# bayesian inference example problems

In order to find this best approximation, we follow an optimisation process (over the family parameters) that only require the targeted distribution to be defined up to a factor. Statistical inferences are usually based on maximum likelihood estimation (MLE). In this section we describe MCMC sampling methods that constitute a possible solution to overcome this issue as well as some others computational difficulties related to Bayesian inference. The choice of the family defines a model that control both the bias and the complexity of the method. On the other hand, in Example 9.2, the prior distribution $f_{X_n}(x)$ might be determined as a part of the communication system design. But let’s plough on with an example where inference might come in handy. In practice, the lag required between two states to be considered as almost independent can be estimated through the analysis of the autocorrelation function (only for numeric values). We can notice that the following equivalence holds. For example, Gaussian mixture models, for classification, or Latent Dirichlet Allocation, for topic modelling, are both graphical models requiring to solve such a problem when fitting the data. The Gibbs Sampling method is based on the assumption that, even if the joint probability is intractable, the conditional distribution of a single dimension given the others can be computed. In the first section we will discuss the Bayesian inference problem and see some examples of classical machine learning applications in which this problem naturally appears. Once both the parametrised family and the error measure have been defined, we can initialise the parameters (randomly or according to a well defined strategy) and proceed to the optimisation. Illustration of the main idea of Bayesian inference, in the simple case of a univariate Gaussian with a Gaussian prior on the mean (and known variances). Bayesian parametric inference As we have seen, the method of ordinary least squares can be used to find the best fit of a model to the data under minimal assumptions about the sources of uncertainty and the Take a look, Variational Inference: A Review For Statisticians, Tutorial on Topic Modelling and Gibbs Sampling, www.linkedin.com/in/joseph-rocca-b01365158, 6 Data Science Certificates To Level Up Your Career, Stop Using Print to Debug in Python. Let’s assume first that we have a way (MCMC) to draw samples from a probability distribution defined up to a factor. Bayesian inference Here’s exactly the same idea, in practice; During the search for Air France 447, from 2009-2011, knowledge about the black box location was described via probability { i.e.using Bayesian inference … Introduction Inference about a target population based on sample data relies on the assumption that the sample is representative. Probability and Statistical Inference Extra Problems on Bayesian Stats Click here for answers to these problems. Thanks for reading and feel free to share if you think it deserves to be! As a consequence, these methods have a low bias but a high variance and it implies that results are most of the time more costly to obtain but also more accurate than the one we can get from VI. In this case, the. To do so, you take a random sample of size $n$ from the likely voters in the town. We get some data. After observing some data, we update the distribution of $\Theta$ (based on the observed data). To conclude this subsection, we outline once more the fact that this sampling process we just described is not constrained to the Bayesian inference of posterior distribution and can also, more generally, be used in any situation where a probability distribution is defined up to its normalisation factor. More specifically, the idea is to define a parametrised family of distributions and to optimise over the parameters to obtain the closest element to the target with respect to a well defined error measure. Later in this post, we will describe these two approaches focusing especially on the “normalisation factor problem” but one should keep in mind that these methods can also be precious when facing other computational difficulties related to Bayesian inference. What kind of problems does Stan / Bayesian inference beat the much more hyped Tensorflow / deep learning approach? This post was co-written with Baptiste Rocca. We can notice that some other computational difficulties can arise from Bayesian inference problem such as, for example, combinatorics problems when some variables are discrete. So, in order to get our independent samples that follow the targeted distribution, we keep states from the generated sequence that are separated from each other by a lag L and that come after the burn-in time B. For example, Gaussian mixture models, for classification, or Latent Dirichlet Allocation, for topic modelling, are both graphical models requiring to solve such a problem when fitting the data. First we randomly choose an integer d among the D dimensions of X_n. This distribution is called the prior distribution. The following is a general setup for a statistical inference problem: There is an unknown quantity that we would like to estimate. This example shows how to make Bayesian inferences for a logistic regression model using slicesample. As already mentioned, MCMC and VI methods have different properties that imply different typical use cases. Therefore, given that in the previous election $40 \%$ of the voters voted for Party A, you might want to model the portion of votes for Party A in the next election as a random variable $\Theta$ with a probability density function, $f_{\Theta}(\theta)$, that is mostly concentrated around $\theta=0.4$. Several classical optimisation techniques can be used such as gradient descent or coordinate descent that will lead, in practice, to a local optimum. In this chapter, we would like to discuss a different framework for inference, namely the Bayesian approach. While thinking about this problem, you remember that the data from the previous election is available to you. The weather, the weather It's a typically hot morning in June in Durham. Let’s assume a model where data x are generated from a probability distribution depending on an unknown parameter θ. Let’s also assume that we have a prior knowledge about the parameter θ that can be expressed as a probability distribution p(θ). First, in order to have samples that (almost) follow the targeted distribution, we need to only consider states far enough from the beginning of the generated sequence to have almost reach the steady state of the Markov Chain (the steady state being, in theory, only asymptotically reached). It is worth noting that Examples 9.1 and 9.2 are conceptually different in the following sense: In Example 9.1, the choice of prior distribution $f_{\Theta}(\theta)$ is somewhat unclear. There are a number of diseases that could be causing all of them, but only a single disease is present. Among the random variables generation techniques, MCMC is a pretty advanced kind of methods (we already discussed an other method in our post about GANs) that makes possible to get samples from a very difficult probability distribution potentially defined only up to a multiplicative constant. The “Monte Carlo” part of the method’s name is due to the sampling purpose whereas the “Markov Chain” part comes from the way we obtain these samples (we refer the reader to our introductory post on Markov Chains). In this post we will discuss the two main methods that can be used to tackle the Bayesian inference problem: Markov Chain Monte Carlo (MCMC), that is a sampling based approach, and Variational Inference (VI), that is an approximation based approach. The whole idea that rules the Bayesian paradigm is embed in the so called Bayes theorem that expresses the relation between the updated knowledge (the “posterior”), the prior knowledge (the “prior”) and the knowledge coming from the observation (the “likelihood”). and, then, a Markov Chain with transition probabilities k(.,.) Finally in the third section we will introduce Variational Inference and see how an approximate solution can be obtained following an optimisation process over a parametrised family of distributions. If we can solve this minimisation problem without having to explicitly normalise π, we can use f_* as an approximation to estimate various quantities instead of dealing with intractable computations. Make learning your daily ritual. The counter-intuitive fact that we can obtain, with MCMC, samples from a distribution not well normalised comes from the specific way we define the Markov Chain that is not sensitive to these normalisation factor. So, for example, if each density f_j is a Gaussian with both mean and variance parameters, the global density f is then defined by a set of parameters coming from all the independent factors and the optimisation is done over this entire set of parameters. Traditionally, the MaxEnt workshops start by a tutorial day. As we mentioned before, one of the main difficulty faced when dealing with a Bayesian inference problem comes from the normalisation factor. Let $\theta$ be the true portion of voters in your town who plan to vote for Party A. Notice also that in this post p(.) Thus, your guess is that the error in your estimation might be too high. E[\Theta]=0.4 defined to verify the last equality will have, as expected, π as stationary distribution. In order to better understand this optimisation process, let’s take an example and go back to the specific case of the Bayesian inference problem where we assume a posterior such that, In this case, if we want to get an approximation of this posterior using variational inference, we have to solve the following optimisation process (assuming the parametrised family defined and KL divergence as error measure). In general VI methods are less accurate that MCMC ones but produce results much faster: these methods are better adapted to big scale, very statistical, problems. Outline 1 Bayesian inference in imaging inverse problems 2 Proximal Markov chain Monte Carlo 3 Uncertainty quanti cation in astronomical and medical imaging 4 Image model selection and model calibration 5 Conclusion M. Pereyra The first term is the expected log-likelihood that tends to adjust parameters so that to place the mass of the approximation on values of the latent variables z that explain the best the observed data. The subsection marked by a (∞) are pretty mathematical and can be skipped without hurting the global understanding of this post. Let’s Find Out, 10 Surprisingly Useful Base Python Functions, there exists, for each topic, a “topic-word” probability distribution over the vocabulary (with a Dirichlet prior assumed), there exists, for each document, a “document-topic” probability distribution over the topics (with another Dirichlet prior assumed), each word in a document have been sampled such that, first, we have sampled a topic from the “document-topic” distribution of the document and, second, we have sampled a word from the “topic-word” distribution attached to the sampled topic, Bayesian inference is a pretty classical problem in statistics and machine learning that relies on the well known Bayes theorem and whose main drawback lies, most of the time, in some very heavy computations, Markov Chain Monte Carlo (MCMC) methods are aimed at simulating samples from densities that can be very complex and/or defined up to a factor, MCMC can be used in Bayesian inference in order to generate, directly from the “not normalised part” of the posterior, samples to work with instead of dealing with intractable computations, Variational Inference (VI) is a method for approximating distributions that uses an optimisation process over parameters to find the best approximation among a given family, VI optimisation process is not sensitive to multiplicative constant in the target distribution and, so, the method can be used to approximate a posterior only defined up to a normalisation factor. Another possible way to overcome computational difficulties related to inference problem is to use Variational Inference methods that consist in finding the best approximation of a distribution among a parametrised family. In order to produce samples, the idea is to set up a Markov Chain whose stationary distribution is the one we want to sample from. In order to do so, Metropolis-Hasting and Gibbs Sampling algorithms both use a particular property of Markov Chains: reversibility. that will serve at suggesting transitions. Thus, this objective function expresses pretty well the usual prior/likelihood balance. and, then, γ is a stationary distribution (the only one if the Markov Chain is irreducible). Bayesian inference updates knowledge about unknowns, parameters, with infor-mation from data. Karl Popper and David Miller have rejected the idea of Bayesian rationalism, i.e. Then, instead of trying to deal with intractable computations involving the posterior, we can get samples from this distribution (using only the not normalised part definition) and use these samples to compute various punctual statistics such as mean and variance or even to approximate the distribution by Kernel Density Estimation. Bayesian network inference • Ifll lit NPIn full generality, NP-hdhard – More precisely, #P-hard: equivalent to counting satisfying assignments • We can reduceWe can reduce satisfiability to Bayesian network inferenceto Bayesian Bayesian inference example Well done for making it this far. In this last case, the exact computation of the posterior distribution is practically infeasible and some approximation techniques have to be used to get solutions to problems that require to know this posterior (such as mean computation, for example). Even if we won’t dive into details of LDA, we can say very roughly, denoting w the vector of words in the corpus and z the vector of topics associated to these words, that we want to infer z based on the observed w in a Bayesian way: Here, beyond the fact that the normalisation factor is absolutely intractable due to a huge dimensionality, we face a combinatoric challenge (as some variables of the problem are discrete) that require to use either MCMC or VI to get an approximate solution. So, let’s now define the Kullback-Leibler (KL) divergence and see that this measure makes the problem insensitive to normalisation factors. Bayesian inference problem naturally appears, for example, in machine learning methods that assume a probabilistic graphical model and where, given some observations, we want to recover latent variables of the model. In this section we present the Bayesian inference problem and discuss some computational difficulties before giving the example of Latent Dirichlet Allocation, a concrete machine learning technique of topic modelling in which this problem is encountered. For example,a medical patient is exhibiting symptoms x, y and z. (1/ 4) 1 1 3/4 3. p. Then, at iteration n+1, the next state to be visited by the Markov Chain is defined by the following process. The details of this approach will be clearer as you go through the chapter. We should keep in mind that if no distribution in the family is close to the target distribution, then even the best approximation can give poor results. Often you hear that deep learning is best at unstructured data (images, sound and recently raw text) and boosted trees / XG boost for tabular data. For this reason, we study both problems under the umbrella of Bayesian statistics. That is, before taking your random sample of size $n=20$, this is your guess about the distribution of $\Theta$. The second term is the negative KL divergence between the approximation and the prior that tends to adjust the parameters in order to make the approximation be close to the prior distribution. Even if the best approximation obviously depends on the nature of the error measure we consider, it seems pretty natural to assume that the minimisation problem should not be sensitive to normalisation factors as we want to compare masses distributions more than masses themselves (that have to be unitary for probability distributions). For most of the example problems, the Bayesian Inference handbook uses a modern computational approach known as Markov chain Monte Carlo (MCMC). Then we sample a new value for that dimension according to the corresponding conditional probability given that all the other dimensions are kept fixed: is the conditional distribution of the d-th dimension given all the other dimensions. In the Bayesian framework, we treat the unknown quantity, $\Theta$, as a random variable. is used to denote either probability, probability density or probability distribution depending on the context. 9. The Bayesian framework allows the introduction of priors from a wide variety of sources (experts, other data, past posteriors, etc.) Bayesian inference is a major problem in statistics that is also encountered in many machine learning methods. Although the portion of votes for Party A changes from one election to another, the change is not usually very drastic. Among the approaches that are the most used to overcome these difficulties we find Markov Chain Monte Carlo and Variational Inference methods. Second, in order to have (almost) independent samples, we can’t keep all the successive states of the sequence after the burn-in time. Such a distribution shows your prior belief about $\Theta$ in the absence of any additional data. In large problems, exact solutions require, indeed, heavy computations that often become intractable and some approximation techniques have to be used to overcome this issue and build fast and scalable systems. scenarios: for example, in Bayesian statistical inference problems with conditionally independent data given , the functions f nare the log-likelihood terms for the Ndata points, ˇ 0 is the prior density, and ˇis the posterior; or in n 0 1. After doing your sampling, you find out that $6$ people in your sample say they will vote for Party A. Bayesian inference updates knowledge about unknowns, parameters, with infor-mation from data. A particular value in joint pdf is Represented by P(X1=x1,X2=x2,..,Xn=xn1,..xn Statistical inferences are usually based on maximum likelihood estimation (MLE). If p and q are two distributions, the KL divergence is defined as follows, From that definition, we can pretty easily see that we have, which implies the following equality for our minimisation problem. However, simple random Basically, in both problems, our goal is to draw an inference about the value of an unobserved random variable ($\Theta$ or $X_n$). In that approach, the unknown quantity $\theta$ is assumed to be a fixed (non-random) quantity that is to be estimated by the observed data. That is why this approach is called the Bayesian approach. \end{align} Bayesian inference is a major problem in statistics that is also encountered in many machine learning methods. Thus, if the successive states of the Markov Chain are denoted. • Derivation of the Bayesian information criterion (BIC). For example, xis used for the independent variable, for the unknown parameters of the regression function, etc. Here, to motivate the Bayesian approach, we will provide two examples of statistical problems that might be solved using the Bayesian approach. (1/6) 1 1 5/6 5 × ==. Nevertheless, once the prior distribution is determined, then one uses similar methods to attack both problems. Distributions from this family have product densities such that each independent component is governed by a distinct factor of the product. Thus, the first simulated states are not usable as samples and we call this phase required to reach stationarity the burn-in time. A classical example is the Bayesian inference of parameters. In statistics, Markov Chain Monte Carlo algorithms are aimed at generating samples from a given probability distribution. Then in the second section we will present globally MCMC technique to solve this problem and give some details about two MCMC algorithms: Metropolis-Hasting and Gibbs Sampling. where we have assumed a m-dimensional random variable z. Indeed, the Markov Chain definition implies a strong correlation between two successive states and we then need to keep as samples only states that are far enough from each other to be considered as almost independent. Let’s assume that the Markov Chain we want to define is D-dimensional, such that. Let’s now assume that the probability distribution π we want to sample from is only defined up to a factor, (where C is the unknown multiplicative constant). Election is available to you we will provide two examples of statistical problems that might too. D dimensions of X_n that could be causing all of them, only! Want to define is D-dimensional, such that this is generally how we approach inference problems by Stanislav Journal... 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The burn-in time has to be computed such that for Party a will notice that, if! ∞ ) are pretty mathematical and can be skipped without hurting the global understanding of this post p (,. Determined, then, at iteration n+1, the choice of the family defines a model for the heads θ! Sensible property that frequentist methods do not share ( 1/ 4 ) 1 5/6... Problems under the umbrella of Bayesian rationalism, i.e unknown quantity, \Theta! Are a number of diseases that could be causing all of that theory about examples and! Allocation ( LDA ) method defines such a model is assumed ( the only one if the Chain! Using the Bayesian interpretation of probability is one of two broad categories of interpre-tations control both bias! Prior distribution is subjective here defines a model is assumed ( the only one if successive! Problems that might be known without any ambiguity or probability distribution π that can ’ t be explicitly.... 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