ML Interview Q Series: Estimating Uniform Distribution Bounds (a, b) using Maximum Likelihood Estimation.

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Say you draw n samples from a uniform distribution U(a, b). What is the MLE estimate of a and b?

MLE FOR A UNIFORM DISTRIBUTION U(a, b)

To understand how to derive the Maximum Likelihood Estimators (MLE) for the parameters a and b, consider that we have n i.i.d. samples drawn from a uniform distribution on the interval [a, b]. Let these samples be denoted by x₁, x₂, …, xₙ. The probability density function (pdf) for a single sample xᵢ under the uniform distribution U(a, b) is

• 1/(b - a) if a ≤ xᵢ ≤ b

• 0 otherwise

The joint likelihood function for all n samples is the product of individual probabilities. When all xᵢ lie in [a, b], the likelihood is

However, this expression for the likelihood is valid only if a ≤ xᵢ ≤ b for every i in {1, 2, …, n}. Otherwise, the likelihood is zero. Therefore, we have the constraints:

a ≤ min(x₁, x₂, …, xₙ) b ≥ max(x₁, x₂, …, xₙ)

Under these constraints, the factor 1/(b - a) is constant across all samples. Maximizing L(a, b) = 1/(b - a)ⁿ is equivalent to minimizing (b - a). Given that a cannot exceed the smallest sample without invalidating the likelihood, and b cannot be smaller than the largest sample, the maximum likelihood occurs by choosing:

â = min(x₁, x₂, …, xₙ) b̂ = max(x₁, x₂, …, xₙ)

This choice satisfies the requirement â ≤ all samples ≤ b̂, and it keeps the interval [a, b] as small as possible, thereby maximizing 1/(b - a)ⁿ.

THE REASONING IN DETAIL

One core principle of MLE is that we want to choose parameter values that maximize the product of the probabilities of observing the data as it was actually observed. For the uniform distribution on [a, b]:

• If any sample xᵢ is less than a or greater than b, the probability (density) for that xᵢ under U(a, b) is zero, which makes the entire likelihood zero.

• Therefore, a must be at most the smallest observation, and b must be at least the largest observation. Otherwise, the likelihood is zero.

• Within the region of parameter space where all observations lie in [a, b], the likelihood is proportional to 1/(b - a)ⁿ.

• To maximize 1/(b - a)ⁿ, we want to minimize (b - a) while respecting a ≤ min(xᵢ) and b ≥ max(xᵢ). The unique solution is a = min(xᵢ) and b = max(xᵢ).

CODE EXAMPLE (PYTHON)

Below is a Python snippet to illustrate how to compute these estimates:

import numpy as np

def mle_uniform(samples):
    """
    Returns the MLE estimates a_hat and b_hat for a uniform distribution
    given a list/array of samples.
    """
    a_hat = np.min(samples)
    b_hat = np.max(samples)
    return a_hat, b_hat

# Example usage:
samples = [2.3, 3.7, 2.9, 2.5, 3.1]
a_est, b_est = mle_uniform(samples)
print("MLE for a:", a_est)
print("MLE for b:", b_est)

This code determines the minimum and maximum of the sample array, which are the MLE estimates for a and b respectively.

SUBTLE POINTS AND POTENTIAL PITFALLS

Outliers. Since b is estimated as the maximum sample and a is the minimum sample, any extreme outlier in the dataset directly shifts the MLE estimates. If your dataset has contamination or outliers, the MLE will expand the [a, b] interval to accommodate them.

Support mismatch. If you incorrectly assume the distribution is uniform when the data is not truly uniform, the MLE estimates can be misleading or might not capture the true underlying distribution.

Small sample sizes. With very few samples, the gap between min and max might not represent the entire potential range of the true distribution. This can produce a wide confidence interval for the parameters a and b.

Are these estimates biased or unbiased?

The MLE estimates â = min(x) and b̂ = max(x) are biased estimators for the true parameters a and b. For instance, on average, min(x) will tend to be larger than the true a, and max(x) will tend to be smaller than the true b. Intuitively, there is a nonzero chance that you have not sampled the actual extreme ends of the true distribution. In fact, in many statistical texts, you’ll see unbiased estimators for a and b that involve corrected terms to account for this bias. However, those adjusted estimators come from methods other than pure MLE, such as the method of moments or applying a bias correction to the MLE.

Could you derive the likelihood and constraints another way?

Yes. One approach is to write the likelihood function and introduce an indicator function I[a ≤ xᵢ ≤ b]. The complete likelihood can be expressed as:

L(a, b) = Π (1/(b - a)) ⋅ I[a ≤ xᵢ ≤ b]

where I[...] is 1 if the condition is true for all i and 0 otherwise. From this perspective, only the range [a, b] that covers all xᵢ yields a nonzero likelihood. Minimizing (b - a) subject to covering all data points yields the same conclusion: pick a = min(xᵢ) and b = max(xᵢ).

What happens if we have prior knowledge about a and b?

If we bring Bayesian reasoning into the picture, we would place priors on a and b (e.g., uniform priors or something else). We would then derive the posterior distribution for a and b given the data. The MLE approach ignores priors and simply uses the data likelihood, whereas a Bayesian approach would integrate the likelihood with a prior. Even so, for many simple priors, the maximum a posteriori (MAP) estimate might still look similar to the sample min and max but adjusted by the prior’s influence.

How would the MLE approach be influenced by data scaling or transformations?

For a uniform distribution U(a, b), any linear transformation of the data that transforms x to y = c·x + d simply rescales the problem. The MLE for the new distribution’s parameters would be the corresponding linear transformation of the original min and max. Non-linear transformations would require re-expressing the uniform distribution in the transformed space, which might no longer remain uniform unless the transformation is carefully chosen.

How do we implement this in real-world pipelines?

In many data workflows, you might do the following in practice:

• Gather your dataset into a NumPy array or a similar structure.

• Ensure that your data is valid and free of anomalies (or handle outliers explicitly).

• Compute the sample minimum and sample maximum.

• Assign these as your estimates for a and b.

If outliers are suspected, you might adopt a robust approach, such as ignoring extreme percentiles or applying domain knowledge to clamp your data. However, that approach ceases to be pure MLE but can be more practical in certain real-world applications.

When you answer real interview questions about the uniform distribution MLE, make sure to emphasize the conceptual reasoning (constrained optimization problem and the nature of the uniform likelihood) and be prepared to discuss biases, alternative estimators, or modifications to handle outliers.

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Below are additional follow-up questions

What if we consider discrete uniform distributions instead of continuous ones? How does the MLE estimation for a and b change in the discrete case?

In a discrete uniform distribution, the support is a set of integer points from a to b (where a and b are integers, and a ≤ b). The probability mass function (pmf) for any integer xᵢ in [a, b] is p(xᵢ) = 1 / (b − a + 1), provided xᵢ is an integer in that range, and 0 otherwise.

The maximum likelihood principle still applies:

1. We must have a ≤ all xᵢ and b ≥ all xᵢ so that no sample falls outside [a, b], otherwise the likelihood is zero.

2. Among all valid intervals [a, b] that contain every sample, the pmf for each xᵢ is 1 / (b − a + 1).

3. The joint likelihood is (1 / (b − a + 1))ⁿ.

4. To maximize that likelihood, we minimize b − a + 1.

Hence, the MLE remains:

â = min(xᵢ) b̂ = max(xᵢ)

The only difference is we treat these as integer values. If the data are integer-valued, then min and max samples will also be integers, making the interpretation straightforward. However, subtle points can arise if your data might appear integer-valued but actually contain measurement noise or rounding errors.

Potential pitfalls:

• Rounding: If real-valued measurements get truncated or rounded to integers before you model them as a discrete uniform, you might inadvertently shift your estimates. • Ties: In discrete distributions, having many repeated values at the min or max can cause suspicion about whether your distribution assumption is correct. Still, the MLE formula remains the same.

How do we handle multi-dimensional uniform distributions? For example, U in [a₁, b₁] × [a₂, b₂] × … × [a_d, b_d]?

In d dimensions, a uniform distribution on a hyper-rectangle [a₁, b₁] × … × [a_d, b_d] has a pdf:

pdf(x₁, x₂, …, x_d) = 1 / ∏(bᵢ − aᵢ) for xᵢ ∈ [aᵢ, bᵢ] for i in 1..d.

To apply MLE:

1. Each dimension must fully contain the observed sample points. In other words, for each dimension j, we need aⱼ ≤ xᵢⱼ for every sample i, and bⱼ ≥ xᵢⱼ for every sample i.

2. The joint likelihood over n samples is proportional to 1 / ∏(bⱼ − aⱼ)ⁿ, where j=1..d.

3. To maximize the likelihood, we minimize each (bⱼ − aⱼ), subject to containing all xᵢⱼ.

Hence the MLE in each dimension j is:

âⱼ = min over all i of xᵢⱼ b̂ⱼ = max over all i of xᵢⱼ

Potential pitfalls:

• Sparse data in high dimensions might not adequately represent the full range, leading to underestimation of the true bounding region. • If the data exhibit correlations in each dimension, a rectangular bounding region might be a poor model. The uniform distribution in a rectangle implies independence and equal density across that entire box. • Outliers in any single dimension can blow up the volume of the bounding region significantly.

How do we interpret the likelihood function when there is measurement error or known data truncation?

In real-world scenarios, measurements can be truncated or censored. For instance, if a sensor only reports values above a certain threshold, or there is a known maximum reading limit:

1. Truncation means you only see samples within [T₁, T₂] even if the true distribution extends beyond that range. In that case, the likelihood for each observed sample changes because you need to account for the conditional probability of seeing that sample given that it falls in [T₁, T₂].

2. You might have to modify the uniform distribution assumption to something like: “Given that X ∈ [T₁, T₂], X is uniformly distributed on [a, b].” But if T₁ > a or T₂ < b, then your maximum-likelihood approach needs to factor in the probability that you never see anything below T₁ or above T₂.

Practical implications:

• If T₁ ≤ min(xᵢ) and T₂ ≥ max(xᵢ), your truncated data range might not affect the MLE because all data still fit comfortably within [a, b]. • If T₁ > a in reality, you cannot directly observe how far below T₁ the true distribution extends; the MLE might push â closer to T₁. Similarly for b with T₂. • Standard MLE derivations assume full observability of samples across the entire support. With truncation, you should consider a truncated likelihood approach, leading to more complex parameter estimation equations.

What if some samples are identical? How do repeated values affect the MLE?

When samples have duplicates:

• min(xᵢ) and max(xᵢ) remain valid. Even if the min or max value occurs multiple times, it does not change the fact that the MLE is still min for a and max for b. • Duplicates do not fundamentally alter the uniform-likelihood structure because the uniform pdf or pmf is still constant within [a, b]. • If the minimum sample value occurs for many data points, that might raise questions about whether the data is truly uniform or if there is a boundary effect. But strictly from the MLE perspective, we simply take the smallest observed value as â and the largest observed value as b̂, regardless of how many times they repeat.

Potential pitfalls:

• Duplicate min and max values reduce variability in the sample, so your interval might look artificially small if you have only a narrow cluster of data. But that is a sample-specific artifact rather than an MLE formula change. • If the entire dataset is identical (all xᵢ the same), then min(x) = max(x), making the MLE degenerate (b̂ = â). The uniform distribution’s pdf becomes undefined (division by zero) for that scenario. In practice, you might say the MLE does not exist or that the data do not support a range.

Suppose we have partial knowledge: we know the true a but not b. How does that affect the MLE for b?

If a is known and fixed, then:

1. The uniform distribution is U(a, b).

2. The likelihood for n samples xᵢ is 1 / (b − a)ⁿ, provided a ≤ xᵢ ≤ b for all i. Otherwise, the likelihood is zero.

3. We only need to estimate b.

4. Since we must cover all data, b ≥ max(xᵢ).

5. Minimizing (b − a) while still covering all data yields b̂ = max(xᵢ).

Hence, the MLE for b is max(xᵢ), given that a is known.

Potential pitfalls:

• You must be absolutely certain about the known a. If that knowledge is incorrect, then the MLE b̂ may not reflect the true distribution support. • If a is known but the data includes values less than a (due to measurement or noise), that indicates a mismatch between the assumed model and actual data.

How do we perform hypothesis testing on whether a proposed [a, b] is valid compared to the MLE [â, b̂]?

You might want to test a hypothesis H₀: [a, b] = [a₀, b₀] vs. H₁: [a, b] ≠ [a₀, b₀] and see how that compares with the MLE. A straightforward approach:

1. Compute the likelihood under H₀: L(a₀, b₀). If any sample lies outside [a₀, b₀], then L(a₀, b₀) = 0, so H₀ is immediately implausible.

2. Compare that with L(â, b̂).

3. Often a likelihood ratio test (LRT) can be used: Λ = L(a₀, b₀) / L(â, b̂). Because (b₀ − a₀) ≥ (b̂ − â), L(a₀, b₀) ≤ L(â, b̂).

4. The ratio might be used to derive a p-value if you have a relevant asymptotic distribution for the test statistic.

Potential pitfalls:

• Uniform distributions yield piecewise-defined likelihoods that can become zero if the proposed [a₀, b₀] does not contain the entire sample. That discontinuity can complicate classical test procedures. • If [a₀, b₀] strictly contains the entire sample, but is much bigger, the likelihood ratio might be very small. That can make it easy to reject H₀ if your sample is large enough.

Could we estimate a and b if we apply transformations or normalizations to the samples first?

Sometimes, data is better modeled if we transform it to a new variable y = g(x) that might appear more uniform over a known range. Then you can estimate parameters on y and back-transform to x. But for a uniform distribution:

1. A linear transformation y = c·x + d is straightforward: if x ~ U(a, b), then y ~ U(c·a + d, c·b + d). The MLE for y would simply be min(yᵢ) and max(yᵢ), which correspond back to c·min(xᵢ)+d and c·max(xᵢ)+d.

2. A non-linear transformation g(x) can distort uniformity. After transformation, the distribution might not remain uniform, so you can’t just apply the same min–max approach.

3. If you do want to apply MLE in a transformed space, you need the Jacobian of the transformation for the pdf, which can make the likelihood function more complex.

Potential pitfalls:

• Arbitrary transformations might invalidate the uniformity assumption in the transformed space. • In practice, transformations are usually done to better match a known distribution (like normal or exponential). Doing so to achieve uniformity is less common, unless you’re looking at probability integral transforms in goodness-of-fit tests.

In practical data science, do we often use a uniform distribution to model real-world data?

Uniform distributions are usually used as a simplistic model or for bounding scenarios, not typically for intricate real-world phenomena. That said, they are still important:

1. Sometimes, you have a scenario where every outcome in [a, b] is equally likely (e.g., selecting a random point in an interval).

2. Uniform distributions can serve as building blocks in simulation or as non-informative priors in Bayesian contexts.

3. For real data, the uniform assumption might be a placeholder before you gather deeper insight.

Real-world pitfalls:

• Real data rarely has a strictly constant density over a perfect interval. Even if it looks roughly uniform, small deviations from uniformity might matter, especially at the boundaries. • MLE min–max is highly sensitive to outliers. One unexpected data point far outside the main cluster can drastically expand the estimated interval.

How would you robustify the MLE approach against outliers or data errors?

Since the standard MLE includes the min and max samples:

1. If you suspect outliers, you could consider a robust approach (though that is no longer the pure MLE). For instance, you might decide to discard any data points beyond some quantile threshold. Then estimate â and b̂ using those trimmed samples.

2. Alternatively, a Bayesian approach might impose a prior that shrinks the support, penalizing large (b − a). Then the posterior mode (MAP) might not always expand to accommodate a single extreme outlier.

3. Another possibility is to build a mixture model where most data is uniform on [a, b], plus a small proportion for outliers. That is more complex but can handle contamination better.

Potential pitfalls:

• Trimming or ignoring outliers must be justified carefully, or you risk discarding legitimate data. • Mixture models can become complicated to fit, and identifiability might be an issue if you only have a small dataset.

How can we derive confidence intervals for a and b from these MLE estimates?

Constructing confidence intervals (CIs) for uniform distribution parameters can be tricky because:

1. The sampling distribution of (min(X), max(X)) is known: – min(X) has distribution F_min(t) = 1 − (1 − F(t))ⁿ, etc. But turning that into a joint CI for a and b is more nuanced.

2. One classical approach for continuous uniform distributions is using order statistics: – The distribution of the smallest order statistic X_(1) (the min) and the largest order statistic X_(n) (the max) are well-known. – If you invert those distributions, you can form intervals that with a certain probability contain the true a and b.

3. Another approach is to use bootstrap methods: sample with replacement from your data, recompute â and b̂ each time, and observe the empirical distribution of the estimates. This yields approximate intervals, although they might be optimistic or pessimistic if the sample is not truly uniform.

Potential pitfalls:

• If the sample size is small, the distribution of min and max might be highly variable, so your CI could be very wide. • Analytical formulas often assume perfect uniformity and i.i.d. samples. If the data deviate from that, the nominal coverage of your CI can be incorrect.

How does knowledge of sub-samples or stratification affect the MLE?

Sometimes, you might split your dataset into subgroups (strata), each presumably uniform with possibly different [a, b]. If you do that:

1. You might compute MLE separately for each subgroup. For subgroup j, âⱼ = min(xᵢ in subgroup j), b̂ⱼ = max(xᵢ in subgroup j).

2. If there is overlap in the intervals, or if you believe the distribution could be uniform for the entire combined set, you can also check the single-interval MLE for the pooled data. Compare which approach is more appropriate with domain knowledge.

3. If sub-samples come from different underlying distributions, forcing a single [a, b] over all data might be too restrictive.

Potential pitfalls:

• You may not have enough samples in each subgroup to estimate an interval reliably, leading to wide intervals or zero-likelihood anomalies if outliers appear in small subgroups. • Combining subgroups that truly have distinct distributions can produce an MLE that poorly fits each subgroup individually.

How does the method of moments estimation compare to the MLE for uniform distributions?

Method of moments (MoM) is a different estimation technique where you match sample moments (like mean, variance) to theoretical moments of the distribution. For a uniform distribution U(a, b):

• Mean: (a + b) / 2 • Variance: (b − a)² / 12

You can solve the system:

mean(sample) = (a + b) / 2 var(sample) = (b − a)² / 12

In principle, that yields:

b − a = √(12 · var(sample)) (a + b) / 2 = mean(sample)

One can solve for a and b from these equations. That is an alternative to min–max:

a_mom = mean(sample) − (1/2)√(12 · var(sample)) b_mom = mean(sample) + (1/2)√(12 · var(sample))

Potential pitfalls:

• The method of moments can yield intervals [a, b] that do not necessarily contain all data points. In that case, the uniform pdf at any data point outside [a, b] would be zero, conflicting with the data. • MoM can produce an estimate that is “in the middle” of the data rather than pegged at the extremes, which might be nonsensical for an actual uniform distribution over all observed values. • In general, the MLE for the uniform distribution is min–max, whereas the MoM is not guaranteed to align with the extremes of the sample, so it can be considered “invalid” if any xᵢ < a_mom or xᵢ > b_mom.

How would you diagnose that the uniform assumption might be incorrect?

To diagnose correctness:

1. Plot a histogram of the data and see if it appears relatively flat across [min(x), max(x)].

2. Use goodness-of-fit tests: one approach is to transform the data to yᵢ = (xᵢ − â) / (b̂ − â). If the data truly follow U(a, b), then yᵢ ~ U(0, 1). Check if the empirical distribution of yᵢ is close to uniform on [0, 1].

3. Look for boundary clustering. If many points cluster at the boundaries, this might be suspicious for a uniform assumption.

Potential pitfalls:

• Even if the histogram is roughly flat, small sample sizes can mislead. • If there is a systematic shape (peaks or troughs), it indicates non-uniformity. • If you see that â or b̂ changes drastically with a single outlier, it might mean your data is not truly uniform but has a heavier tail.

How can we handle a scenario where we suspect the data is uniform, but we also suspect noise outside the range [a, b]?

This is like a contaminated model:

1. You might propose that with high probability p, X ∼ U(a, b), but with small probability (1 − p), it falls outside [a, b].

2. The likelihood function then becomes a mixture model. You do partial MLE or EM-based fitting to estimate a, b, and p.

3. Alternatively, if you strongly suspect only a handful of points are noise, you might do iterative cleaning: estimate min–max, remove extreme outliers that deviate too far, and re-estimate. This is no longer the pure MLE but might be more practical.

Potential pitfalls:

• Mixture models can be hard to fit if the “out of range” data doesn’t follow a well-defined distribution. • A small fraction of outliers can dramatically affect the uniform MLE unless you explicitly model that contamination.

If a and b are random variables themselves, can we do a hierarchical or Bayesian model?

Yes. In a Bayesian setting:

1. Place priors on a and b (e.g., a prior that expects them to be near certain values).

2. The posterior distribution p(a, b | x₁, …, xₙ) = (likelihood) × (prior).

3. If the prior is conjugate-like or simplified, you can get closed-form posteriors in some special cases, but typically you use sampling methods (MCMC).

Potential pitfalls:

• If your prior is too narrow (e.g., you strongly believe a < 0 or b > 10), data that contradicts this might need many samples to override the prior. • Misspecified priors can skew your posterior, so domain knowledge is crucial.

Could the MLE fail in degenerate cases (e.g., all samples are the same single value)?

Yes, in a degenerate scenario where all samples xᵢ = c for some constant c:

1. min(xᵢ) = c and max(xᵢ) = c, so â = b̂ = c.

2. Then the likelihood function involves 1 / (b − a)ⁿ, but (b − a) = 0, so the pdf is not well-defined.

In strict terms, the MLE does not exist or is infinite for the uniform distribution if all data are identical. The uniform model on an interval of zero length is not a valid pdf. Realistically, you might interpret that the data do not provide any information about the range. One might artificially expand the interval by an infinitesimal amount or consider an alternative model that allows for degenerate distributions.

Potential pitfalls:

• This is a boundary case in uniform distribution theory that reveals how the MLE formula can break down if the data lack variability. • If there is even a tiny variation in the data, the usual min–max approach is valid.

What are common numerical issues one might encounter when computing the uniform MLE in large-scale applications?

Although the actual formula for MLE is straightforward, large-scale or high-dimensional scenarios can lead to:

• Floating-point precision: If b − a is extremely large or extremely small, 1 / (b − a)ⁿ might overflow or underflow. • Data storage: Keeping track of min and max in a streaming fashion requires a reliable algorithm but is usually trivial (one pass). However, extreme values might be lost if the data are partially summarized incorrectly. • Parallelization: If data are distributed across many machines, you need a correct global min and global max. This typically involves a reduce operation over all nodes.

Potential pitfalls:

• Failing to collect the global min and max across all shards can lead to a misestimation of the MLE. • If min and max are computed in floating-point with rounding, slight inaccuracies are typically inconsequential, but you must be consistent across the pipeline.

What if the true distribution is slightly bigger than [min, max], but due to sampling variability we never see the extremes?

In reality, if the process truly generated data from [a_true, b_true], but your sample min and max are strictly inside that range, the MLE will estimate â = min(xᵢ) > a_true and b̂ = max(xᵢ) < b_true. This is expected because the MLE is the best point estimate given your observed sample. However:

1. The MLE is systematically biased for uniform distributions, especially with small samples.

2. If you need interval estimates that capture a_true and b_true with high probability, you need confidence intervals or Bayesian posterior intervals, which typically expand beyond the sample min and sample max to account for the possibility that you simply haven’t observed the true boundary extremes.

Potential pitfalls:

• Overconfidence in â and b̂ for small n can lead to errors in subsequent analysis. For instance, a forecast or simulation that assumes [â, b̂] might incorrectly exclude possible out-of-range future data. • In many engineering contexts, you might want to build in a safety margin around min and max to handle the possibility of unobserved extremes.

If the data comes from a time series, does the MLE formula still apply?

If the data are i.i.d. from a uniform distribution, time-series context doesn’t affect the math. But in many time-series scenarios, the data might exhibit autocorrelation, trends, or non-stationarity. That can break the uniform i.i.d. assumption:

1. If Xₜ are correlated across time, the joint likelihood is no longer simply the product of 1 / (b − a) for each sample.

2. If there is a trend, your min and max might shift over time, so a single [a, b] for all time might be inappropriate.

3. You might choose a rolling min–max approach or a piecewise uniform model over time segments.

Potential pitfalls:

• If the underlying process systematically drifts upward, the sample max in early data might underestimate future extremes. • Non-stationarity makes the uniform model questionable. You might need additional parameters to describe how the bounds evolve over time.