Stress testing MLE

Maximum likelihood estimation (MLE) is the workhorse of parameter estimation. It is versatile and comes with helpful bounds on its performance.

Deriving MLE

One way to define it is through Bayesian inference. In Bayesian inference, we are usually interested in the posterior distribution of the parameters given the data. \(P(\theta \mid D) = \frac{P(D \mid \theta) P(\theta)}{P(D)}\)

Although knowing the posterior would be very helpful, also in terms of uncertainty quantification, many times it is intractable to calculate the denominator \(P(D)\), called the evidence.

The evidence is an integral over the parameter space, \(P(D) = \int P(D \mid \theta) P(\theta)\, d\theta\).

So in the likely case that we cannot compute it analytically, we are left with three options: we can either approximate it, bound it or drop it.

To compute it, we need to evaluate the integral over the entire parameter space, which is usually impossible if the parameter space is large and high-dimensional: think every possible set of weights and biases that our neural network could have.

In the case of dropping it, we are left with the unnormalized posterior \(P(D \mid \theta) P(\theta)\), which is proportional to the true posterior. We are allowed to drop the evidence because it does not depend on the parameters \(\theta\), so it does not move the location of the maximum. Maximizing this product gives the maximum a posteriori (MAP) estimate \(\hat{\theta}_{MAP} = \arg\max_{\theta} P(D \mid \theta) P(\theta)\).

The prior is helpful if we have some educated guess about our parameters. But what if we do not have any educated guess? In that case, we can use a uniform prior and \(P(\theta)\) will be constant, hence it cancels out, and the MAP estimate collapses onto the maximum likelihood estimate \(\hat{\theta}_{MLE} = \arg\max_{\theta} P(D \mid \theta)\).

Now suppose the prior is normally distributed (or any other distribution for that matter) with mean \(\mu_\theta\) and variance \(\sigma_\theta^2\), then we essentially impose a constraint on our parameters.

Now what happens if we optimize this? So essentially, we are interested in maximizing the posterior distribution. If we assume that our data is independent and identically distributed (i.i.d.), then we can write the likelihood as a product over the data points.

In practice, it is not so convenient to take the product over many terms, hence usually we apply the negative logarithm, which turns the product into a sum and the maximization into a minimization problem. Note that the logarithm is a monotonically increasing function, so we do not alter the position of maxima and minima.

So now we essentially have two terms to optimize. The first term is the negative log-likelihood of the data, and the second term is the negative log-prior of the parameters (a quadratic penalty, exactly \(L_2\) regularization). The first one grows with the number of data points, while the second one is independent of \(n\) and controlled by the variance \(\sigma_{\theta}^2\). What happens if our number of data points \(n\) goes to infinity? To see this, let us divide by the number of data points \(n\) as we let \(n \to \infty\).

\(\lim_{n \to \infty} \frac{1}{n} \left( - \sum_{i=1}^{n} \log (P(x_i \mid \theta)) + \frac{(\theta - \mu_{\theta})^2}{2\sigma_{\theta}^2} \right) = - \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \log (P(x_i \mid \theta)) + \lim_{n \to \infty} \frac{1}{n}\cdot \frac{(\theta - \mu_{\theta})^2}{2\sigma_{\theta}^2}\) As \(n \to \infty\), the second term goes to \(0\), because the prior penalty is a fixed, finite quantity that gets divided by an ever-growing \(n\). So we are left with the first term (the average negative log-likelihood) and we can see that in the limit MAP approaches MLE: with enough data, the likelihood overwhelms the prior, so the regularized estimate collapses onto the unregularized one. Conversely, in the small-data regime the prior matters most, which is exactly when regularization helps.

KL-divergence

Another related concept is Kullback-Leibler (KL) divergence; a measure that aims to quantify the dissimilarity between two probability distributions.

Note that KL-divergence is not a metric, because it is not symmetric, i.e. \(D_{KL}(P \| Q) \neq D_{KL}(Q \| P)\).

However, it is very helpful in the context of information theory and machine learning, where we often want to minimize the difference between two probability distributions.

We can rewrite the KL-divergence as:

where \(H(P) = -\sum_x P(x)\log P(x)\) is the entropy of \(P\) and \(H(P,Q) = -\sum_x P(x)\log Q(x)\) is the cross-entropy between \(P\) and \(Q\).

If our data distribution is not changing, the entropy \(H(P)\) is constant. In that case, minimizing the KL-divergence is equivalent to minimizing the cross-entropy \(H(P, Q)\).

Looking closer at the cross-entropy, we can connect it to the negative log-likelihood. Let \(P\) be the true data distribution and \(Q = P(\cdot \mid \theta)\) our model. We never have access to \(P\) directly, but we do have samples \(x_1, \dots, x_n\) from it, which define the empirical distribution \(\hat{P}(x) = \frac{1}{n}\sum_{i=1}^n \mathbf{1}[x = x_i]\). Replacing \(P\) with \(\hat{P}\):

This is exactly the (averaged) negative log-likelihood. So, under the assumption that the entropy of the data is constant, minimizing the KL-divergence is equivalent to minimizing the negative log-likelihood. In other words, by maximizing the likelihood, we are minimizing the KL-divergence between the true distribution and our model. The two objectives differ only by the constant \(H(P)\), so the equivalence is exact in terms of the location of the optimum.

Convergence Guarantees of MLE

A large part of why MLE is trusted comes from its asymptotic guarantees. Under suitable regularity conditions (a correctly specified, identifiable model with smooth likelihood, and the true parameter \(\theta^*\) in the interior of the parameter space), the MLE has three desirable properties as \(n \to \infty\):

• Consistency. The estimator converges in probability to the true parameter: \(\hat{\theta}_{MLE} \xrightarrow{p} \theta^*\). Intuitively, the average log-likelihood converges to \(\mathbb{E}_{P}[\log P(x \mid \theta)]\), which (by the KL argument above) is maximized at the true \(\theta^*\).
• Asymptotic normality. The estimation error, suitably scaled, is Gaussian: \(\sqrt{n}\,(\hat{\theta}_{MLE} - \theta^*) \xrightarrow{d} \mathcal{N}\!\big(0,\ I(\theta^*)^{-1}\big),\) where \(I(\theta^*)\) is the Fisher information per sample. This is what justifies the usual confidence intervals and standard errors.
• Efficiency. The asymptotic variance \(I(\theta^*)^{-1}\) attains the Cramér-Rao lower bound: no (consistent, asymptotically unbiased) estimator can do better in the limit.

The important caveat is that all three rest on the model being correctly specified. The experiments below are essentially an exploration of what happens as we chip away at these assumptions.

Negative log-likelihood and cross-entropy loss

The previous two sections gave us two views of the same objective. From MLE we got the negative log-likelihood by taking logs of the product of per-sample likelihoods. From KL-divergence we got the cross-entropy by measuring the dissimilarity between the data distribution and our model. We saw that these coincide. This section makes the practical loss explicit, since the negative log-likelihood is the cross-entropy loss that we minimize when training most modern models.

Recall the starting point. For i.i.d. data, maximizing the likelihood is the same as minimizing its negative logarithm, averaged over the dataset:

This is the negative log-likelihood (NLL). It is a general objective: plug in whatever likelihood your model defines and you get a concrete loss. The name cross-entropy loss comes from rewriting this average as a cross-entropy between the empirical data distribution \(\hat{P}\) and the model \(Q = P(\cdot \mid \theta)\), exactly as we did above:

Classification: the categorical case

In a \(K\)-class classification problem the model outputs a probability vector \(\hat{y} = (\hat{y}_1, \dots, \hat{y}_K)\) over the classes, usually produced by a softmax on top of the network’s logits \(z\):

The label of a single example is a one-hot vector \(y\), which puts all its mass on the true class \(c\). The cross-entropy between the label distribution \(y\) and the prediction \(\hat{y}\) is

where the sum collapses to a single term because \(y_k = 0\) for every class except the true one. So for a one-hot label the cross-entropy loss is simply the negative log-probability the model assigns to the correct class. Averaged over the dataset this is precisely the NLL of a categorical likelihood:

Minimizing it pushes the predicted probability of the true class toward \(1\), and because of the softmax normalization this simultaneously pushes the other classes toward \(0\).

Binary case

For two classes the categorical loss reduces to the familiar binary cross-entropy. With a single sigmoid output \(\hat{y} = \sigma(z) \in (0, 1)\) interpreted as \(P(y = 1)\) and a label \(y \in \{0, 1\}\), the likelihood of one example is the Bernoulli \(\hat{y}^{y} (1 - \hat{y})^{1 - y}\), whose negative log is

Regression as a special case

The same recipe recovers the squared-error loss. If we model the target with a Gaussian likelihood \(P(y \mid x, \theta) = \mathcal{N}(y; f_\theta(x), \sigma^2)\) of fixed variance, the negative log-likelihood of one example is

and dropping the additive constant leaves the mean squared error. So MSE for regression and cross-entropy for classification are not two unrelated heuristics: they are the same NLL objective under a Gaussian and a categorical likelihood respectively. This is why the “cross-entropy loss” of deep learning is just MLE wearing a different hat, and why all the convergence guarantees above apply to the standard training loop.

Approximating the evidence

We already discussed that most of the time, for the complex models we usually work with, we can’t calculate the evidence exactly. However, we still can approximate it.

Evidence Lower Bound (ELBO)

One way to handle the evidence is to lower-bound it. We introduce an arbitrary distribution \(q(\theta)\) over the parameters (the variational distribution) and use Jensen’s inequality on the concave logarithm:

The right-hand side is the ELBO:

The first term, \(\mathbb{E}_{q}[\log P(D, \theta)]\), is the expected (complete-data) log-joint, often called the energy term. The second term, \(-\mathbb{E}_{q}[\log q(\theta)] = H(q)\), is the entropy of the variational distribution. So the ELBO trades off fitting the data (energy) against keeping \(q\) spread out (entropy).

How tight is this bound? The gap between the true log-evidence and the ELBO is exactly the KL-divergence between our variational distribution and the true posterior:

This is a beautiful result: because \(\log P(D)\) does not depend on \(q\), maximizing the ELBO is equivalent to minimizing the KL-divergence to the true posterior. So the ELBO does double duty: it both approximates the evidence and pushes \(q\) toward the posterior. This is the foundation of variational inference and, with a reparametrized \(q\), of the variational autoencoder.

Markov Chain Monte Carlo (MCMC)

Instead of approximating the evidence with a bound, MCMC sidesteps it entirely by sampling from the posterior. The key observation is that the posterior \(P(\theta \mid D) \propto P(D \mid \theta) P(\theta)\) is known up to the normalizing constant \(P(D)\), and many sampling schemes only need the unnormalized density.

The idea is to construct a Markov chain whose stationary distribution is the posterior. The classic recipe is Metropolis-Hastings: from the current state \(\theta\), propose a move \(\theta'\) from a proposal distribution \(g(\theta' \mid \theta)\) and accept it with probability

Notice that \(P(D)\) appears in both numerator and denominator and cancels, which is exactly why we never need the evidence. If the chain satisfies detailed balance and is ergodic, its samples converge in distribution to the posterior, and we can estimate any expectation \(\mathbb{E}_{P(\theta\mid D)}[f(\theta)]\) by averaging \(f\) over the samples. In high dimensions, gradient-aware variants such as Hamiltonian Monte Carlo and the No-U-Turn Sampler (NUTS) explore the space far more efficiently than random-walk proposals.

Importance sampling

Unlike the ELBO, which only lower-bounds the evidence, importance sampling gives an unbiased estimate of it. The trick is to rewrite the evidence integral as an expectation under an arbitrary proposal distribution \(q(\theta)\) (which we can sample from and evaluate):

Drawing \(\theta_1, \dots, \theta_K \sim q\), the Monte Carlo estimator is simply the average of the importance weights \(w_k = \frac{P(D, \theta_k)}{q(\theta_k)}\):

This is unbiased for any valid \(q\) (as long as \(q(\theta) > 0\) wherever \(P(D,\theta) > 0\)), and its variance is what makes or breaks it. The estimator is best when \(q\) closely matches the shape of the integrand \(P(D,\theta)\); if \(q\) is too narrow or misplaced, a single sample landing in a high-density region produces a gigantic weight, and the estimate is dominated by a handful of samples, giving catastrophically high variance, which is the usual failure mode in high dimensions. This sensitivity to the proposal is exactly what more elaborate schemes (annealed importance sampling, bridge/path sampling, sequential Monte Carlo) are designed to tame, by annealing from an easy distribution toward the posterior through a sequence of intermediate steps.

Importance-weighted ELBO (the IWAE bound)

The standard ELBO and importance sampling can be combined into a single estimator that gives a tighter lower bound. Notice that the plain ELBO is the expectation of the log of a single importance weight, \(\mathbb{E}_q\!\left[\log \frac{P(D,\theta)}{q(\theta)}\right]\). The importance-weighted bound (IWAE) instead averages \(K\) weights inside the log:

The intuition: the argument of the log is exactly the unbiased importance-sampling estimator of \(P(D)\) from above. By Jensen’s inequality (the concave-log version), taking the log of an average underestimates the average of the log, so \(\mathcal{L}_K \leq \log P(D)\) remains a valid lower bound. But averaging more samples reduces the variance inside the log, which shrinks the Jensen gap. The result is a chain of increasingly tight bounds:

So \(\mathcal{L}_1\) recovers the ordinary ELBO, and cranking up \(K\) interpolates smoothly toward the true log-evidence. The cost is \(K\) times more samples per gradient step, and the gradient signal-to-noise ratio degrades for very large \(K\), so in practice \(K\) is a modest number (e.g. 5 to 50). This is the bound used to train importance-weighted autoencoders.

Laplace approximation

A much cheaper, deterministic alternative is to approximate the posterior by a Gaussian centered at its mode. We already know how to find that mode: it is the MAP estimate \(\hat\theta_{MAP}\). The idea is to Taylor-expand the log-joint \(\ell(\theta) = \log P(D, \theta)\) to second order around \(\hat\theta\). Since \(\hat\theta\) is a maximum, the gradient vanishes, leaving

where \(\mathbf{H} = -\nabla^2_\theta \ell(\theta)\big\rvert_{\hat\theta}\) is the Hessian of the negative log-joint at the mode (its curvature, closely related to the Fisher information). Exponentiating, the integrand looks locally like an unnormalized Gaussian, and we can use the known Gaussian normalizing constant to evaluate the evidence integral in closed form:

where \(d\) is the dimension of \(\theta\). Taking logs gives the workhorse formula

The approximation is essentially free once we have the MAP estimate and the Hessian, and it is accurate whenever the posterior is well approximated by a single Gaussian bump, which, by the asymptotic normality of MLE/MAP, becomes increasingly true as \(n \to \infty\). It breaks down for multimodal, heavily skewed, or fat-tailed posteriors, where a single Gaussian centered at one mode misses most of the mass. (Dropping the prior and curvature terms and keeping only the leading \(n\)-dependence recovers the Bayesian Information Criterion, \(-2\log P(D) \approx -2\log P(D\mid\hat\theta) + d\log n\).)

Independent and identically distributed data

The assumption of independent and identically distributed (i.i.d.) data is a strong one, yet something that we can usually impose in practice (even in cases where it does not perfectly hold).

Defining independence mathematically is straightforward. \(P(X \cap Y) = P(X) \cdot P(Y)\) Essentially, two random variables are independent iff knowing the value of one does not give us any information about the other; their joint distribution is the product of their marginal distributions.

How would the definition of the joint distribution change if the random variables are dependent?

That means, if we want to know the joint distribution of our two dependent random variables, we need to know the conditional distribution of one given the other.

What about identically distributed? With that we mean that the random variables come from the same distribution with the same parametrization. Mathematically, this means:

\(F_X(x) = F_Y(x) \quad \forall x \in \mathbb{R}\) where \(F_X(x)\) and \(F_Y(x)\) are the cumulative distribution functions of \(X\) and \(Y\).

Please note that identically distributed does not mean that the random variables are independent, nor that they are equal.

Let us consider an example where we have identically distributed random variables that are not independent: \(X \sim \mathcal{N}(0,1)\) \(Y \sim \mathcal{N}(0,1)\) If \(X\) and \(Y\) are correlated (e.g., \(\text{Cov}(X, Y) = 0.5\)), they are still identically distributed as standard normals, but knowing \(X\) provides information about \(Y\). A particularly sharp version takes \(Y = Z X\) with \(Z\) a random sign: \(X\) and \(Y\) then have the same standard-normal marginal and are even uncorrelated, yet \(Y\) is completely determined by \(X\) (up to sign). The figure below shows exactly this: identical, uncorrelated marginals on the left, but a joint distribution that collapses onto the two lines \(y = \pm x\) on the right.

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wcWmQ73/G/x0s3iUFg3b+Nl24Sg8DSv8uhRbbz/f4/GQRWDi2y8T9iEFg5tMjuvxKDwMqhRdY/PzVeukkM2r+TGARWDi3Sv65H4XoUrsc/AiuHFtnO4z8ZBFYOLbLVv6AaL90kBvE/EoPAyqFF8r/6fmq8dJN4P2IQWDm0yARAJQaBlUOLzD8hsHJoke32P1pkO99Pjbc/ZDvfT42X4j/Jdr6fGi+tv8P1KFyPwsW/IbByaJHt6L+F61G4HoXbP90kBoGVQ+u/HVpkO99P5T9cj8L1KFzxP7ByaJHtfNc/5/up8dJN0r91kxgEVg7dPy2yne+nxtO/wcqhRbbz7T+yne+nxkv9vxsv3SQGgdW/QmDl0CLb/7/sUbgehev3v23n+6nx0vU/pHA9Ctej7D81XrpJDALnvz81XrpJDPi/uB6F61G4B8DHSzeJQWDhv1yPwvUoXANA16NwPQrX2z/sUbgehevhv8P1KFyPwt0/YOXQItv5+L81XrpJDALTv/LSTWIQWLk/arx0kxgEtr8dWmQ730/pPxSuR+F6FNY/x0s3iUFg5T9qvHSTGATmvyPb+X5qvOw/qvHSTWIQ+j8K16NwPQrvv9Ei2/l+auy/x0s3iUFg9T+mm8QgsHLIvxBYObTIdvY/JzEIrBxa3L9I4XoUrkf3P/hT46WbxMA/UI2XbhKD0D8K16NwPQr5P166SQwCK9e/6SYxCKwc7r8j2/l+arzcvyGwcmiR7dw/dZMYBFYO+b/8qfHSTWLkP3Noke18P+G/ke18PzVe8j/0/dR46SYDwMl2vp8aL+m/YOXQItv5+r8730+Nl27qv1g5tMh2vs8/tvP91Hjpxr/+1HjpJjHQvycxCKwcWsS/lkOLbOf7yT++nxov3STwvzm0yHa+n+Y/nu+nxks35T+kcD0K16PYP23n+6nx0uE/i2zn+6nx0j9I4XoUrkcAQBKDwMqhRba/2c73U+Ol079U46WbxCDov1CNl24Sg/C/gZVDi2zn87+mm8QgsHLsv5MYBFYOLbK/x0s3iUFg1T/pJjEIrByqP1CNl24Sg+i/zczMzMzM7D/ZzvdT46XnP3sUrkfhesS/TDeJQWDl5L/wp8ZLN4nhP6rx0k1iEMg/XrpJDAIr97+cxCCwcmixv/hT46WbxNA/zczMzMzM7L9SuB6F61HIP0jhehSuR/e/x0s3iUFg9T8rhxbZzvfzP/7UeOkmMdC/CKwcWmQ71z9I4XoUrkcDwOXQItv5fvK/N4lBYOXQ0r/0/dR46SbxvwwCK4cW2eY/wcqhRbbz/z+iRbbz/dTyv2IQWDm0yOq/FK5H4XoUzj8tsp3vp8b5P1pkO99PjfO/rBxaZDvfzz+JQWDl0CL9PzvfT42Xbvq/5dAi2/l+9L+JQWDl0CLbv3npJjEIrOC/arx0kxgE6j9g5dAi2/nOP2ZmZmZmZvy/exSuR+F64D/l0CLb+X7iv/yp8dJNYvQ/f2q8dJMY5L/8qfHSTWLkvx1aZDvfT+E/0SLb+X5q6D956SYxCKzcP2Dl0CLb+fq/nu+nxks34T8lBoGVQ4vwvxSuR+F6FM4/YhBYObTI9r8lBoGVQ4vcP23n+6nx0um/ukkMAiuH9L8MAiuHFtnmP2Dl0CLb+c4/2c73U+Ol47+e76fGSzf3P9NNYhBYOdy//Knx0k1ioD+e76fGSzfRP9ejcD0K1+O/vp8aL90k3j91kxgEVg7hv5HtfD81Xtq/AiuHFtnO9T8OLbKd76fwv+f7qfHSTQPALbKd76fG+T8xCKwcWmQEQOxRuB6F69m/y6FFtvP9/r/Xo3A9CtfTv+f7qfHSTdK/UrgehetRyL8CK4cW2c7xv4/C9Shcj+I/zczMzMzM4D+BlUOLbOf3v5HtfD81XuY/BoGVQ4tsAEBqvHSTGATGP8UgsHJokdW/kxgEVg4twr+uR+F6FK7jPxkEVg4tsvu/y6FFtvP9xD/2KFyPwvXYv8UgsHJokf0/EoPAyqFFxr/jpZvEILD6P0SLbOf7qfG/YhBYObTI4j/RItv5fmrUPwIrhxbZzuu/Di2yne+nxj+cxCCwcmjzPyPb+X5qvNS/30+Nl24S+78730+Nl26Sv3e+nxov3ey/F9nO91Pj1b/LoUW28/20v7Kd76fGS/u/QmDl0CLb+b8MAiuHFtnePyPb+X5qvOC/hetRuB6FBMAfhetRuB7pP5zEILByaNE/DAIrhxbZ5r/fT42XbhL1v42XbhKDwOo/vHSTGARW1j+q8dJNYhADQOF6FK5H4do/okW28/3U2D/HSzeJQWDFv76fGi/dJOo/AAAAAAAA5D/VeOkmMQj0P3npJjEIrOC/16NwPQrX278OLbKd76fevx1aZDvfT+k/K4cW2c739z/HSzeJQWDdvzMzMzMzM9u/f2q8dJMY1D+wcmiR7XzPvxBYObTIdu4/arx0kxgEAsAQWDm0yHbqv6AaL90kBum/j8L1KFyPAsAMAiuHFtnuv0jhehSuR+2/7nw/NV66yb8CK4cW2c7xP1yPwvUoXM+/5dAi2/l+8L/Jdr6fGi+tv2IQWDm0yPA/CKwcWmQ777/0/dR46SbtvxfZzvdT4+E/0SLb+X5qzL9OYhBYObTkP3npJjEIrIy/EFg5tMh25j+PwvUoXI/wv/p+arx0k4i/NV66SQwCy7/b+X5qvHTzvzVeukkMAvm/wcqhRbbz5T8QWDm0yHbWvycxCKwcWvC/+n5qvHSTuL8/NV66SQzyP3Noke18PwLAwcqhRbbz97/wp8ZLN4ntv8dLN4lBYPc/6SYxCKwc0j8lBoGVQ4voPz0K16NwPfK/7FG4HoXr8b956SYxCKzcPxkEVg4tsq0/xSCwcmiR4T+q8dJNYhDIv5huEoPAytE/001iEFg5xD9MN4lBYOXoP23n+6nx0uk/7FG4HoXr+b9g5dAi2/kBwNV46SYxCPA/NV66SQwC8z+8dJMYBFbwv8P1KFyPwv2/8tJNYhBYuT+YbhKDwMrtP/hT46WbxPw/UI2XbhKD4D8CK4cW2c7Xv/p+arx0k+y/ukkMAiuHhj+JQWDl0CLTvz0K16NwPfC/MQisHFpkAECPwvUoXI/8P5MYBFYOLfI/Rrbz/dR47b9cj8L1KFzrP3sUrkfheuQ/JzEIrBxa3D8AAAAAAAD0P1K4HoXrUeQ/rkfhehSu5z/8qfHSTWLkP23n+6nx0tU/JQaBlUOL/L8bL90kBoG1P5huEoPAyuG/exSuR+F69L+28/3UeOn6P0SLbOf7qfs/WDm0yHa+9T9SuB6F61HQPwRWDi2ynfU/+n5qvHSTiD+WQ4ts5/vJP7ByaJHtfPG/nMQgsHJo2T+4HoXrUbiuPwwCK4cW2fS/6SYxCKwcqr97FK5H4Xq0v/hT46WbxPw/z/dT46Wb7D+6SQwCK4eGP6wcWmQ7388/ukkMAiuHpj+WQ4ts5/vJvx1aZDvfT/G/hxbZzvdTw7+sHFpkO9/nvwAAAAAAAPS/JzEIrBxa4D+gGi/dJAbZP2Q730+Nl/y/sHJoke18v7+sHFpkO9/vP4ts5/up8fA/0SLb+X5q8D8rhxbZzvejPwrXo3A9Cuu/8tJNYhBY8b8UrkfhehTWP6jGSzeJQdg/ZDvfT42X9D+amZmZmZnxP0w3iUFg5cC/gZVDi2zn87/RItv5fmrUvy2yne+nxss/QmDl0CLbyb/l0CLb+X7iv/7UeOkmMdA/VOOlm8Qg4L9/arx0kxjkv4GVQ4ts59M/7nw/NV662b8IrBxaZDvPP0a28/3UeNE/001iEFg58r9iEFg5tMjevzVeukkMAvc/ZDvfT42X8r9WDi2yne8AwJhuEoPAyv2/GQRWDi2ynT9YObTIdr6fP2iR7Xw/Nb6/BoGVQ4ts8z/8qfHSTWIFwPLSTWIQWNk/YOXQItv5+D8/NV66SQzyv1K4HoXrUdi/f2q8dJMY6L/P91PjpZvsv3e+nxov3dS/okW28/3U9j8xCKwcWmT9Pxsv3SQGgdW/exSuR+F6/j8730+Nl26iP6rx0k1iEPw/AiuHFtnOt7/4U+Olm8TAP1yPwvUoXNc/O99PjZduCUAIrBxaZDvrPzm0yHa+n+a/NV66SQwC7z9eukkMAivXv1yPwvUoXN+/SgwCK4cW7T9YObTIdr6fP0Jg5dAi29E/eekmMQisjD/FILByaJHVPzMzMzMzM9s/y6FFtvP9/r/y0k1iEFjlPwaBlUOLbO+/30+Nl24S978ZBFYOLbKtvxsv3SQGgbU/aJHtfD815r9kO99PjZfqP0a28/3UePW/PzVeukkM2r+4HoXrUbjiv6rx0k1iEKi/QmDl0CLb0T9U46WbxCDwv166SQwCK+c/VOOlm8QgsD8Sg8DKoUX+v/LSTWIQWP+/+n5qvHSTiL99PzVeuknMv5HtfD81Xrq/eekmMQisnL8hsHJoke3MP7x0kxgEVu4/LbKd76fG8b+Nl24Sg8Dyv7ByaJHtfPG/5dAi2/l+0j/sUbgehevzPy/dJAaBldu/arx0kxgEBMDdJAaBlUP7vw4tsp3vp+q/QmDl0CLb4b/pJjEIrBzavyuHFtnO96M/K4cW2c73079iEFg5tMjwP28Sg8DKoeW/L90kBoGV678OLbKd76feP/p+arx0k/i/9ihcj8L12D+R7Xw/NV66PzeJQWDl0MK/nMQgsHJo+T+BlUOLbOfjv3sUrkfhegBAzczMzMzMzL8730+Nl270v+xRuB6F67E/nu+nxks38b+q8dJNYhDov5zEILByaNk/mG4Sg8DK4T+BlUOLbOfjvy/dJAaBle8/ukkMAiuH8j9g5dAi2/n2PyGwcmiR7eA/AiuHFtnO9T9/arx0kxj+vycxCKwcWtS/WDm0yHa+678QWDm0yHa+PxKDwMqhReK/c2iR7Xw/xb/pJjEIrBz+P8P1KFyPwsW/5dAi2/l+2j8ZBFYOLbLNv9v5fmq8dLM/+n5qvHSTeD956SYxCKzcP6RwPQrXo/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"},"y":{"dtype":"f8","bdata":"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In MLE, the i.i.d. assumption is powerful because it allows us to write the joint probability of the data as a product of individual probabilities: \(P(x_1, x_2, \ldots, x_n) = \prod_{i=1}^n P(x_i)\)

If we, on the other hand, have non-identically distributed random variables, the joint probability changes to: \(P(x_1, x_2, \ldots, x_n) = \prod_{i=1}^n P_i(x_i)\) Instead of having the same distribution for all random variables, we now have different distributions for potentially every random variable. That would mean we need \(n\) different models, one for each random variable. However, it is common practice to make use here of parameter sharing and essentially fit one model to all random variables, predicting a mean and potentially a variance for each random variable.

What about the dependent case? Now the joint distribution is modeled by using the chain rule: \(P(x_1, x_2, \ldots, x_n) = P(x_1) \prod_{i=2}^n P(x_i \mid x_{i-1}, x_{i-2}, \ldots, x_1)\) and that means we need to model the conditional distribution of each random variable given all previous ones, which has an exponential complexity of \(O(k^n)\), where \(k\) is the number of states a node can be in. Sometimes it is practical to assume the Markov property, i.e., the state at time \(t\) only depends on the state at time \(t-1\). This effectively reduces the computation to \(P(x_1, x_2, \ldots, x_n) = P(x_1) \prod_{i=2}^n P(x_i \mid x_{i-1})\) and has a linear complexity of \(O(k^2 n)\).

Experiments: General i.i.d. case

Note: the sections below describe the experimental setup and the behavior we expect MLE to exhibit. The empirical curves from running all four regimes are collected in the Results subsection at the end.

Let us first do some experiments for the general i.i.d. case. Consider a dataset consisting of images of cats and dogs.

We can surely argue that all of these images are identically distributed, as they all come from the same distribution, i.e., the distribution of cat and dog images. Now if we also make sure we shuffle the data, each time we feed it to the model we additionally impose independence.

Setup. Train a standard classifier with the negative-log-likelihood (cross-entropy) loss on shuffled mini-batches. This is the textbook regime, so it serves as our control: with the i.i.d. assumption satisfied and the model well specified, we expect MLE to behave exactly as the theory predicts: the average NLL on held-out data should track the training NLL, and the test error should decrease smoothly with \(n\). Expected outcome: clean convergence, calibrated estimates, no surprises. Everything after this is a controlled departure from this baseline.

From i.i.d. to n.i.i.d.

Now let us consider a dataset where we have identically distributed random variables, but they are not independent. Taking our example from before, we can quite easily achieve this by deciding to not shuffle the data. Thus, we create a dependency between the consecutive data points. Now, knowing the black-cat image at place 100, we can predict the white-collie image at place 101 with high certainty: essentially, we introduced a dependency between the data points.

But wait, you might say, did we not also break the assumption of identically distributed random variables? In a sorted/blocked ordering, yes: each mini-batch is then dominated by one class, so the batch distribution shifts over training, breaking identical distribution as well. And note that simply freezing a random permutation does not help either: a fixed shuffle of i.i.d. draws is still i.i.d., so the order alone introduces no statistical dependence (it only removes the re-randomization across epochs, which is an optimization effect, not a property of the data).

To isolate genuine dependence while keeping the marginal identical, we instead order the samples with a Markov chain over the class label. We walk through the dataset by staying in the current class with probability \(p_{\text{stick}}\) (say \(0.9\)) and switching otherwise, drawing an image of the current class at each step. Because the transition probabilities are symmetric, the stationary distribution of the chain is \(50/50\), which matches the balanced dataset, so every sample is still marginally drawn from the same class distribution (identical). But consecutive samples are now strongly class-correlated: knowing that step \(100\) is a cat makes step \(101\) very likely a cat too. That is real sample-to-sample dependence, and we feed the data in this fixed Markov order (no shuffling).

Expected outcome: with shared parameters and SGD, breaking independence mainly hurts the optimization (correlated gradients within a batch increase gradient variance across batches and slow/destabilize convergence) rather than the estimator’s target. The MLE solution it converges to is similar, but it gets there more slowly and noisily.

What about i.n.i.d?

Now let us consider a dataset where we have non-identically distributed random variables that are independent. This we can achieve by introducing different distributions as noise to our data points. In other words, we sample noise from different distributions (e.g., per-sample Gaussian noise with a different variance for each data point) and simply add it to our data points.

Expected outcome: because the samples are still independent, the likelihood still factorizes, so MLE remains well-behaved if the model accounts for the per-sample distribution (e.g., heteroscedastic noise via a predicted variance, as in the parameter-sharing comment above). If instead we (incorrectly) assume a single shared noise level, the model is misspecified: it will fit some compromise variance, the estimates remain consistent for the mean but the uncertainty estimates are wrong.

The endboss: n.i.n.i.d.

Now let’s lose both: identically distributed and independence. We can simply combine our approaches from before. We do not shuffle at all (this should already be sufficient to break both assumptions), and additionally we introduce different distributions as noise to our data points.

Expected outcome: this is the worst case for vanilla MLE, since both the factorized likelihood and the constant-distribution assumption are violated at once. We expect the largest gap between train and test NLL, the most unstable optimization, and the most poorly calibrated uncertainty. This sets up the random-walk model, which is a clean, analyzable instance of exactly this regime.

Results

We train the same small convolutional classifier under all four regimes, with \(5\) random seeds each, and plot the mean curves with \(\pm 1\) standard-deviation bands across seeds. The i.i.d. baseline (blue) trains smoothly and generalizes best. Breaking independence (N.I.I.D, green) mainly adds optimization noise, since correlated batches give noisier gradients, while the target is unchanged. Breaking identical distribution through per-sample heteroscedastic noise (I.N.I.D, orange) slows learning and widens the train/val gap. Losing both at once (N.I.N.I.D, red), where the class-sorted ordering makes each batch drift from all-cats to all-dogs, is the most unstable and the worst generalizing of the four, exactly as anticipated.

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Random walk model

Let us now consider a random walk model. Here the data we deal with is neither identically distributed nor independent, and on top of that it is non-stationary. This is a common scenario in time series analysis and, for instance, in modeling stock prices.

The model is simply \(x_t = x_{t-1} + \epsilon_t, \qquad \epsilon_t \sim \mathcal{N}(0, \sigma^2),\) so that, starting from \(x_0 = 0\), \(x_t = \sum_{s=1}^{t} \epsilon_s \sim \mathcal{N}(0,\ t\sigma^2).\)

Two things are immediately visible and both break our assumptions:

• Not identically distributed. The variance \(\text{Var}(x_t) = t\sigma^2\) grows linearly with \(t\). The marginal distribution of \(x_t\) literally changes with \(t\), which is non-stationarity.
• Not independent. \(\text{Cov}(x_s, x_t) = \min(s, t)\,\sigma^2 \neq 0\), so points are strongly correlated, and more so the longer the series.

Simulating a single 2D walk makes both points tangible: the trajectory (the positions) drifts and never settles, while the per-step increments form a clean, stationary Gaussian cloud whose MLE standard deviation matches the true step size.

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"},"y":{"dtype":"f8","bdata":"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"},"y":{"dtype":"f8","bdata":"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The very nature of this problem space makes it hard for MLE to find a good fit if we fit the wrong model. Suppose we naively treat the observations as i.i.d. \(\mathcal{N}(\mu, \sigma^2)\) and estimate a single mean and variance. The sample mean is not even a consistent estimator of anything meaningful here: its variance does not shrink with more data but actually grows, \(\text{Var}(\bar{x}) = \frac{\sigma^2}{n^2}\sum_{s,t}\min(s,t) \approx \frac{n\sigma^2}{3} \to \infty\), because the strong correlations mean we effectively have far fewer than \(n\) independent observations. The estimated variance, meanwhile, is dominated by the late, high-variance part of the series.

The right move is to fit MLE to the increments \(\Delta_t = x_t - x_{t-1} = \epsilon_t\), which are i.i.d. \(\mathcal{N}(0, \sigma^2)\). Then MLE recovers \(\hat{\sigma}^2 = \frac{1}{n}\sum_t \Delta_t^2\) and all the nice guarantees come back. This is the key practical lesson: MLE is not broken by dependence/non-stationarity per se; it is broken by misspecifying the likelihood. Choose the right factorization (here, condition on the previous step via the Markov property) and the problem becomes well-posed again.

Side by side, the two fits make the lesson concrete. Fitting a Gaussian to the increments recovers the true step size, while fitting the same Gaussian to the positions produces a mean and variance that describe nothing in particular: the histogram of positions is not even bell-shaped, and its spread is set by how far the walk happened to wander.

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MC28/3UeOkbQIXrUbgexSXATmIQWDk0HkDByqFFtrMkwJHtfD813h5AGy/dJAaBJcDP91PjpZseQFpkO99PzSXA0SLb+X7qHUACK4cW2U4owDMzMzMzMxxAc2iR7Xx/KMAIrBxaZDsfQEoMAiuHVijA+FPjpZsEIUAIrBxaZLsowGq8dJMYxCBAbxKDwMqhLMDufD81XjohQC/dJAaBFSzAy6FFtvP9IkBSuB6F65EswLx0kxgEFiNALbKd76dGLcApXI/C9eghQD0K16Nw/S3AN4lBYOWQIUD8qfHSTaIswMUgsHJokSFABoGVQ4tsLcD2KFyPwrUhQGiR7Xw/NS3A1XjpJjEIIUCq8dJNYhAswH0/NV66SR5AukkMAitHLMAfhetRuJ4fQARWDi2ynS3A7FG4HoUrIEDfT42XblIswDm0yHa+nyBA3SQGgZUDLsCYbhKDwMofQEjhehSuxyzAi2zn+6kxIEAK16NwPcoswIlBYOXQoiBAUrgehesRLMBCYOXQItsfQFyPwvUoXCzAUrgehesRIECBlUOLbOcswKwcWmQ73x9AQmDl0CKbK8BxPQrXo/AdQK5H4XoUrinADAIrhxbZIEAxCKwcWmQrwGiR7Xw/tSBAzczMzMwMK8AxCKwcWuQfQJhuEoPAyivA7FG4HoVrH0BaZDvfTw0rwEw3iUFgZR5AYOXQIts5KcBg5dAi2/keQIcW2c73UynA46WbxCDwIEDByqFFtrMowHWTGARWTiFA1XjpJjEIKcAfhetRuB4jQEa28/3UOCjA0SLb+X7qIkA/NV66ScwpwArXo3A9SiNA/tR46SbxKsBt5/up8ZIhQEa28/3UOCvAQmDl0CLbIUDjpZvEIPApwBfZzvdTIyJAbxKDwMohKcC28/3UeOkgQA4tsp3vJynA8KfGSzcJIUCYbhKDwEoowPT91HjpJiFAsHJoke18J8DZzvdT46UgQOF6FK5HISfAHVpkO98PIUDhehSuR2EowLByaJHtPCFAMzMzMzOzKMBeukkMAqseQMuhRbbzvSfAi2zn+6nxHUDy0k1iEJgowP7UeOkmMR1A2/l+arx0KMC0yHa+nxogQKwcWmQ7nyjA+n5qvHSTIEAAAAAAAEAnwFyPwvUoHCFArkfhehQuJsBvEoPAyqEgQClcj8L1aCXAbef7qfGSIEC8dJMYBFYkwKRwPQrXIx9Ad76fGi8dJcDXo3A9CtcgQI/C9ShcTyXAexSuR+F6IECWQ4ts5zslwC/dJAaBlR9A9P3UeOnmI8Av3SQGgRUdQFpkO99PDSTASOF6FK7HHUAOLbKd7yckwHnpJjEILBxAoBov3SQGJcCgGi/dJAYdQHWTGARWDibAN4lBYOVQHkAUrkfhepQnwGiR7Xw/NR1AmG4Sg8CKJ8BOYhBYObQaQKRwPQrX4ybARIts5/upGkAhsHJoke0nwFTjpZvEoBdAy6FFtvN9KcDufD81XroXQGZmZmZmpirAAAAAAAAAFUDhehSuR+EqwHWTGARWjhlAtMh2vp+aKsAUrkfhehQbQKRwPQrXYyrA2c73U+OlHECWQ4ts57srwGIQWDm0yBtARrbz/dR4K8BzaJHtfL8bQESLbOf7qSvA7FG4HoVrGkB56SYxCOwrwJHtfD813hhA8tJNYhBYLsB7FK5H4XoWQJHtfD813i3Af2q8dJOYGUB1kxgEVg4swH0/NV66yRlAtvP91HgpK8C0yHa+n5obQLTIdr6f2ivAaJHtfD81GEBt5/up8dIrwMl2vp8arxRAvp8aL90kLMBmZmZmZuYVQKrx0k1ikC3AEFg5tMj2FUBt5/up8VIswOOlm8QgsBZAzczMzMzMK8AAAAAAAIAYQMUgsHJoESrA5/up8dLNGEBKDAIrh9YowJMYBFYOrRhA/Knx0k1iKcDn+6nx0k0ZQD0K16Nw/SnAKVyPwvUoGEAMAiuHFpkqwMUgsHJokRNAy6FFtvN9KsAK16NwPQoRQJqZmZmZmSrAcT0K16PwE0AfhetRuB4rwEJg5dAi2xJAqMZLN4nBKcCmm8QgsPITQEjhehSuxyjAPQrXo3A9E0DVeOkmMQgowJHtfD813hFAaJHtfD+1KMCPwvUoXA8TQHWTGARWTinAZDvfT42XFEBEi2zn++kmwLgehetROBFAXrpJDAKrJ8D2KFyPwnUTQB1aZDvfzyfAYhBYObTIFUDfT42XbtIowIts5/upcRZARrbz/dT4KMDufD81XroWQDEIrBxapCjAfT81XrpJFECe76fGS7cpwC/dJAaBFRdAeekmMQhsKcCyne+nxksXQJ7vp8ZLdynAUI2XbhIDGECsHFpkO58qwFCNl24SAxZAxSCwcmhRKcA3iUFg5VAWQCuHFtnOdyjAtMh2vp8aFUBkO99PjRcnwLKd76fGyxVAXI/C9SicJsAxCKwcWuQWQA4tsp3vZyfAKVyPwvUoE0AIrBxaZHsowOxRuB6FaxZAO99PjZcuJ8DufD81XroVQAisHFpkeyfAzczMzMzMF0D0/dR46aYnwE5iEFg5NBVAiUFg5dBiJsAOLbKd7ycWQPT91Hjp5ijA1XjpJjGIFUA1XrpJDMIowMuhRbbzfRZAQmDl0CKbKMC4HoXrUTgVQEa28/3UuCjAHVpkO9/PFUA9CtejcP0owMUgsHJoERVAsHJoke28J8CTGARWDi0TQEoMAiuHFijA001iEFg5DkCyne+nxosnwPhT46WbxBBAc2iR7Xz/JsC0yHa+n5oSQAIrhxbZjibAfT81XrpJE0Av3SQGgRUmwI2XbhKDwBJAvp8aL93kJMB1kxgEVg4SQM/3U+OlWybAqMZLN4nBE0Coxks3iYEkwLx0kxgE1hFAz/dT46WbJMAQWDm0yHYQQLByaJHt/CTAdZMYBFaOEEB7FK5H4TomwBsv3SQGARBAGQRWDi2yJ8DufD81XroNQG8Sg8DK4SjA5dAi2/l+CUAnMQisHJoqwMdLN4lBYA5A8tJNYhAYKsBOYhBYObQGQHE9CtejsCrAoBov3SQGBEBvEoPAymErwBBYObTIdvw/x0s3iUGgKsCyne+nxkv5P5zEILByKCrAhetRuB6F/T9iEFg5tMgowJZDi2zn++0/JQaBlUMLJ8D0/dR46Sb5P0a28/3UeCbAtMh2vp8aBkBqvHSTGEQmwDEIrBxaZAlAGy/dJAYBJ8AhsHJoke0NQEoMAiuH1ibAVOOlm8QgDEC4HoXrUbgkwIGVQ4ts5wpAg8DKoUW2JMAfhetRuB4KQJ7vp8ZLdyXAPQrXo3A9DEDufD81XrokwA4tsp3vpw1A16NwPQoXJ8Dy0k1iENgQQHE9CtejcCfAbef7qfHSDEC8dJMYBNYmwBSuR+F6FA9A4XoUrkfhJ8CWQ4ts53sUQEoMAiuHFinAarx0kxgEEUDLoUW28z0qwFyPwvUo3BNAXrpJDAJrKsAEVg4tsh0TQAisHFpkeyrA7FG4HoXrEUB56SYxCCwrwKRwPQrXoxBAEFg5tMh2KsCuR+F6FK4SQPhT46WbhCvAKVyPwvUoE0B7FK5H4boqwOXQItv5/hBACKwcWmQ7K8DXo3A9CtcNQKAaL90khizAc2iR7Xw/DkDD9Shcj0ItwPyp8dJNYhBACtejcD1KLcCDwMqhRTYRQKJFtvP9lC3A4XoUrkfhD0B56SYxCKwtwHe+nxov3Q9AVOOlm8RgLsBiEFg5tMgQQG8Sg8DKoS3AmpmZmZkZEUBeukkMAusrwHnpJjEIrA9A2c73U+NlK8DVeOkmMQgOQLbz/dR4KSvAg8DKoUW2CkA/NV66SQwqwNEi2/l+agtAQmDl0CLbKMAnMQisHFoGQAwCK4cWWSnAYhBYObTIAkDhehSuR+EpwGq8dJMYBABAjZduEoMAKsAX2c73U+P/P5huEoPACirAcT0K16Nw+z9CYOXQItspwNNNYhBYOQFANV66SQzCKsAIrBxaZDsBQKJFtvP91CrAtMh2vp8aA0BMN4lBYOUqwArXo3A9CgNAiUFg5dDiK8BU46WbxCADQAIrhxbZzizA/tR46SYxBUC28/3UeOkswCcxCKwcWgVAWmQ730/NLcDFILByaJEIQFyPwvUoHC3ANV66SQwCDEBvEoPAyuEqwM3MzMzMzAtArkfhehSuKcDn+6nx0k0LQPCnxks3SSnA+FPjpZvEDECF61G4HgUpwGZmZmZmZg9Abef7qfFSKcCPwvUoXI8PQArXo3A9yijALbKd76dGE0BvEoPAyqEpwIcW2c73Uw9ABFYOLbIdKsDjpZvEILAPQHE9Ctej8CnAukkMAisHEEA1XrpJDMIowHsUrkfh+hFAAiuHFtnOKMAlBoGVQwsRQIGVQ4tsJynAhxbZzvdTEEDhehSuR2EpwJHtfD81XgpA16NwPQrXKcB56SYxCKwIQLpJDAIrByrADi2yne+nB0BCYOXQIlspwLpJDAIrhwVAsp3vp8bLKcAZBFYOLbIHQEjhehSuByrAj8L1KFyPCEDpJjEIrFwpwHNoke18PwpA6SYxCKwcKcAQWDm0yHYJQK5H4XoUbijAKVyPwvUoCkDJdr6fGu8owI/C9ShcjwlA9P3UeOkmKsCDwMqhRbYFQPCnxks3CSzAy6FFtvP9AkAzMzMzM7MqwPhT46WbxABAtvP91HjpKcCNl24Sg8AAQOkmMQisnCrAtMh2vp8aBkBI4XoUrgcqwP7UeOkmMf4/PQrXo3D9KMCamZmZmZn1P166SQwCKynAi2zn+6nx+j+e76fGS3cpwMHKoUW28wFA2/l+ary0KMBI4XoUrkf3PyGwcmiRLSnA8KfGSzeJ+T8bL90kBsEpwAIrhxbZzvc/TmIQWDn0KMC8dJMYBFb6Pzm0yHa+3ynAWDm0yHa++z8j2/l+anwqwGq8dJMYBPI/BFYOLbKdK8BSuB6F61H0P76fGi/dpCvAJzEIrBxa9D8K16NwPcoswOF6FK5H4fI/hxbZzvfTLcCLbOf7qfHmP8uhRbbz/S3AmpmZmZmZmT9t5/up8VItwOf7qfHSTfQ/d76fGi/dLMCBlUOLbOfnP4lBYOXQIi3AuB6F61G4nr/dJAaBlYMtwCUGgZVDi8w/BoGVQ4vsLMDjpZvEILDSv8dLN4lBoCzA5/up8dJN0j8fhetRuN4qwKRwPQrXo7C/lkOLbOe7K8CLbOf7qfGiv7bz/dR4qS7AMQisHFpk07/Jdr6fGu8uwHWTGARWDuG//Knx0k1CMMD2KFyPwvXkvxKDwMqhpTDA7nw/NV660b9CYOXQIhsxwI2XbhKDwNq/DAIrhxY5McD0/dR46SbBv+XQItv5fjLA2/l+arx0078EVg4tsh0ywArXo3A9Ct+/Gy/dJAbhMsDP91PjpZvUv7x0kxgEtjLAc2iR7Xw/3b/RItv5fkozwOXQItv5fuq/ppvEILByM8A9CtejcD30v5zEILByaDTAZmZmZmZm+r+q8dJNYnA0wKabxCCwcv6/YOXQItt5NMCYbhKDwMr9v7pJDAIrBzTAUrgehetR6L+28/3UeKkzwDEIrBxaZPe/BFYOLbL9NMC+nxov3ST4vxSuR+F69DTAuB6F61G4AMCLbOf7qdE0wJhuEoPAygPAke18PzWeNMBKDAIrhxYFwCUGgZVDazTAqvHSTWIQDMAX2c73U6M0wA4tsp3vpwzAYOXQIttZNcBU46WbxCAFwOXQItv5njXAiUFg5dAi/7+HFtnO9/M1wLByaJHtfADAI9v5fmpcNcAlBoGVQ4v+v6RwPQrXQzXA0SLb+X5q/L+mm8QgsPI1wPCnxks3if+/kxgEVg5tNsDByqFFtvP9v6abxCCwsjbA5/up8dJN/r/pJjEIrLw2wGQ730+Nl/6/Gy/dJAbBNsC28/3UeOkBwA4tsp3vRzfAxSCwcmiRBMDl0CLb+b42wN9PjZduEgPAwcqhRbZzNsC0yHa+nxr7v5huEoPACjfAoBov3SQG979OYhBYOZQ3wBkEVg4tsvm/g8DKoUVWN8DFILByaJHpv+58PzVeujfA30+Nl24S679mZmZmZiY3wNv5fmq8dPm/8tJNYhD4NsDsUbgehev7vylcj8L1aDfAjZduEoPA+r+wcmiR7Rw3wKAaL90kBgHAPzVeuknsNsCHFtnO91MAwCUGgZVDizfAPQrXo3A99L85tMh2vv82wLTIdr6fGvu/mG4Sg8BKN8CLbOf7qfH0vwrXo3A9ijfAokW28/3UAcCkcD0K1wM3wM3MzMzMzAHAHVpkO98PNsCR7Xw/NV4AwJ7vp8ZL9zXAsHJoke18479aZDvfTw02wPLSTWIQWPG/46WbxCDwNcAlBoGVQ4vgv4XrUbgehTbA30+Nl24S77/sUbgehUs2wFpkO99PjQTAZDvfT43XNsDHSzeJQWABwMP1KFyPIjfA4XoUrkfhB8B56SYxCCw2wIXrUbgehQ3AyXa+nxpvNsBqvHSTGAQPwDEIrBxaZDbAWDm0yHY+EMACK4cW2W42wG8Sg8DKIRDAYOXQItt5NsAX2c73U+MPwNEi2/l+qjfAH4XrUbgeDsCR7Xw/NV44wEw3iUFgZRPAWmQ730+tN8AQWDm0yPYVwIPAyqFFljfAUI2XbhKDF8CTGARWDu03wCUGgZVDixbAMzMzMzNzOMDNzMzMzMwVwMP1KFyPQjjAK4cW2c53E8AhsHJokW04wMdLN4lBYBHAxSCwcmiRN8CcxCCwcmgMwGZmZmZmRjfAz/dT46WbCsCamZmZmdk1wK5H4XoUrgHAz/dT46U7NsArhxbZzvf7vzvfT42X7jXAMQisHFpk+79CYOXQIts1wLByaJHtfPe/UI2XbhJjNcBYObTIdr71v2Q730+NNzXAnu+nxks35b8CK4cW2Q41wPYoXI/C9di/46WbxCCQNMDdJAaBlUPbPzMzMzMz8zPArkfhehSu4z9/arx0k9gzwPp+arx0k2g/MzMzMzOzM8B7FK5H4Xrgv4PAyqFFNjTAUrgehetRyL9QjZduEkM0wFpkO99Pjac/+n5qvHSTNMBOYhBYObTkPxsv3SQGgTTA16NwPQrX8z+YbhKDwCo0wF66SQwCK/U/Di2yne9nNMAGgZVDi2z1P8uhRbbz/TPANV66SQwC+z/Xo3A9Cpc0wJqZmZmZmQFA30+Nl26yNMB3vp8aL90EQLKd76fGCzXA0SLb+X5qB0CLbOf7qXE1wFYOLbKd7wdAK4cW2c6XNcCYbhKDwMoGQMdLN4lB4DXAkxgEVg4tBkCsHFpkO782wK5H4XoUrglAarx0kxhENsC28/3UeOkHQNv5fmq89DbAhxbZzvdTBUAIrBxaZHs2wG3n+6nx0gdAbxKDwMqhNsDRItv5fmoLQLpJDAIrJzfAK4cW2c73CEAK16NwPeo2wC/dJAaBlQhA9ihcj8I1N8DfT42XbhL9Pzm0yHa+3zbARIts5/up/z/l0CLb+b43wAisHFpkOwJAVOOlm8TAN8Dy0k1iEFgDQA4tsp3vRzfA/Knx0k1iAEAGgZVDi+w2wB+F61G4HgNAXrpJDAKLNsBEi2zn+6kJQBBYObTINjbA0SLb+X5qB0Cq8dJNYlA1wMuhRbbz/QJAsp3vp8aLNcCoxks3iUH+P0a28/3UmDXAEFg5tMh2A0Cmm8QgsPI0wB1aZDvfTwVAVOOlm8RANcC0yHa+nxoDQK5H4XoUjjXAtvP91HjpBkA1XrpJDAI2wJHtfD81XgpAoBov3SQGNsB1kxgEVg4LQPhT46WbRDXAKVyPwvUoCUAdWmQ73280wIXrUbgehQNAmpmZmZnZNMDRItv5fmr6PwwCK4cWOTXA30+Nl24S/z+0yHa+n9o0wMP1KFyPwgBAf2q8dJO4NMC+nxov3ST4P/yp8dJNYjXA+n5qvHST+j/2KFyPwrU0wESLbOf7qfc/uB6F61FYNcBiEFg5tMjmP4/C9ShcrzXA1XjpJjEI+D9Ei2zn+8k1wNv5fmq8dO8/8tJNYhAYNsCDwMqhRbbnP7x0kxgE9jXA5/up8dJN2j8OLbKd72c2wH0/NV66SdQ/sp3vp8YrNsBvEoPAyqHlP0w3iUFg5TXAz/dT46Wb3D9YObTIdr42wPp+arx0k6g/JzEIrByaNsACK4cW2c7jv5qZmZmZeTbAc2iR7Xw/7b9qvHSTGMQ2wA4tsp3vp9a/1XjpJjGoNsDJdr6fGi/VP42XbhKDIDfAeekmMQis7D956SYxCKw2wAisHFpkO+O/nu+nxku3NsDD9Shcj8LFP8HKoUW2UzbA6SYxCKwcqr+LbOf7qXE2wOXQItv5fua/16NwPQpXNsAv3SQGgZX5vwwCK4cW2TbApHA9Ctej8L/P91Pjpds2wPCnxks3ifO/8KfGSzfpNsAzMzMzMzPDP76fGi/dZDbAEFg5tMh2zr+PwvUoXI81wOkmMQisHNK/cT0K16MwNcA730+Nl27iv2IQWDm0qDXAZmZmZmZmxr8lBoGVQws2wOkmMQisHOK/uB6F61GYNcAnMQisHFryv65H4XoUTjbAK4cW2c73678pXI/C9Ug3wAAAAAAAAPS/ZDvfT413N8D4U+Olm8T+vy2yne+nBjjAcT0K16Nw+b/hehSuR+E3wOXQItv5fgDAPQrXo3CdN8AMAiuHFtn0v6jGSzeJQTjAJzEIrBxa9L/4U+Olm0Q3wIts5/up8e6/I9v5fmrcNsDfT42XbhLxv8P1KFyPQjfArkfhehSu378CK4cW2e42wHnpJjEIrOC/0SLb+X4qN8BSuB6F61H2v0oMAiuHFjfAQmDl0CLb+7+BlUOLbCc3wHsUrkfheva/WDm0yHa+NsAj2/l+arzwvwRWDi2ynTXAbxKDwMqh3b89CtejcP01wIGVQ4ts5+u/MzMzMzPTNMDdJAaBlUPzvwwCK4cW2TTAyXa+nxov1b89CtejcP0zwGDl0CLb+da/WDm0yHbeM8A3iUFg5dDyv6abxCCwUjTAWDm0yHa+9b/6fmq8dFM0wLx0kxgEVvi/AAAAAABANMCJQWDl0CLbv/hT46WbBDTASOF6FK5H0b8xCKwcWgQ0wGIQWDm0yNa/ZmZmZmZmM8AIrBxaZDvrv0oMAiuHdjPA/Knx0k1i9r8lBoGVQ2szwHE9CtejcPW/L90kBoH1MsCDwMqhRbbnv1TjpZvEoDLA7FG4HoXrsT9oke18P9UxwMl2vp8aL80/EoPAyqGlMcCuR+F6FK7nv+kmMQisfDHALbKd76fG57/b+X5qvFQywA4tsp3vp+6/JzEIrBzaMcBzaJHtfD/5v9v5fmq8NDHAcT0K16Nw9b9SuB6F61ExwGQ730+Nl/a/SOF6FK7nMMBqvHSTGAQBwPhT46WbxC/AEoPAyqFFBMAxCKwcWkQwwGIQWDm0yP6/WDm0yHa+L8CTGARWDi30v6abxCCwsi7AXrpJDAIr8b/TTWIQWPkvwIts5/up8fS/LbKd76fGL8BKDAIrhxb1v0SLbOf76TDARrbz/dR46b8GgZVDiwwwwPyp8dJNYrC/AiuHFtmOMMCDwMqhRbbDP+XQItv5njHAK4cW2c737z+e76fGS5cywEjhehSuR/M/w/UoXI8CMsCamZmZmZn9P0oMAiuH1jHAaJHtfD818j8X2c73U+MxwD81XrpJDPA/TmIQWDm0McDByqFFtvP3P3Noke18XzHAEFg5tMh2BEBeukkMAgsxwLbz/dR46QJA7nw/NV46MMDZzvdT46UBQIPAyqFFFjDAx0s3iUFgAkDP91PjpTsxwHsUrkfhegNAEFg5tMgWMcCkcD0K16MCQGIQWDm0KDHANV66SQwC+z+R7Xw/NR4xwIXrUbgehQNAokW28/00McA730+Nl24BQMP1KFyPgjHA5dAi2/l+CUDjpZvEIDAywOF6FK5H4QxAbxKDwMohMsBI4XoUrkcSQCcxCKwcejHAtMh2vp8aEkBzaJHtfD8ywBBYObTI9hRA+FPjpZuEMcCNl24Sg8ASQEw3iUFgJTHA9P3UeOkmEUBiEFg5tIgxwHe+nxov3QtAxSCwcmgRMcDByqFFtvMMQFK4HoXr0TDA4XoUrkdhEUAj2/l+anwwwD81XrpJjBJAI9v5fmocMMDsUbgehWsQQBsv3SQGITDAarx0kxgEDECHFtnO95MwwNnO91PjpQtAPzVeukkMMMBxPQrXo/ARQNejcD0KlzDAfT81XrpJEkDHSzeJQaAwwNejcD0KVxNAL90kBoFVMMDFILByaBEUQMUgsHJo0TDAQmDl0CLbFECsHFpkO18wwIcW2c730xNAarx0kxjELsBcj8L1KFwTQGZmZmZmZi/AwcqhRbbzE0AOLbKd76cvwJ7vp8ZLNxFAyXa+nxqvL8CcxCCwcmgUQK5H4XoUri3ALbKd76dGE0Atsp3vp8YswArXo3A9ihNAy6FFtvN9LMD6fmq8dBMWQKRwPQrXoyvA/Knx0k1iFkDFILByaJEswIlBYOXQIhRAbxKDwMqhLMBiEFg5tMgUQFTjpZvEICzADAIrhxZZFkBmZmZmZiYrwArXo3A9ihFA5dAi2/k+KsAX2c73U+MOQG8Sg8DKISnAH4XrUbgeEECNl24Sg4AowN9PjZduEg5AK4cW2c43KMDP91PjpZsGQDm0yHa+HyfASOF6FK5HBUCDwMqhRbYpwLpJDAIrhwlA8KfGSzcJKMCgGi/dJAYGQN9PjZdu0ibAoBov3SQGA0DXo3A9ChcpwGQ730+NlwFA3SQGgZXDKMBU46WbxCACQClcj8L1qCjAHVpkO99PA0CLbOf7qXEowKabxCCwcgJAJQaBlUMLKcD8qfHSTWIBQEw3iUFgZSnAkxgEVg4tAEAAAAAAAIAqwAAAAAAAAPw/0SLb+X5qKsB7FK5H4Xr+PwisHFpkeyrANV66SQwC/T8ZBFYOLfIpwMP1KFyPwgFAarx0kxhEKcCF61G4HoX1P0a28/3UuCnAXrpJDAIr+z/b+X5qvDQpwOf7qfHSTQBAMQisHFrkKMAtsp3vp8b5P+xRuB6FqyfAbef7qfHS7T/Jdr6fGq8mwPCnxks3ie0/Di2yne8nKMAzMzMzMzPnP8HKoUW28ybAL90kBoGV2z9Ei2zn+ykowJHtfD81XtI/CKwcWmT7KMAZBFYOLbLVvyUGgZVDiyjACKwcWmQ7rz9/arx0kxgpwM/3U+Olm9Q/CKwcWmR7KcBGtvP91HjRP1yPwvUoXCnA5/up8dJN8D8nMQisHNoowKRwPQrXo+A/vHSTGARWKMCTGARWDi3aP1pkO99PzSjAukkMAiuHhr+HFtnO99MowMuhRbbz/aS/exSuR+E6KcAv3SQGgZXnv+Olm8Qg8CrA8KfGSzeJ5b/D9Shcj8IqwDeJQWDl0No/46WbxCDwKsDD9Shcj8LtP1g5tMh2vinA+FPjpZvE7D/RItv5fqopwBfZzvdT4+U/MQisHFrkJ8DsUbgeheuxv65H4XoU7ifAgZVDi2zn4z99PzVeugkowMUgsHJokdU/dZMYBFbOJ8AzMzMzMzPjP7ByaJHtPCfAg8DKoUW25z+wcmiR7XwmwAaBlUOLbPM/8tJNYhCYJ8D8qfHSTWLwP2Dl0CLbuSbA5/up8dJN9D89CtejcH0mwK5H4XoUrvk/fT81XrqJJcA3iUFg5dDyP2q8dJMYxCbAXrpJDAIr4z8OLbKd7+ckwDvfT42XbrI/MQisHFokJcAbL90kBoGVv1yPwvUoHCTAexSuR+F67L9mZmZmZiYkwOkmMQisHNK/YhBYObQII8ArhxbZzvezv7Kd76fGSyPAbxKDwMqh6T8rhxbZzvckwA4tsp3vp/Q/9ihcj8J1JMAxCKwcWmT5P4ts5/upMSTAoBov3SQG+z9QjZduEkMjwAIrhxbZzvE/w/UoXI+CJMC0yHa+nxrzP2iR7Xw/dSTA0SLb+X5q9D8EVg4tst0iwFpkO99Pje8/BFYOLbIdIsAMAiuHFtnyP+XQItv5fh/AukkMAiuH3j8lBoGVQ0sgwKwcWmQ7388/ppvEILDyHcDP91PjpZvEv28Sg8DKoRrA0SLb+X5q5D9mZmZmZuYYwAaBlUOLbOs/UI2XbhIDHMC8dJMYBFbuP8dLN4lB4B7AUI2XbhKD6D8rhxbZzncewC/dJAaBleM/xSCwcmiRHMAbL90kBoHhP0SLbOf7qR/AnMQgsHJokT9CYOXQIhshwD81XrpJDMK/nu+nxku3H8B56SYxCKy8P/Cnxks3CR/Ay6FFtvP94L/NzMzMzEwawHE9CtejcOG/EFg5tMh2FcCHFtnO91Pvv5qZmZmZmRbAJQaBlUOL1D+Nl24Sg8AYwMuhRbbz/eA/IbByaJFtGcBYObTIdr6/P7TIdr6fGhXA/tR46SYx2L8dWmQ7308TwBKDwMqhRcY/vHSTGARWFcDNzMzMzMzov0w3iUFg5RjAYhBYObTI8r/ByqFFtnMYwOkmMQisHPC/hxbZzvfTGMCLbOf7qfHwv+F6FK5H4RbAmpmZmZmZ5b85tMh2vh8WwLByaJHtfO+/xSCwcmiRFMD6fmq8dJPsv6rx0k1iEBbAtvP91Hjp8r/P91PjpZsTwPp+arx0k9C/RIts5/upFMD4U+Olm8T0v8uhRbbzfRPAAAAAAAAA/L+gGi/dJAYSwMuhRbbz/f6/iUFg5dAiC8B/arx0kxj+v4XrUbgehQfAtvP91HjpA8BmZmZmZmYRwJHtfD81XgPAx0s3iUHgEMCwcmiR7XwAwJZDi2zn+xHADi2yne+nBcDJdr6fGi8VwHE9CtejcAPA3SQGgZXDE8CLbOf7qfEGwFK4HoXr0RTAQmDl0CLbCcDP91PjpZsYwKrx0k1iEAnA0SLb+X7qFsD2KFyPwvUJwN0kBoGVwxnAjZduEoPACsDwp8ZLNwkXwAisHFpkOwLA5dAi2/l+FsD8qfHSTWIFwCuHFtnOdxrAxSCwcmiRC8C+nxov3aQawESLbOf7qQrAbxKDwMohGsC4HoXrUbgLwJ7vp8ZLNx3AUI2XbhIDEcAGgZVDi+wcwClcj8L1qBLAHVpkO99PGsAUrkfhepQVwHE9Ctej8BbAtvP91HjpFsDLoUW2830WwBBYObTIdhbArkfhehQuF8DLoUW2830awL6fGi/dJBjAppvEILByF8Boke18P7UawG8Sg8DKIRvAvp8aL90kG8Db+X5qvPQbwAisHFpkux3A9ihcj8J1HcD+1HjpJrEewH0/NV66SR7Ake18PzVeHsDHSzeJQeAcwGQ730+Nlx3A/tR46SYxHMAIrBxaZDsawAAAAAAAABrASOF6FK4HIMDufD81XroawEJg5dAimyHAtMh2vp8aHMAv3SQGgdUgwDVeukkMghzA3SQGgZVDIcCsHFpkO98ZwGq8dJMYxCLAj8L1KFwPGsArhxbZzjciwFCNl24SAxrAbxKDwMphI8B1kxgEVg4awG3n+6nxkiLA8tJNYhBYGcCoxks3iUEiwO58PzVeuhjArBxaZDsfIsCDwMqhRTYbwK5H4XoUbiLADAIrhxZZGsAlBoGVQ4slwLx0kxgE1hjAaJHtfD81JcAbL90kBgEbwE5iEFg59CXAtvP91HhpG8AK16NwPYolwClcj8L1KBzAObTIdr5fJMCiRbbz/VQbwClcj8L1KCXALbKd76dGHMA730+Nl+4kwFK4HoXr0RrAjZduEoNAJMBQjZduEgMbwCcxCKwcGiTAqvHSTWIQG8BI4XoUrkcjwPCnxks3CR3AIbByaJGtIsAfhetRuF4gwGDl0CLbOSLAd76fGi8dIMDufD81XjoiwOXQItv5PiLAKVyPwvWoIMAMAiuHFlkhwIlBYOXQoiDApHA9CtfjIMBt5/up8ZIhwFTjpZvEYCDA3SQGgZUDIMB1kxgEVo4hwL6fGi/dpBrAy6FFtvP9IcCwcmiR7fwYwMUgsHJo0SLAaJHtfD81GMBkO99PjRckwKAaL90khhnA30+Nl25SJMC8dJMYBNYawB1aZDvfjyTA/Knx0k3iGMDXo3A9CtclwC2yne+nRhvAmG4Sg8DKJcCBlUOLbGccwLKd76fGyybApHA9CtcjHMBcj8L1KFwnwCuHFtnOdxvAZmZmZmZmJ8CsHFpkO98bwCuHFtnOdyfAbef7qfHSGcCBlUOLbCcmwL6fGi/dJBrAyXa+nxovJ8BeukkMAisdwF66SQwCaynAhetRuB6FH8CF61G4HkUqwDm0yHa+nx/AZmZmZmYmLMDdJAaBlcMcwEoMAiuHFivAEoPAyqFFGsCamZmZmZkrwGIQWDm0yBnAPzVeukmMLMDjpZvEILAWwMP1KFyPwi3Ag8DKoUW2F8BKDAIrh1YswAisHFpkuxLAIbByaJFtLMCq8dJNYpATwFYOLbKd7y7ApHA9CtcjEMBU46WbxKAuwJzEILBy6BPAqMZLN4nBLcCuR+F6FK4SwFyPwvUoHC7Asp3vp8bLE8DhehSuR2EtwIcW2c73UxPAvp8aL93kLMDHSzeJQeAQwDMzMzMzsyvAg8DKoUW2DsAEVg4tsh0rwLbz/dR46RHACKwcWmQ7LMAtsp3vp0YQwC/dJAaBlSvAexSuR+F6DsBYObTIdr4qwAwCK4cWWRDAIbByaJEtKsBt5/up8VISwARWDi2yHSrAEFg5tMh2EMAlBoGVQ8sqwB+F61G4HgnAnMQgsHKoKcBvEoPAyqEDwB+F61G43inAyXa+nxovCcDFILByaJEqwGDl0CLb+QbADAIrhxbZKsB/arx0kxgEwKabxCCwsirA2/l+arx0BsDdJAaBlcMpwD81XrpJDAHA5dAi2/l+KsAAAAAAAAD+v0oMAiuHVirAWDm0yHa++7/+1HjpJnEqwBfZzvdT4wLAYhBYObQIKcCq8dJNYhAHwIGVQ4ts5ynAO99PjZduBMDTTWIQWLkpwARWDi2ynQfAbxKDwMphKsCoxks3iUEIwMUgsHJo0SrARIts5/upBcAIrBxaZPsrwJ7vp8ZLNwTAw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How does this affect the way we use MLE? It tells us that the assumptions live in the model, not in MLE itself. The work is in choosing a likelihood whose factorization matches the dependence structure of the data.

Theoretical Background

Variance of a random variable

The variance of a random variable \(X\) is defined as: \(\text{Var}(X) = \mathbb{E}[(X - \mu_X)^2]\) where \(\mu_X = \mathbb{E}[X]\) is the expected value of \(X\).

We can further express the variance as: \(\text{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2\)

Derivation

Metric

A metric is a function that defines a distance between any two elements of a set. The following properties must hold for a metric \(d(x, y)\):

• Non-negativity: \(d(x, y) \geq 0\)
• Identity of indiscernibles: \(d(x, y) = 0 \iff x = y\)
• Symmetry: \(d(x, y) = d(y, x)\)
• Triangle inequality: \(d(x, z) \leq d(x, y) + d(y, z)\)

Entropy

The entropy of a random variable \(X\) is defined as: \(H(X) = - \sum_{x} P(x) \log (P(x))\) It is a measure of the uncertainty of a random variable. The higher the entropy, the more uncertain the random variable is.

How can we see this? Let’s consider a fair coin flip (using \(\log_2\), so entropy is measured in bits). Then \(P(H) = P(T) = 0.5\). So the entropy is: \(H(X) = - (0.5 \log_2 0.5 + 0.5 \log_2 0.5) = - (0.5 \cdot (-1) + 0.5 \cdot (-1)) = 1 \text{ bit}\) Now consider a fully biased coin. Then \(P(H) = 1\) and \(P(T) = 0\). So the entropy is (with the convention \(0 \log 0 = 0\)): \(H(X) = - (1 \log_2 1 + 0 \log_2 0) = - (1 \cdot 0 + 0) = 0\) This means that the entropy is \(0\) when the random variable is deterministic (no uncertainty), and maximal when the random variable is uniformly distributed (maximum uncertainty). For a variable with \(k\) outcomes the maximum is \(\log_2 k\); for the binary coin that maximum is \(\log_2 2 = 1\) bit.

Softmax

The softmax function turns a vector of real-valued scores (logits) \(z = (z_1, \dots, z_K)\) into a probability distribution over \(K\) classes: \(\text{softmax}(z)_k = \frac{e^{z_k}}{\sum_{j=1}^{K} e^{z_j}}.\) Each output lies in \((0, 1)\) and the outputs sum to \(1\), so they can be read as class probabilities. Exponentiating makes every score positive and accentuates differences: a logit that is larger than the rest dominates the distribution, while equal logits give the uniform distribution. The function is shift-invariant, \(\text{softmax}(z + c) = \text{softmax}(z)\) for any constant \(c\), which is why implementations subtract \(\max_k z_k\) before exponentiating for numerical stability. It is the multi-class generalization of the sigmoid \(\sigma(z) = \frac{1}{1 + e^{-z}}\), which is the \(K = 2\) case, and it is the standard final layer pairing with the cross-entropy loss.

Jensen Inequality

Jensen’s inequality is a property of convex functions that can also be used in the context of random variables. It states that for a convex function \(f\) and a random variable \(X\), the following inequality holds: \(f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]\)

Proof. By definition of a convex function, at any point \(\mu\) in the domain of \(f\) there is a supporting line (tangent) \(g(x) = f(\mu) + f'(\mu)(x - \mu)\) that lies below \(f\) everywhere: \(g(x) \leq f(x)\) for all \(x\).

Now choose the anchor point \(\mu = \mathbb{E}[X]\) and take expectations of both sides of \(g(X) \leq f(X)\): \(\mathbb{E}[f(X)] \geq \mathbb{E}[g(X)] = f(\mathbb{E}[X]) + f'(\mathbb{E}[X])\big(\mathbb{E}[X] - \mathbb{E}[X]\big) = f(\mathbb{E}[X]).\) The middle term vanishes because \(\mathbb{E}[X] - \mathbb{E}[X] = 0\), leaving \(\mathbb{E}[f(X)] \geq f(\mathbb{E}[X])\), which is the claim. (For a concave function such as \(\log\), the inequality flips, which is the version we used for the ELBO.)

Markov Inequality

The Markov inequality states that for a non-negative random variable \(X\) and any \(a > 0\), the probability that \(X\) takes on a value greater than or equal to \(a\) is at most \(\frac{\mathbb{E}[X]}{a}\). \(P(X \geq a) \leq \frac{\mathbb{E}[X]}{a}\)

It is most useful for tail bounds when only the mean is known. For example, if household income is non-negative with mean $50,000, then the probability that a household earns at least $200,000 is at most \(\frac{50000}{200000} = 0.25\), without assuming anything about the shape of the distribution. The bound is tight (achieved with equality) for the two-point distribution that puts mass \(\frac{\mathbb{E}[X]}{a}\) on \(a\) and the rest on \(0\); for most smooth distributions it is quite loose, which is why sharper bounds (e.g., Chebyshev, which uses the variance) are often preferred when more is known.

Cauchy-Schwarz Inequality for Random Variables

Generally, the Cauchy-Schwarz inequality provides an upper bound on the absolute value of the inner product between two vectors.

Let \(u\) and \(v\) be two vectors in \(\mathbb{R}^n\). Then the Cauchy-Schwarz inequality states that: \(\mid \langle u, v \rangle\mid \leq \|u\| \|v\|\) where \(\langle u, v \rangle\) is the inner product of \(u\) and \(v\), and \(\|u\|\) and \(\|v\|\) are the Euclidean norms of \(u\) and \(v\), respectively.

We can use the Cauchy-Schwarz inequality to bound the covariance of two random variables \(X\) and \(Y\): \(\mid \text{Cov}(X, Y)\mid \leq \sqrt{\text{Var}(X) \text{Var}(Y)} = \sigma_X \sigma_Y\)

Proof: Let \(\mu_x = \mathbb{E}[X]\) and \(\mu_y = \mathbb{E}[Y]\). Then we can write \(\text{Cov}(X, Y) = \mathbb{E}[(X - \mu_x)(Y - \mu_y)]\) \(\text{Cov}(X, Y)^2 = \mathbb{E}[(X - \mu_x)(Y - \mu_y)]^2 \leq \mathbb{E}[(X - \mu_x)^2] \, \mathbb{E}[(Y - \mu_y)^2] = \text{Var}(X) \text{Var}(Y)\) where the inequality follows from the Cauchy-Schwarz inequality (treating the centered variables as vectors under the inner product \(\langle U, V \rangle = \mathbb{E}[UV]\)). Taking square roots gives \(\mid \text{Cov}(X,Y)\mid \leq \sigma_X \sigma_Y\).

Thus we have a bound on the covariance of two random variables. Dividing through, this is exactly the statement that the correlation coefficient \(\rho = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}\) lies in \([-1, 1]\).

Leibniz Rule

The Leibniz rule states that the derivative of an integral with respect to a parameter is equal to the integral of the derivative of the integrand with respect to the parameter. \(\frac{d}{d\theta} \int f(x; \theta)\, dx = \int \frac{d}{d\theta} f(x; \theta)\, dx\)

This rule holds as long as the integrand is differentiable with respect to the parameter and suitable regularity conditions hold (so that differentiation and integration can be interchanged, e.g., dominated by an integrable function, and the domain of integration not depending on \(\theta\)). It is a powerful tool in calculus and is often used in the context of probability theory and statistics, especially when dealing with expectations and likelihood functions, as it allows us to interchange differentiation and integration.

Score Function

The score function is defined as the gradient of the log-likelihood function with respect to the parameters: \(S(\theta) = \nabla_{\theta} \log p(x; \theta) = \frac{\partial \ln p(x; \theta)}{\partial \theta}\) where \(p(x; \theta)\) is the probability density function of the data \(x\) given the parameters \(\theta\). It indicates how sensitive the likelihood function is to changes in the parameters. The score function has the nice property that its expected value is zero, which can be shown using the Leibniz rule.

Fisher Information

The variance of the score function is called the Fisher information. The score function is the gradient of the log-likelihood function with respect to the parameters, and it has the nice property that its expected value is zero.

Proof

Let \(p(x; \theta)\) be the probability density function of the data \(x\) given the parameters \(\theta\). Then we can write \(S(\theta) = \nabla_{\theta} \log p(x; \theta) = \frac{\partial \ln p(x; \theta)}{\partial \theta} = \frac{1}{p(x;\theta)}\frac{\partial p(x;\theta)}{\partial \theta}\)

The expectation of the score is: \(\mathbb{E}[S(\theta)] = \int S(\theta) p(x; \theta)\, dx = \int \frac{1}{p(x;\theta)}\frac{\partial p(x; \theta)}{\partial \theta} p(x; \theta)\, dx = \int \frac{\partial p(x; \theta)}{\partial \theta}\, dx\)

Under the assumption that \(p(x; \theta)\) is regular (differentiable, with continuous derivative), we can use the Leibniz rule to pull the derivative outside the integral:

Given that the mean of the score is zero, the Fisher information is simply its second moment:

It can also be shown (again via the Leibniz rule, differentiating once more) that \(I(\theta) = -\mathbb{E}\!\left[\frac{\partial^2}{\partial \theta^2} \log p(x;\theta)\right]\), i.e., the expected curvature of the negative log-likelihood, so a sharply peaked likelihood carries more information about \(\theta\).

Cramer-Rao Lower Bound

The Cramér-Rao lower bound (CRLB) gives a fundamental floor on how well any unbiased estimator can do. For an unbiased estimator \(\hat{\theta}\) of \(\theta\) based on a single observation, \(\text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)},\) and for \(n\) i.i.d. observations, since Fisher information is additive, \(\text{Var}(\hat{\theta}) \geq \frac{1}{n\, I(\theta)}.\)

In words: no unbiased estimator can have variance lower than the inverse Fisher information. An estimator that attains this bound is called efficient. The connection to MLE is the headline result of this whole post: the MLE is asymptotically efficient, meaning its variance approaches the CRLB as \(n \to \infty\) (cf. asymptotic normality, where the asymptotic variance was exactly \(I(\theta^*)^{-1}\)).

Sketch. Let \(\hat\theta\) be unbiased, so \(\mathbb{E}[\hat\theta] = \theta\). Differentiating \(\mathbb{E}[(\hat\theta - \theta) ] = 0\) and using \(\mathbb{E}[S(\theta)]=0\), one finds \(\text{Cov}(\hat\theta, S(\theta)) = 1\). Applying the Cauchy-Schwarz inequality, \(1 = \text{Cov}(\hat\theta, S(\theta))^2 \leq \text{Var}(\hat\theta)\,\text{Var}(S(\theta)) = \text{Var}(\hat\theta)\, I(\theta),\) which rearranges to \(\text{Var}(\hat\theta) \geq 1/I(\theta)\).

Lipschitz-Continuity

We can also try to control how fast a function can change using Lipschitz continuity. A function \(f\) is Lipschitz continuous with constant \(L \geq 0\) if, for all \(x\) and \(y\), \(\| f(x) - f(y) \| \leq L\, \|x - y\|.\)

In words: the output can change by at most \(L\) times the change in input, so the function has bounded “slope.” For differentiable \(f\) this is equivalent to \(\|\nabla f\| \leq L\) everywhere. Lipschitz constants show up all over learning theory: they bound how much a loss can change between nearby parameters (useful for convergence rates of gradient descent, where a Lipschitz gradient sets a safe step size), and they appear in generalization and robustness arguments.

Cite This Article

@misc{hatzky2026mle,
   title        = Stress testing MLE,
   author       = Julian Hatzky,
   year         = 2026,
   month        = Jun,
   url          = https://ju2ez.github.io/blog/2026/stress-testing-mle/
}
  

Julian Hatzky. (2026, Jun). Stress testing MLE. . https://ju2ez.github.io/blog/2026/stress-testing-mle/

Julian Hatzky. "Stress testing MLE." , 18 Jun 2026, https://ju2ez.github.io/blog/2026/stress-testing-mle/.

Julian Hatzky. "Stress testing MLE." . Jun 18, 2026. https://ju2ez.github.io/blog/2026/stress-testing-mle/