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Efficient Likelihood Evaluation of State-Space Representations

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How can the evaluation of state-space integrals be made more efficient? “Efficient Likelihood Evaluation of State-Space Representations” introduces the EIS filter, building on previous particle filter methods, to provide continuous and adapted approximations that enhance the computation of these integrals, particularly in applications like DSGE models.

Image courtesy of interviewee. November 21, 2024

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This video discusses numerical techniques used to calculate integrals that cannot be calculated analytically, with a focus on Bayesian inference and importance sampling. When direct sampling from the posterior distribution is not feasible, importance sampling can provide estimates by introducing a sampling density that draws parameters which are then weighted according to their relevance to the posterior distribution. This video method is mentioned in the context of calculating expectations and allows for approximating characteristics of complex models by generating samples.

This video also introduces the concept of filtering within dynamic models, where observable and unobservable state variables interact. Filtering helps infer the unobservable states based on observed data over time. The limitations of traditional filters, such as particle filters, are highlighted, noting their inefficiency particularly when the observed data suggests a certain state distribution that does not align well with the underlying model.

Lastly, the efficient importance sampler is proposed as a solution to these inefficiencies by combining these distributions to refine the sampling process. This video strategy leverages linear normal approximations to hone in on the correct parameter space, broadening the scope of models that can be effectively analyzed while minimizing numerical error. Efficient inference allows for more complex models to be evaluated feasibly by reducing variations that arise from numerical approximations.

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Here are five key points from the video:

1. **Need for Numerical Techniques**: Numerical techniques are essential for approximating integrals and characteristics of posterior distributions in Bayesian inference when analytical solutions are not feasible. These techniques, such as importance sampling, allow for approximating complicated functions of model parameters.

2. **Importance Sampling Methodology**: Importance sampling is a technique used when the posterior distribution cannot be directly sampled. It involves calculating a weighted average of function values derived from parameter draws, where the weights account for the ratio between the posterior distribution and the importance sample distribution, thus ensuring that the approximation remains unbiased.

3. **Influence of Numerical Error**: The accuracy of numerical estimates diminishes with numerical error, which can arise from the random draws in sampling processes. Increasing the number of draws can reduce numerical error, while the efficiency and alignment of the sampling distribution with the posterior distribution significantly impact the overall estimation error.

4. **Dynamic Models and Filtering**: In the convideo of dynamic models that evolve over time, filtering is necessary for making inferences about unobserved state variables based on observed data. The challenge lies in integrating over possible realizations of these states, especially when dealing with non-linear and non-normal distributions.

5. **Advancements through Particle Filtering**: Particle filtering techniques help manage the inefficiencies in traditional methods by using a swarm of particles to estimate state probabilities over time. By combining linear normal approximations of sampling densities with target distributions, it becomes possible to navigate more complex models beyond the constraints of linear and normal assumptions, enhancing the scope of viable analytical models.

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Here are some key quotes extracted from the video:

1. **On the necessity of numerical techniques:**
- "Numerical techniques arise, given the need to calculate integrals that cannot be calculated analytically, but rather need to be approximated."

2. **Regarding Bayesian inference:**
- "When I can't calculate the characteristics of the posterior distribution of those functions analytically, I can resort to numerical techniques to do so."

3. **On sampling from the posterior distribution:**
- "In that case, I can approximate the intervals that I'm after by obtaining random draws from that normal distribution."

4. **On the law of large numbers:**
- "By the law of large numbers, the estimates that I get will be arbitrarily close to the true value."

5. **On importance sampling:**
- "Importance sampling is a tool that can be used when the posterior distribution itself is not available as a random number generator."

6. **On the efficiency of the importance sampler:**
- "The quality of the sampler that we use has a huge impact on the numerical error."

7. **On filtering and state variables:**
- "Filtering is to make inferences about the value of the state period by period."

8. **On the limitations of linear and normal assumptions:**
- "The shortcoming of that is the linearity and the normality."

9. **On particle filtering:**
- "In a particle filtering scenario, we've made an inference about the value of the state in period T minus one."

10. **On the challenges of particle filtering:**
- "The problem can arise in that setting... the suggested values of the states in time T aren't well aligned with what the model would tell us."

11. **On combining distributions for efficiency:**
- "What the efficient importance sampler does then is it takes these two distributions and combines them."

12. **On the expansion of models that can be evaluated:**
- "This unleashes the shackles of the linear normal assumption... the universe of models that we can look at are greatly expanded."

These quotes capture the essence of the discussion around numerical techniques, Bayesian inference, importance sampling, filtering in dynamic models, and the challenges of working with non-linear models.

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Numerical techniques, arise, given the need

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to calculate integrals that, cannot be,

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calculated analytically, but rather need to be approximated.

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So as a lead example, is, is Bayesian inference of

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complicated functions

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of the underlying parameters of a model.

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Um, when I can't calculate the characteristics

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of the posterior distribution of those functions

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analytically, I can resort to numerical techniques to do so.

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As a, as a simple, lead example.

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In this context, suppose

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that I can use the posterior distribution itself to generate

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artificial realizations of model parameters.

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So, for example, suppose the posterior distribution

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is a normal distribution.

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In that case, I can approximate the intervals that I'm

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after by obtaining random draws from

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that normal distribution.

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For each draw of the parameters,

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calculate the underlying function of interest,

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and then simply calculate the sample average of the,

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of those, function values.

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And by the law of large numbers, the estimates

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that I get will be arbitrarily close to the true value.

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Important sampling is a tool that can be used

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when the posterior distribution itself is not available

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as a random number generator, which is typically the case.

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So instead of being able

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to draw parameters from the posterior distribution,

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I introduce an important sampling density, which needs

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to have several characteristics.

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But one is that I can obtain parameter draws

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from that distribution.

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In that case, just like in that simple case,

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for every parameter that I draw,

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I calculate the function of interest.

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But in this case, instead of a simple sample average,

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I'm gonna calculate a weighted average where the weight

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is determined by the ratio of the posterior distribution

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of the parameters, which is important

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'cause it tells me the post posterior probability

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of any particular parameterization divided

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by the important sampler itself evaluated at that drawing.

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That's important because the important

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sampler is just a tool.

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It should not influence the answer that I get

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to my, my calculation.

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If you write down a different distribution than me,

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we still need to get the same answer.

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So by calculating that weighted average, the, the value

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of the importance sampler in the denominator offsets the

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direct influence that the sampler has in generating

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a drawing in the first place.

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It's the same idea. At the end of the day,

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I've got a numerical approximation for my target.

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Now I've just used a tool that gets around the problem

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of not being able to draw from the

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posterior here's efficiency.

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At the end of the day, we won't get the exact answer due

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to nu numerical approximation.

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So in that first example, if you

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and I both use the posterior distribution

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to obtain sample drawings, we go home

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and we generate 800 million draws.

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We're gonna get, we're each gonna get an answer,

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and our answers will slightly differ.

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And the reason for that difference is

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simply numerical error.

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Um, the more draws we get in that first example,

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the smaller will be the numer numerical error.

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And in principle, we can get our answers

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as close together as possible.

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We can drive that numerical error almost to zero.

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We can't ever eliminate it,

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but we can get it arbitrarily small with important sampling.

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Same idea, except that the quality of the sampler

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that we use has a huge impact on

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the, that numerical error.

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So in the extreme case in which I can write down an

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important sampler that perfectly mimics the,

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the actual posterior distribution, so

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that the relative weights are always identically one,

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then I've gotten to the exact context

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of essentially being able to draw right from the posterior.

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But if the two distributions are not well aligned, then

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we'll get very different weights.

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Draw by draw. Many of the weights that I, that many

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of the draws I obtained from the important sampler just

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won't get any importance at all from the

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posterior distribution.

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And that induces a much higher level of numerical error

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into our estimates.

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So then the idea behind efficiency in the important sampler

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is to tailor the importance sampler as closely

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as possible to the posterior.

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Let's talk about filtering for a second.

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The, the context here is we're working with dynamic models,

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models that evolve over time,

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and they involve the interface of two different types

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of variables.

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Variables that I can observe directly, like in the context

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of a macro application.

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I can observe output consumption, investment, labor hours,

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but there are other variables that I can't observe directly,

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such as a productivity shock.

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Things that cause stochastic wiggling beyond what I can see

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for a reason beyond what I can see in the data.

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Um, so we've got an interface of variables

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that can be observed

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and variables that cannot, the variables

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that cannot are called state variables,

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and they evolve from one period to the next in, in ways

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that the model would tell us,

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but we can't directly observe them.

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When I make inferences about

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how the data reflect on the model, the likelihood

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of the model given the data, the inferences that I make

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cannot depend on a particular realization of the state

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because that's unknown.

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So what I have to do is to make inferences about the value

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of the state period by period.

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And that's called filtering.

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When I, when I, when I have an understanding

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of the probability of the state at any given value

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of the state over every period of the sample,

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then given those probabilities,

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I can make inferences about the model itself.

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In state-based evaluations,

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I'm making inferences about the states

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and about the, the likelihood function of the model.

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And both of those then entail integrating over

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possible realizations of the state.

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In the case in which variables are normally distributed

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and the models are linear, we can,

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we can do likelihood calculations

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and filtering analytically,

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and we can get billions of evaluations

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of the likelihood function per second.

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Um, and so we can very, very quickly

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make inferences about the likelihood function.

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Um, the shortcoming of that is the linearity

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and the normality.

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So for example, if I'm interested in models on in which the,

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the non-linearity is itself, what is interesting,

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for example, in macro models where it's costly

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to adjust production technologies, for example,

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to scale up a production technology, the cost of doing

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that would be exponential

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or shifting my investment from one sector to another.

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Those non-linearities are inherently important in

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the research that we're doing.

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To the extent that observations, we, we get lots

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and lots of things that look like outliers from the lens

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of a normal distribution, which have very thin tails.

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It might be that the normal distribution is not a good

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assumption, that we need fatter tailed distributions

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to account for surprising results

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or, you know, what, what would otherwise look like outliers.

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So, so that's the context we wanna break away from that,

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that linear normal specification.

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What got us involved in efficient, important sampling,

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quite frankly, is that the filtering techniques

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that were being that,

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that were being implemented in these contexts, turn out

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to be quite inefficient numerically.

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Let me quickly explain the, how the particle filter works

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and then talk about why they're inefficient

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and then what we do to get around that.

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So in a part particle filtering, scenario,

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we've made an inference about the value of the state

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in period T minus one.

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And that inference is in the form of a swarm of particles

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that give us a probability of a particular state at time,

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T minus one, given the data observed up till time T minus

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one, we then use that particle as a type

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of important sampler to make inferences in period T.

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Um, and the way it works is from that swarm of particles

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and a state transition density, we can make

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predictions about what the time t values

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of the states would be without taking into account

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what the ober, the observed varia the observed variables

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looked like in period T.

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So it's sort of a blind sampler.

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The problem two problems can arise in that setting.

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One is that because we had potentially an unusual

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realization of the observables in time t

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that the suggested values of the states in time t aren't,

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aren't well aligned with what the model would tell us.

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If you think about those weighted averages

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that I just talked about, you get a lot of weights

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that just don't get much importance from, from the data.

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The other, scenario that can arise is that the data

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are extremely informative about the values

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of the states at period T so that the distribution

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that we're trying to find is very, very tightly distributed.

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And you're groping around with a big fat swarm of particles

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and it's hard to find it.

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I I like to use the following analogy.

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Suppose we're trying to find a needle in a haystack,

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but we don't know which haystack the needle is in.

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So we're getting suggestions of haystacks.

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For example, I'm from the Midwest in the United States, so

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the suggestions are coming from the western part

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of Nebraska, but the haystack lives in

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the eastern part of Iowa.

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And as we get suggestions about these haystacks in

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Nebraska, we're wasting our time.

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We're getting, you know, we're getting lots of draws

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that just don't have any information in them.

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And so we, we end up being very, very inefficient.

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So what in, what the efficient, important sampler does then

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is it takes these two distributions and combines them.

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Um, and, and we make, we make,

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we linear normal approximations of the two of, of the,

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of the sampling density

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and the target, which, if they're very far apart,

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like one's in the northwest northeast corridor

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and one's in the southwest corridor, we know

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that their weighted average lives somewhere in the middle

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of the parameter space in a two dimensional space.

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So what we do is we build a linear,

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normal approximation in that middle.

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We know the truth is non-linear and non-normal,

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but we, that's where we start to look.

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And so we're in the neighborhood of the right haystack

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when we start looking for that needle.

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This unleashes the shackles of the linear normal assumption.

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We can tackle much more interesting models,

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or much more complicated models may maybe

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Not interesting, that's a pejorative, you know, that's a,

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but just the, the, the, the, the universe of models

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that we can look at, are, are just greatly expanded.

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Um, and

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because we can, obtain inferences more efficiently,

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they're actually feasible to, to evaluate.

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Um, in the inefficient case, you know, you

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and I could come up with estimates of our models, but,

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but if our estimators are inefficient,

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they're gonna be very different reflecting a lot

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of numerical error.

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And so driving the numerical error as small as we can out

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of the inference just makes it, more,

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more feasible to, to tackle more complicated models.

Approximating Integrals: Bayesian Inference and Numerical Techniques

00:00:00 - Numerical techniques, arise, given the need to calculate integrals that, cannot be, calculated analytically, but rather need to be approximated. So as a lead example, is, is Bayesian inference of complicated functions of the underlying parameters of a model. when I can't calculate the characteristics of the posterior distribution of those functions analytically, I can resort to numerical techniques to do so.

00:00:37 - In this context, suppose that I can use the posterior distribution itself to generate artificial realizations of model parameters. So, for example, suppose the posterior distribution is a normal distribution. In that case, I can approximate the intervals that I'm after by obtaining random draws from that normal distribution. For each draw of the parameters, calculate the underlying function of interest,

00:01:09 - of those, function values. And by the law of large numbers, the estimates that I get will be arbitrarily close to the true value. Important sampling is a tool that can be used when the posterior distribution itself is not available as a random number generator, which is typically the case. So instead of being able to draw parameters from the posterior distribution,

Weighted Averages in Bayesian Parameter Sampling

00:01:40 - to have several characteristics. But one is that I can obtain parameter draws from that distribution. In that case, just like in that simple case, for every parameter that I draw, I calculate the function of interest. But in this case, instead of a simple sample average, I'm gonna calculate a weighted average where the weight is determined by the ratio of the posterior distribution of the parameters, which is important

00:02:10 - of any particular parameterization divided by the important sampler itself evaluated at that drawing. That's important because the important sampler is just a tool. It should not influence the answer that I get to my, my calculation. If you write down a different distribution than me, we still need to get the same answer. So by calculating that weighted average, the, the value of the importance sampler in the denominator offsets the

00:02:44 - a drawing in the first place. It's the same idea. At the end of the day, I've got a numerical approximation for my target. Now I've just used a tool that gets around the problem of not being able to draw from the posterior here's efficiency. At the end of the day, we won't get the exact answer due to nu numerical approximation. So in that first example, if you and I both use the posterior distribution

Minimizing Numerical Error in Sampling Techniques

00:03:12 - and we generate 800 million draws. We're gonna get, we're each gonna get an answer, and our answers will slightly differ. And the reason for that difference is simply numerical error. the more draws we get in that first example, the smaller will be the numer numerical error. And in principle, we can get our answers as close together as possible. We can drive that numerical error almost to zero.

00:03:42 - but we can get it arbitrarily small with important sampling. Same idea, except that the quality of the sampler that we use has a huge impact on the, that numerical error. So in the extreme case in which I can write down an important sampler that perfectly mimics the, the actual posterior distribution, so that the relative weights are always identically one,

00:04:18 - of essentially being able to draw right from the posterior. But if the two distributions are not well aligned, then we'll get very different weights. Draw by draw. Many of the weights that I, that many of the draws I obtained from the important sampler just won't get any importance at all from the posterior distribution. And that induces a much higher level of numerical error into our estimates.

Dynamic Models and State Filtering in Bayesian Inference

00:04:49 - is to tailor the importance sampler as closely as possible to the posterior. Let's talk about filtering for a second. The, the context here is we're working with dynamic models, models that evolve over time, and they involve the interface of two different types of variables. Variables that I can observe directly, like in the context of a macro application. I can observe output consumption, investment, labor hours,

00:05:24 - such as a productivity shock. Things that cause stochastic wiggling beyond what I can see for a reason beyond what I can see in the data. so we've got an interface of variables that can be observed and variables that cannot, the variables that cannot are called state variables, and they evolve from one period to the next in, in ways that the model would tell us, but we can't directly observe them. When I make inferences about

00:05:56 - of the model given the data, the inferences that I make cannot depend on a particular realization of the state because that's unknown. So what I have to do is to make inferences about the value of the state period by period. And that's called filtering. When I, when I, when I have an understanding of the probability of the state at any given value of the state over every period of the sample,

Challenges in Linear Normal Models and Non-linearity in Inference

00:06:26 - I can make inferences about the model itself. In state-based evaluations, I'm making inferences about the states and about the, the likelihood function of the model. And both of those then entail integrating over possible realizations of the state. In the case in which variables are normally distributed and the models are linear, we can, we can do likelihood calculations and filtering analytically,

00:06:57 - of the likelihood function per second. and so we can very, very quickly make inferences about the likelihood function. the shortcoming of that is the linearity and the normality. So for example, if I'm interested in models on in which the, the non-linearity is itself, what is interesting, for example, in macro models where it's costly to adjust production technologies, for example,

00:07:30 - that would be exponential or shifting my investment from one sector to another. Those non-linearities are inherently important in the research that we're doing. To the extent that observations, we, we get lots and lots of things that look like outliers from the lens of a normal distribution, which have very thin tails. It might be that the normal distribution is not a good assumption, that we need fatter tailed distributions

Enhancing Filtering Techniques through Efficient Importance Sampling

00:08:00 - or, you know, what, what would otherwise look like outliers. So, so that's the context we wanna break away from that, that linear normal specification. What got us involved in efficient, important sampling, quite frankly, is that the filtering techniques that were being that, that were being implemented in these contexts, turn out to be quite inefficient numerically. Let me quickly explain the, how the particle filter works

00:08:31 - and then what we do to get around that. So in a part particle filtering, scenario, we've made an inference about the value of the state in period T minus one. And that inference is in the form of a swarm of particles that give us a probability of a particular state at time, T minus one, given the data observed up till time T minus one, we then use that particle as a type

00:09:04 - and the way it works is from that swarm of particles and a state transition density, we can make predictions about what the time t values of the states would be without taking into account what the ober, the observed varia the observed variables looked like in period T. So it's sort of a blind sampler. The problem two problems can arise in that setting.

Challenges in State Variable Estimation and Efficient Sampling Techniques

00:09:36 - realization of the observables in time t that the suggested values of the states in time t aren't, aren't well aligned with what the model would tell us. If you think about those weighted averages that I just talked about, you get a lot of weights that just don't get much importance from, from the data. The other, scenario that can arise is that the data are extremely informative about the values

00:10:06 - that we're trying to find is very, very tightly distributed. And you're groping around with a big fat swarm of particles and it's hard to find it. I I like to use the following analogy. Suppose we're trying to find a needle in a haystack, but we don't know which haystack the needle is in. So we're getting suggestions of haystacks. For example, I'm from the Midwest in the United States, so the suggestions are coming from the western part

00:10:36 - the eastern part of Iowa. And as we get suggestions about these haystacks in Nebraska, we're wasting our time. We're getting, you know, we're getting lots of draws that just don't have any information in them. And so we, we end up being very, very inefficient. So what in, what the efficient, important sampler does then is it takes these two distributions and combines them.

Overcoming Linear Constraints in Model Estimation

00:11:06 - we linear normal approximations of the two of, of the, of the sampling density and the target, which, if they're very far apart, like one's in the northwest northeast corridor and one's in the southwest corridor, we know that their weighted average lives somewhere in the middle of the parameter space in a two dimensional space. So what we do is we build a linear, normal approximation in that middle. We know the truth is non-linear and non-normal,

00:11:36 - And so we're in the neighborhood of the right haystack when we start looking for that needle. This unleashes the shackles of the linear normal assumption. We can tackle much more interesting models, or much more complicated models may maybe Not interesting, that's a pejorative, you know, that's a, but just the, the, the, the, the universe of models that we can look at, are, are just greatly expanded. and

00:12:10 - they're actually feasible to, to evaluate. in the inefficient case, you know, you and I could come up with estimates of our models, but, but if our estimators are inefficient, they're gonna be very different reflecting a lot of numerical error. And so driving the numerical error as small as we can out of the inference just makes it, more,