Bayesian Inference: Evidence Computation & Challenges

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Questions and Answers

In Bayesian inference, how is the evidence $P(x)$ rigorously computed, considering both the likelihood $P(x/\omega_j)$ and prior probabilities $P(\omega_j)$ across all classes?

Through the integration of the product of the likelihood $P(x/\omega_j)$ and the prior probability $P(\omega_j)$ over a subset of classes defined by a domain expert.
By maximizing the likelihood $P(x/\omega_j)$ for the most probable class $\omega_j$, effectively disregarding the prior probabilities.
Using a weighted average of the likelihoods $P(x/\omega_j)$, where weights are determined by an external validation dataset, and the prior probabilities are ignored.
By computing the sum of the product of the likelihood $P(x/\omega_j)$ and the prior probability $P(\omega_j)$ over all classes. (correct)

Considering a multi-class Bayesian classification problem with non-uniform priors, how does the aggregate evidence $P(x)$ influence the posterior probability $P(\omega_i | x)$ for a specific class $\omega_i$?

$P(x)$ acts as a scaling factor, disproportionately amplifying the posterior probability for classes with lower prior probabilities.
$P(x)$ directly maximizes the posterior probability for the class with the highest likelihood $P(x | \omega_i)$, irrespective of prior probabilities.
$P(x)$ serves as a normalizing constant, ensuring that the posterior probabilities across all classes sum to unity, thus calibrating the probabilistic outputs. (correct)
$P(x)$ is only relevant during the training phase and has no bearing on the posterior probability calculation during inference.

When dealing with high-dimensional data in a Bayesian framework, what computational challenges arise in calculating the evidence $P(x)$, and which advanced techniques are employed to mitigate these?

The primary challenge lies in the curse of dimensionality, necessitating the use of Markov Chain Monte Carlo (MCMC) methods or variational inference to approximate the intractable integral or sum. (correct)
High-dimensional data obviates the need for explicit evidence calculation as it becomes asymptotically equivalent to maximum likelihood estimation due to data sparsity.
The computational bottleneck is due to the need for high-precision arithmetic, which is addressed by using specialized hardware accelerators like quantum annealers to compute exact posteriors.
The only significant challenge is the requirement for large memory to store the prior probabilities, mitigated through lossless compression algorithms tailored to probability distributions.

In the context of Bayesian model comparison, how does the evidence $P(x)$ (also known as the marginal likelihood) serve as a critical metric for selecting the optimal model from a set of candidate models $M_i$?

The evidence $P(x|M_i)$ quantifies the probability of observing the data $x$ under the model $M_i$, inherently penalizing over-complex models through Occam’s razor by distributing probability across more parameter values. (A) Signup and view all the answers

Considering a scenario where the likelihood $P(x | \omega_j)$ is modeled using a Gaussian Mixture Model (GMM), and the prior $P(\omega_j)$ is a Dirichlet distribution, what analytical or computational method is typically employed to evaluate the evidence $P(x)$?

Variational inference techniques are applied to derive a lower bound on the evidence by optimizing a tractable approximation to the posterior distribution, thereby avoiding direct computation of $P(x)$. (A) Signup and view all the answers

In the context of Bayesian decision theory, what constitutes the optimal decision rule when minimizing the expected risk $R(\omega_j / x)$?

Selecting the class $\omega_j$ that minimizes the expected loss $R(\omega_j / x)$, effectively balancing the probabilities and associated losses for each class. (A) Signup and view all the answers

Within the risk function $R(\omega_j / x) = \sum_{i=0} L(\omega_i / \omega_j) P(\omega_i / x)$, how does the loss function $L(\omega_i / \omega_j)$ influence the decision-making process?

It quantifies the penalty incurred when classifying an instance as $\omega_j$ when its true class is $\omega_i$, thereby weighting the impact of classification errors. (D) Signup and view all the answers

Given the risk function $R(\omega_j / x) = \sum_{i=0} L(\omega_i / \omega_j) P(\omega_i / x)$, what is the implication of setting $L(\omega_i / \omega_j) = 0$ for all $i = j$ and $L(\omega_i / \omega_j) = 1$ for all $i \neq j$?

All classification errors are equally penalized, reducing the decision rule to maximizing the posterior probability $P(\omega_j / x)$. (B) Signup and view all the answers

Consider a scenario where $R(\omega_1 / x) = 0.2$ and $R(\omega_2 / x) = 0.3$. According to Bayesian decision theory, which class should be selected and why?

$\omega_1$ should be chosen because it is the class with the lower expected risk. (D) Signup and view all the answers

In a multi-class classification problem, how does the risk function $R(\omega_j / x)$ change with the introduction of a new class $\omega_{k+1}$ if $P(\omega_{k+1} / x) > 0$ and some $L(\omega_i / \omega_j) \neq L(\omega_i / \omega_{k+1})$?

The risk function is recomputed to account for the redistribution of posterior probabilities across all classes, potentially altering the minimum risk decision boundaries. (A) Signup and view all the answers

Flashcards

What is Evidence P(x)?

The total probability of observing the data x, across all possible classes.

What is Likelihood P(x|ωj)?

P(x|ωj) represents how likely the data x is, given that it belongs to class ωj.

What is Prior Probability P(ωj)?

P(ωj) represents the prior belief or probability of class ωj occurring before observing any data.

How to compute Evidence P(x)?

Compute P(x) by summing the product of the likelihood P(x|ωj) and the prior probability P(ωj) over all classes.