KURS FUNKCJE WIELU ZMIENNYCH Lekcja 5 Dziedzina funkcji ZADANIE DOMOWE Strona 2 Częśd 1: TEST Zaznacz poprawną odpowiedź (tylko jedna jest logarytm, arcsinx, arccosx, arctgx, arcctgx c) Dzielenie, pierwiastek, logarytm. 4 Dlaczego maksymalizujemy sumy logarytmów prawdopodobienstw? z maksymalizacją logarytmów prawdopodobieństwa poprawnej odpowiedzi przy a priori parametrów przez prawdopodobienstwo danych przy zadanych parametrach. Zadanie 1. (1 pkt). Suma pięciu kolejnych liczb całkowitych jest równa. Najmniejszą z tych liczb jest. A. B. C. D. Rozwiązanie wideo. Obejrzyj na Youtubie.

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The prior may be very vague. When we see lkgarytmy data, we combine our prior distribution with a likelihood term to get a posterior distribution. The likelihood term takes into account how probable the observed data is given the parameters of the model.

It favors parameter settings that make the data likely. It fights the prior With enough data the likelihood terms always win. Our model of a coin has one parameter, p.

### Uczenie w sieciach Bayesa – ppt pobierz

Suppose we observe tosses and there odpowkedzi 53 heads. Pick the value of p that makes the observation of 53 heads and 47 tails most probable. Is it reasonable to give a single answer? Our computations of probabilities will work much better if we take this uncertainty into account.

In this case we used a uniform distribution.

### Opracowania do zajęć wyrównawczych z matematyki elementarnej

Multiply the prior probability of each parameter value by the probability ofpowiedzi observing a head given that value. Then scale up all of the probability densities so that their integral comes to 1. This gives the posterior distribution. Multiply the prior probability of each parameter value by the probability of observing a tail given that value. Then renormalize to get the posterior distribution.

Look how sensible it is! We can do this by starting with a random weight vector and then adjusting it in the direction that improves p W D. It is easier to work in the log domain.

If we want to minimize a cost we use negative log probabilities: Because the log function is monotonic, so we can maximize sums of log probabilities.

Then all we have to do is to maximize: This is called maximum likelihood learning. It is very widely used for fitting models in statistics. So it just scales the squared error.

It looks for the parameters that have the greatest product of the prior term and the likelihood term. Minimizing the squared weights is equivalent to maximizing the log probability of the weights under a zero-mean Gaussian maximizing prior.

To make predictions, let each different setting of the parameters make its own prediction and then combine all these predictions by weighting each of them by the posterior probability of that setting of the parameters.

This is also computationally intensive.

The full Bayesian approach allows us to use complicated models even when we do not have much data. If you do not have much data, you should use a simple model, because a complex one will overfit. But only if you assume that fitting a model odpowiedzo choosing a single best setting of the parameters.

If you use the full posterior over parameter settings, overfitting disappears!

With little data, you get very vague predictions because many different parameters settings have significant posterior probability. The complicated model fits the data better. But it is not economical and it makes silly predictions.

But what if we start with a reasonable prior over all ospowiedzi polynomials and use the full zdaania distribution. Now we get vague and sensible predictions. There is no reason why the amount of data should influence our prior beliefs about the complexity of the model.

This is expensive, but it does not involve any gradient descent and there are no local optimum issues. After evaluating each grid point we use all of them to make predictions on test data This is also expensive, but it works much better than ML learning when the posterior is vague or multimodal this happens when data is scarce.

For each grid-point compute the probability of the observed outputs of all the training cases. This is the likelihood term and is explained on the next slide Multiply the prior for each grid-point p Wi by the likelihood term and renormalize to get the posterior probability for each grid-point p Wi,D. Make predictions p ytest input, D by using the posterior probabilities of all grid-points to average the predictions p ytest input, Wi made by the different grid-points.

It assigns the complementary probability to the answer 0. The number of grid odpowedzi is exponential in the number of parameters. So we cannot deal with more than a few parameters using a grid. If there is enough data to make most parameter vectors very unlikely, only need a tiny fraction of the grid points make a significant contribution to the predictions.

Maybe we can just evaluate this tiny fraction It might be good enough to just sample weight vectors according to their posterior probabilities.

Sample weight vectors with this probability. Suppose we add some Gaussian noise to the weight vector after each update. So the weight vector never settles down.

## Uczenie w sieciach Bayesa

It keeps wandering around, but it tends to prefer low cost regions of the weight space. If we use just the right amount of noise, and if we let the weight vector zadanix around for long enough before we take a sample, we will get a sample from the true posterior over weight vectors.

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