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Bayesian Inference, Time Series & Disaster Modeling (Statistics) Solved Questions & Notes (2026) - Apex Rankers

Statistics & Data Science > Statistics > Bayesian Inference, Time Series & Disaster Modeling

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Bayesian Inference, Time Series & Disaster Modeling

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Q. 1 Statistics
Difficulty: easy (1 Mark)
In Bayesian statistical inference, what is Bayes' Theorem for continuous parameter $\theta$ and data $\mathbf{x}$ expressed as?
A
$\pi(\theta \mid \mathbf{x}) = \frac{f(\mathbf{x} \mid \theta) \pi(\theta)}{\int f(\mathbf{x} \mid \theta) \pi(\theta) d\theta} \propto L(\theta; \mathbf{x}) \times \pi(\theta)$
✓ Correct
B
$\pi(\theta \mid \mathbf{x}) = f(\mathbf{x} \mid \theta) + \pi(\theta)$
C
$\pi(\theta \mid \mathbf{x}) = \frac{\pi(\theta)}{f(\mathbf{x} \mid \theta)}$
D
$\pi(\theta \mid \mathbf{x}) = \int f(\mathbf{x} \mid \theta) d\theta$
💡 Step-by-Step Explanation & Concept Rationale
Bayes' Theorem updates the prior probability distribution $\pi(\theta)$ with the observed data likelihood $L(\theta; \mathbf{x})$ to yield the posterior distribution $\pi(\theta \mid \mathbf{x}) \propto \text{Likelihood} \times \text{Prior}$.
Q. 2 Statistics
Difficulty: easy (1 Mark)
What is a 'Conjugate Prior' in Bayesian probability theory?
A
A prior distribution that belongs to the same probability distribution family as the resulting posterior distribution
✓ Correct
B
A prior distribution that is always uniform on $[0, 1]$
C
A prior distribution that produces zero variance
D
A prior distribution chosen by a panel of judges
💡 Step-by-Step Explanation & Concept Rationale
A prior is conjugate to the likelihood if the posterior distribution $\pi(\theta \mid \mathbf{x})$ belongs to the exact same parametric family as the prior $\pi(\theta)$ (e.g., Beta prior is conjugate to Binomial likelihood, yielding a Beta posterior).
Q. 3 Statistics
Difficulty: medium (1 Mark)
If a Binomial likelihood $X \sim B(n, p)$ has a conjugate $\text{Beta}(\alpha, \beta)$ prior on $p$, what is the resulting posterior distribution $\pi(p \mid x)$?
A
$\text{Beta}(\alpha + x, \beta + n - x)$
✓ Correct
B
$\text{Beta}(\alpha + n, \beta + x)$
C
$\text{Gamma}(\alpha + x, \beta + n)$
D
$\text{Normal}(\alpha, \beta)$
💡 Step-by-Step Explanation & Concept Rationale
Multiplying the Beta prior $p^{\alpha-1}(1-p)^{\beta-1}$ by the Binomial likelihood $\binom{n}{x}p^x(1-p)^{n-x}$ yields $p^{(\alpha+x)-1}(1-p)^{(\beta+n-x)-1}$, which is exactly $\text{Beta}(\alpha + x, \beta + n - x)$.
Q. 4 Statistics
Difficulty: medium (1 Mark)
What is the conjugate prior distribution for the mean $\mu$ of a Normal distribution $N(\mu, \sigma^2)$ when variance $\sigma^2$ is known?
A
A Normal distribution $N(\mu_0, \sigma_0^2)$
✓ Correct
B
A Gamma distribution
C
A Uniform distribution
D
A Chi-Square distribution
💡 Step-by-Step Explanation & Concept Rationale
The Normal distribution is conjugate to a Normal likelihood for known variance: if prior is $N(\mu_0, \sigma_0^2)$, the posterior distribution of $\mu$ is also strictly Normal with precision equal to the sum of prior and data precisions.
Q. 5 Statistics
Difficulty: medium (1 Mark)
What is the conjugate prior for the parameter $\lambda$ of a Poisson distribution?
A
Gamma distribution $\text{Gamma}(\alpha, \beta)$
✓ Correct
B
Beta distribution
C
Normal distribution
D
Exponential distribution
💡 Step-by-Step Explanation & Concept Rationale
The Gamma distribution $\text{Gamma}(\alpha, \beta)$ is the conjugate prior for the Poisson rate $\lambda$. With $n$ observations summing to $\sum x_i$, the posterior is $\text{Gamma}(\alpha + \sum x_i, \beta + n)$.
Q. 6 Statistics
Difficulty: hard (1 Mark)
What is Jeffreys' Prior in Bayesian statistics, and what is its defining invariance property?
A
$\pi(\theta) \propto \sqrt{\det I(\theta)}$ (where $I(\theta)$ is Fisher Information); it is invariant under 1-to-1 reparameterization $\phi = g(\theta)$
✓ Correct
B
$\pi(\theta) = 1$ for all parameters
C
$\pi(\theta) = \frac{1}{\theta^2}$
D
$\pi(\theta) = e^{-\theta}$
💡 Step-by-Step Explanation & Concept Rationale
Harold Jeffreys (1946) derived non-informative priors proportional to the square root of the Fisher Information determinant, ensuring that transforming parameters (e.g., from variance to standard deviation) preserves mathematical consistency.
Q. 7 Statistics
Difficulty: easy (1 Mark)
What is the Bayesian Point Estimator under Squared Error Loss $L(\theta, \hat{\theta}) = (\theta - \hat{\theta})^2$?
A
The Posterior Mean $E[\theta \mid \mathbf{x}]$
✓ Correct
B
The Posterior Median
C
The Posterior Mode (MAP)
D
The Prior Mean
💡 Step-by-Step Explanation & Concept Rationale
Minimizing expected squared error loss $E[(\theta - \hat{\theta})^2 \mid \mathbf{x}]$ yields the conditional expectation (Posterior Mean) $\hat{\theta} = \int \theta \pi(\theta \mid \mathbf{x}) d\theta$.
Q. 8 Statistics
Difficulty: medium (1 Mark)
What is the Bayesian Point Estimator under Absolute Error Loss $L(\theta, \hat{\theta}) = |\theta - \hat{\theta}|$?
A
The Posterior Median of $\pi(\theta \mid \mathbf{x})$
✓ Correct
B
The Posterior Mean
C
The Posterior Mode (MAP)
D
The Maximum Likelihood Estimator
💡 Step-by-Step Explanation & Concept Rationale
Minimizing expected absolute error loss $E[|\theta - \hat{\theta}| \mid \mathbf{x}]$ yields the 50th percentile (Posterior Median) of the posterior distribution.
Q. 9 Statistics
Difficulty: medium (1 Mark)
What is the Maximum A Posteriori (MAP) estimator?
A
The mode of the posterior distribution: $\hat{\theta}_{MAP} = \arg\max_\theta \pi(\theta \mid \mathbf{x}) = \arg\max_\theta [\ln L(\theta; \mathbf{x}) + \ln \pi(\theta)]$
✓ Correct
B
The average of the prior and likelihood
C
The maximum value in the sample dataset
D
The median of the prior distribution
💡 Step-by-Step Explanation & Concept Rationale
The MAP estimator is the parameter value that maximizes the posterior density function, corresponding to optimal Bayesian decision under 0-1 (all-or-nothing) loss.
Q. 10 Statistics
Difficulty: hard (1 Mark)
What is a 'Highest Posterior Density' (HPD) Credible Interval in Bayesian analysis?
A
A $100(1-\alpha)\%$ credible region where every parameter point inside the interval has higher posterior density than any point outside, achieving the shortest possible interval width for that coverage
✓ Correct
B
An interval where the two tail areas are exactly equal
C
A frequentist confidence interval calculated on sample means
D
An interval containing 95% of the raw data points
💡 Step-by-Step Explanation & Concept Rationale
An HPD interval is the optimal Bayesian credible set: it encompasses $1-\alpha$ posterior probability while ensuring every point inside has density higher than any point outside, resulting in the minimum interval length.
Q. 11 Statistics
Difficulty: medium (1 Mark)
How does a Bayesian $95\%$ Credible Interval fundamentally differ from a Frequentist $95\%$ Confidence Interval in philosophical interpretation?
A
A Bayesian credible interval means there is a $95\%$ probability that the unknown parameter $\theta$ lies within the fixed interval $[a, b]$ given the observed data; a frequentist CI treats $\theta$ as fixed and the interval bounds as random under hypothetical repeated sampling
✓ Correct
B
There is no difference; they are mathematically identical in all cases
C
Frequentist intervals can only be computed for coin tosses
D
Bayesian intervals do not use data
💡 Step-by-Step Explanation & Concept Rationale
Bayesian inference treats the parameter $\theta$ as a random variable with a posterior probability distribution conditioned on the observed data, enabling direct probabilistic statements about the parameter itself.
Q. 12 Statistics
Difficulty: hard (1 Mark)
What is the Metropolis-Hastings (MH) algorithm in Markov Chain Monte Carlo (MCMC)?
A
An MCMC algorithm that generates samples from a target posterior distribution $\pi(\theta)$ by proposing candidate moves $\theta^* \sim q(\theta^* \mid \theta^{(t)})$ and accepting them with probability $\alpha = \min\left(1, \frac{\pi(\theta^*) q(\theta^{(t)} \mid \theta^*)}{\pi(\theta^{(t)}) q(\theta^* \mid \theta^{(t)})}\right)$
✓ Correct
B
A method for sorting numbers in ascending order
C
A deterministic numerical integration algorithm
D
A linear regression algorithm
💡 Step-by-Step Explanation & Concept Rationale
Metropolis-Hastings generates a stationary Markov chain whose equilibrium distribution is exactly the target posterior $\pi(\theta)$, enabling Bayesian sampling without computing the intractable normalizing denominator $\int L(\theta)\pi(\theta)d\theta$.
Q. 13 Statistics
Difficulty: hard (1 Mark)
What is 'Gibbs Sampling' in multi-parameter Bayesian MCMC?
A
A special case of the Metropolis-Hastings algorithm where each scalar parameter $\theta_j$ is sequentially updated by sampling directly from its Full Conditional Distribution $\pi(\theta_j \mid \boldsymbol{\theta}_{-j}, \mathbf{x})$ with an acceptance probability of exactly 1.0
✓ Correct
B
Sampling with replacement from raw data
C
A method that works only on two variables
D
An optimization method for finding the global minimum
💡 Step-by-Step Explanation & Concept Rationale
Gibbs sampling updates one parameter at a time from its full univariate conditional distribution, accepting every step ($100\%$ acceptance rate) and avoiding manual tuning of proposal widths.
Q. 14 Statistics
Difficulty: hard (1 Mark)
What does the Gelman-Rubin convergence diagnostic statistic ($\hat{R}$ / Potential Scale Reduction Factor) measure in MCMC?
A
The ratio of variance between multiple independent MCMC chains to variance within chains; $\hat{R} \approx 1.0$ (typically $< 1.05$) indicates convergence to the stationary posterior distribution
✓ Correct
B
The number of iterations per second
C
The correlation between prior and posterior means
D
The percentage of missing data in the chain
💡 Step-by-Step Explanation & Concept Rationale
The Gelman-Rubin $\hat{R}$ compares within-chain and between-chain variances across multiple chains initialized from dispersed starting points: values near $1.00$ indicate that all chains have converged and mixed thoroughly.
Q. 15 Statistics
Difficulty: medium (1 Mark)
What is 'Effective Sample Size' (ESS) in MCMC posterior analysis?
A
The number of independent, uncorrelated samples that contain the same amount of information as the $N$ autocorrelated MCMC draws: $\text{ESS} = \frac{N}{1 + 2 \sum_{k=1}^\infty \rho_k}$
✓ Correct
B
The total number of iterations run by the computer
C
The sample size of the original survey
D
The number of parameters in the model
💡 Step-by-Step Explanation & Concept Rationale
Because successive MCMC states are autocorrelated, $N$ iterations contain less independent information than $N$ i.i.d. draws. The ESS quantifies the effective number of independent draws for computing posterior standard errors.
Q. 16 Statistics
Difficulty: easy (1 Mark)
What is 'Covariance Stationarity' (Weak Stationarity) in Time Series Analysis?
A
A stochastic process where the Mean is constant ($E[Y_t] = \mu$), Variance is constant ($\text{Var}(Y_t) = \sigma^2$), and Autocovariance depends ONLY on the time lag $k$, not on actual time $t$: $\text{Cov}(Y_t, Y_{t-k}) = \gamma_k$
✓ Correct
B
A process where all observations are equal to zero
C
A process where the variance increases linearly over time
D
A deterministic sine wave
💡 Step-by-Step Explanation & Concept Rationale
Weak (second-order) stationarity requires constant mean, constant finite variance, and autocovariances $\gamma_k$ that depend solely on the lag separation $k$ rather than the absolute time index $t$.
Q. 17 Statistics
Difficulty: medium (1 Mark)
What is a 'Random Walk' process $Y_t = Y_{t-1} + \epsilon_t$ (where $\epsilon_t \sim \text{WN}(0, \sigma^2)$)?
A
A non-stationary unit root process whose variance increases linearly with time ($\text{Var}(Y_t) = t \sigma^2$), exhibiting infinite memory of past shocks
✓ Correct
B
A stationary white noise process
C
A deterministic trend with constant variance
D
A seasonal process with period 12
💡 Step-by-Step Explanation & Concept Rationale
A random walk has a unit root ($|\phi| = 1$), non-constant variance $\text{Var}(Y_t) = t\sigma^2$, and persistent non-decaying shocks, making it non-stationary.
Q. 18 Statistics
Difficulty: medium (1 Mark)
What is 'Spurious Regression' in time series econometrics (Granger & Newbold 1974)?
A
A misleading regression between two independent non-stationary unit root series ($I(1)$) that yields high $R^2$ and highly statistically significant t-ratios, even when the series have zero causal or economic relationship
✓ Correct
B
A regression with negative $R^2$
C
A regression with perfect collinearity
D
A calculation error caused by dividing by zero
💡 Step-by-Step Explanation & Concept Rationale
Regressing one random walk on another unrelated random walk produces spurious correlation: standard t-ratios diverge to $\pm \infty$ and Durbin-Watson statistic $d \to 0$, creating the illusion of a strong relationship.
Q. 19 Statistics
Difficulty: hard (1 Mark)
What is 'Cointegration' (Robert Engle & Clive Granger 1987)?
A
A property where two or more non-stationary time series $I(1)$ share a common stochastic trend such that a linear combination of them is stationary ($I(0)$): $Y_t - \beta X_t = u_t \sim I(0)$
✓ Correct
B
Two time series having the same sample size
C
Two stationary series that are negatively correlated
D
Two time series with zero variance
💡 Step-by-Step Explanation & Concept Rationale
Cointegrated series drift together over time in a stable long-run equilibrium relationship ($u_t \sim I(0)$), preventing spurious regression and enabling Vector Error Correction Modeling (VECM).
Q. 20 Statistics
Difficulty: medium (1 Mark)
What is the Augmented Dickey-Fuller (ADF) Test used for in time series modeling?
A
Testing for the presence of a Unit Root ($H_0: \gamma = 0$ / non-stationary) against the alternative of Stationarity ($H_1: \gamma < 0$)
✓ Correct
B
Testing for heteroscedasticity in OLS
C
Testing for seasonality in monthly data
D
Testing whether sample mean is zero
💡 Step-by-Step Explanation & Concept Rationale
The ADF test estimates $\Delta Y_t = \alpha + \beta t + \gamma Y_{t-1} + \sum_{i=1}^p \delta_i \Delta Y_{t-i} + \epsilon_t$. If the t-statistic is more negative than the non-standard Dickey-Fuller critical values, $H_0$ (unit root) is rejected.
Q. 21 Statistics
Difficulty: hard (1 Mark)
What is the Null Hypothesis of the KPSS (Kwiatkowski-Phillips-Schmidt-Shin) Test, and how does it contrast with the ADF test?
A
The KPSS test tests the null hypothesis that the series is STATIONARY ($H_0: I(0)$), whereas the ADF test tests the null of a Unit Root ($H_0: I(1)$)
✓ Correct
B
KPSS tests for equality of two variances
C
KPSS tests for cointegration across 10 series
D
They share the exact same null hypothesis
💡 Step-by-Step Explanation & Concept Rationale
KPSS reverses the null hypothesis: $H_0$ is trend/level stationarity. Combining ADF and KPSS (confirmatory analysis) robustly verifies whether a series is genuinely stationary or unit-root non-stationary.
Q. 22 Statistics
Difficulty: easy (1 Mark)
In an Autoregressive model of order 1 ($AR(1)$): $Y_t = \phi Y_{t-1} + \epsilon_t$, what is the condition for weak stationarity?
A
$|\phi| < 1$
✓ Correct
B
$\phi = 1$
C
$\phi > 1$
D
$\phi = 0$
💡 Step-by-Step Explanation & Concept Rationale
For an $AR(1)$ process to be stationary and causal, the autoregressive root must lie outside the unit circle in the complex plane, which translates to the real condition $|\phi| < 1$.
Q. 23 Statistics
Difficulty: medium (1 Mark)
In the Box-Jenkins methodology for time series modeling, what are the theoretical shapes of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for a stationary $AR(p)$ process?
A
ACF decays exponentially / oscillates toward zero; PACF cuts off (drops to zero) abruptly after lag $p$
✓ Correct
B
ACF cuts off after lag $p$; PACF decays exponentially
C
Both ACF and PACF cut off abruptly after lag 1
D
Both ACF and PACF remain constant across all lags
💡 Step-by-Step Explanation & Concept Rationale
For an $AR(p)$ process: the ACF tails off (exponential decay or damped sinusoid), while the PACF cuts off sharply after lag $p$, which uniquely identifies the autoregressive order $p$.
Q. 24 Statistics
Difficulty: medium (1 Mark)
What are the theoretical ACF and PACF patterns for a Moving Average process of order $q$ ($MA(q)$)?
A
ACF cuts off abruptly after lag $q$; PACF decays exponentially / oscillates toward zero
✓ Correct
B
ACF decays exponentially; PACF cuts off after lag $q$
C
Both ACF and PACF decay exponentially
D
Both ACF and PACF cut off at lag 1
💡 Step-by-Step Explanation & Concept Rationale
For an $MA(q)$ process: the ACF truncates to exactly zero for all lags $k > q$, while the PACF decays smoothly (tails off) toward zero.
Q. 25 Statistics
Difficulty: easy (1 Mark)
In an $\text{ARIMA}(p, d, q)$ model, what does the parameter $d$ represent?
A
The order of Differencing applied to the non-stationary series to achieve stationarity ($\Delta^d Y_t = (1 - L)^d Y_t$)
✓ Correct
B
The number of autoregressive lags
C
The number of moving average lags
D
The seasonal period length (e.g., 12 for monthly)
💡 Step-by-Step Explanation & Concept Rationale
The parameter $d$ indicates the number of times the raw series is differenced to remove stochastic trends: $d=1$ means first difference $\Delta Y_t = Y_t - Y_{t-1}$; $d=2$ means second difference.
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