to fulfill those wants; has there been any research done in connecting two separate neural networks to one another (one trained to be emotional, and one trained to be logical) to see if it would result in almost a free-will sort of "brain"?
I don't really know anything about neural networks except that they were modeled after the biological synapses in the human brain, which is why I ask.
I'm not even sure if this would be possible considering that even a trained neural network sometimes doesn't act logically (a.k.a. do what you thought you trained it to do).
First, most modern neural networks aren't really modeled after biological synapses. They use an Artificial Neuron which allowed Back Propagation to work rather than a Perceptron which is a much more accurate representation.
When you feed the output of one network into the input of another network, you've really just created one larger network, not two separate networks. It just happens that in this case portions of the networks would be trained independently.
That said, all neural networks have to be trained. Which means you need sample input and sample output. You are looking to create a decision engine of sorts I suppose. So you would need to create a dataset where it makes sense that there might be an emotional and rational response, such as purchasing an item. You'd have to train the 'rational' network to accept as a set of inputs the output of an 'emotional' network. Which means you are really just training the rational decision engine to be responsive based on the leve of 'distress' caused by the emotional network.
Just my two cents.
I have also heard of one hemisphere being called "divergent" and one "convergent". This may not make any more sense than emotional vs logical, but it does hint at how you might model it more easily. I don't know how the brain achieves some of the impressive computational feats it does, but I wouldn't be very surprised if all revolved around balance, but maybe that is just one of the baises you have when you are a brain with two hemipheres (or any even number) :D
A balance between convergence and divergence is the crux of the creativity inherent in evolution. Replicating this with neural nets sounds promising to me. Suppose you make one learning system that generalizes and keeps representations of only the typical groups of patterns it is shown. Then you take another and make it generate all the in-betweens and mutants of the patterns it is shown. Then you feed them to eachother in a circle, and poof, you have made something really interesting!
It's even more complex than that, unbelievably. The left hemisphere works on a set of logical rules, it uses these to predict its environment and categorize input. It also infers rules and stores them for future use. The right hemisphere is based, as you said, on emotion, but also on memory of single, unique or emotionally relevant occurrences. A software implementation should also be able to retrieve and store these two data types and exchange "opinions" about them.
While the left hemisphere of the brain may be more involved in making emotional decisions, emotion itself is unlikely to occur exclusively in one side of the brain, and the interplay between emotions and rational thought within the brain is likely to be substantially more complex than having two completely separate circuits. For instance, a study on rhesus macaques found that dopamine and other hormones associated with emotional responses essentially implements temporal difference learning within the brain (I'm still looking for a link to it). This suggests that separating emotional and rational thought into two separate neural networks probably wouldn't be practical, even if we had the resources to build neural networks on the scale of brain hemispheres (which we don't, or at least not within most research budgets).
This idea is supported by Sloman and Croucher's suggestion that emotion will likely be an unavoidable emergent property of a sufficiently advanced intelligent system. Such systems (discussed in detail in the paper) will be much more complex than straight-up neural nets. More importantly, though, the emotions won't be something that you can localize to one part of the system.