A + Bayesian Inference Assignment Help

Bayesian inference is a branch of statistics that takes advantage of what we know about probabilities in order to find answers to difficult problems. It has been used extensively in fields such as medicine, engineering, and criminology. Bayesian inference can be a difficult topic, but it’s worth the effort. Are you looking for A + Bayesian Inference Assignment Help? Worry no more! We got you covered!

Bayes theorem

Bayes theorem is at the heart of Bayesian inference. It says that the probability an event occurring is obtained by multiplying the prior probability with likelihood, given some evidence.  It represents how likely it is to see the evidence given that the event has occurred divided by the probability of seeing that evidence.

Example 1:   Let’s say we roll a six-sided die once. What is the probability that we rolled a “1” instead of a “2”? We can use Bayes’ theorem to figure this out.

The probability of rolling a “1” is 1/6 because there’s only one way to roll a “1”. The probability of rolling a “2”, however, is 5/6, because there are five ways to roll a “2”: 2-2, 1-3, 2-1, 0-4, and 0-5.  The likelihood in this case is 5/6 because the only way to roll a “1” is if you rolled a “2”, which means there are five ways for us to see evidence that the event has occurred.

The prior probability is 1/6 because there are six possible outcomes, and only one of them results in us rolling a “1”. Thus, using Bayes’ theorem, we get our probability as 5/12, or 1/2. That means the probability of rolling a “1” is twice the probability of rolling a “2”.

Loss function

The loss function is a common method used in Bayesian inference. It represents the difference between the expected value of a random variable and the observed value. It is an average function. It shows progress when the expected value of a random variable is getting closer to its observed value. The expected value can be expressed as the sum of all possible values our random variable can take multiplied by their probabilities.

Example 2: Bayesian inference on the possible values that randomly drawn samples can have with respect to some distribution. When the losses are increasing, it is easy to obtain extreme values.

Imagine that the possible values are 1-5 and our random samples have a mean of 2. The samples are such as {1,2}, then the loss is 0 because the expected value of these samples is 2. However, if similar samples such as {3}, then the loss becomes 1/2 because 3-2 = 1.

A sample of size 2, gives more extreme values. This means that there’s some other distribution where the samples are less extreme. If it continues in this way, then trying to get samples of size 2 will eventually result in us being stuck in a local minimum.

Example 3:  Taking some function that represents an algorithm’s performance, and f'(x) is the loss over the number of iterations.  A plot on the rate of change from iteration to iteration is done by visualizing the function’s tangent. Progress is indicated by when the algorithm curving downward. This means that it’ll eventually reach a minimum.  Increase in loss is indicated by caving upward or to the left.

  Posterior distribution

The posterior distribution is the probability of any possible outcome given your evidence. With Bayesian inference, you begin with a prior distribution, which represents what you think the probabilities are before seeing any evidence. You then receive some evidence, which is called likelihood. The posterior distribution is the probability of every possible outcome given   your evidence.

Example 4:

Estimation of time period to reach a certain destination. Assuming that the acceleration of your car is constant which is 5 m/s ², time t, the velocity at any time t is a (t).  The probability density function for this kind of distribution can be written as p(t) = 1/ (2²), where a is constant; a (0) = 0. This is called the prior distribution.

Suppose through observation, it takes 4 seconds to reach 10m/s. This new information will be updated on the acceleration of the car. The likelihood in this case will be 10m/s. This probability can be written as p (4) = 1/6. There are 6 possible values you could have reached in 4 seconds: 10, 11, 12, 13, 14, and 15.

The posterior distribution can be written as p(t|4) = [1/ (2 (4) 2)] [ 1/6], which reduces top(t|4) = 1/3. This is because there are three possible outcomes for any value of t given that it took 4 seconds to reach 10m/s: 5, 7, and 9. This means the posterior distribution is a uniform distribution between 5 and 9.

That means if we roll a die now, there’s a 1/3 chance that the result is 5 or less, and a 2/3 chance that it’s 6 or greater. The posterior distribution represents our updated belief in what will happen next based on our new evidence. It’s this information that we use to make predictions about the future.

Types of Priors

 

Conjugate prior

 

It is a prior in which the posterior is of the same form as the likelihood. A special case is when the prior is normal (or approximately so), in which case the posterior will also be normal or approximately so.  They to be conveniently expressed and manipulated, making computations. For example: When the prior is normal, a simple formula for computing the posterior odds is available:

prior probabilities * likelihood ratios = posterior probabilities. This allows probability statements to be expressed as fractions, or as multiples of 1/100. In scenarios where it is difficult to assess the relative plausibility of two possibilities, they can greatly simplify the computation of the posterior odds, and therefore simplify statistical calculations.

 

Non informative priors

 

It is a prior distribution that has minimal effect on the shape of the posterior distribution. It is also called “flat” or “uniform”. For example: A non-informative prior might be an improper uniform distribution which assigns equal probabilities to all possible values of the parameter being estimated.

Non-informative priors are normally used for Bayesian model comparison, where it is desired to test different models against one another. It contains no information about the process being modeled, so comparison with alternative models will not favor one over another due solely to prior information. If two or more models are found to have comparable support, then they are considered “equally good” with regard to the data assumed for the prior distribution.

Informative priors

It is s a non-random probability distribution. It is a set of parameters over some space representing the range of possible values about the parameters. It can be used to express domain knowledge of the parameters being estimated.

Jeffrey’s Prior

Jeffrey’s prior, named after the statistician and geneticist Edwin Jaynes, is a generalization of an “ideal” prior proposed by Laplace. It does not assume equal probabilities for different values of the parameter. Instead it creates a probability density function that assigns weight to each value in proportion to how likely that value should be according to the data.

For example: If you have a priori information that the parameter has a normal distribution with mean 0 and standard deviation 1, you can create a Jeffrey’s prior from this information. In this case, the density will be proportional to exp(-z²). The result is that probabilities decline exponentially with increasing values of z.

Jeffreys’ prior may be thought of as a method for down-weighting or up-weighting certain types of data, in effect allowing some types of data to have greater influence on the final result than other types.

Application of Bayesian inference

Bayes’ theorem is used in a variety of scientific fields. One example of this is in cosmology, where the sun’s gravitational pull on planets allows us to estimate their mass. Astronomers use these measurements to determine the properties of distant stars located throughout the galaxy. The probability distributions are   used to compute the likelihood of these stars’ properties given cosmological models.

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